arXiv:cond-mat/0506035v1 [cond-mat.mes-hall] 1 Jun 2005

FUNDAMENTAL ASPECTS OF ELECTRON
CORRELATIONS AND QUANTUM TRANSPORT IN
ONE-DIMENSIONAL SYSTEMS
Dmitrii L. Maslov
Department of Physics, University of Florida, P. O. Box 118441, Gainesville, FL 32611-8440, USA

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Photo: width 7.5cm height 11cm

Contents
1. Introduction
2. Non-Fermi liquid features of Fermi liquids: 1D physics in higher dimensions
2.1. Long-range effective interaction
2.2. 1D kinematics in higher dimensions
2.3. Infrared catastrophe
2.3.1. 1D
2.3.2. 2D
3. Dzyaloshinskii-Larkin solution of the Tomonaga-Luttinger model
3.1. Hamiltonian, anomalous commutators, and conservation laws
3.2. Reducible and irreducible vertices
3.3. Ward identities
3.4. Effective interaction
3.5. Dyson equation for the Green’s function
3.6. Solution for the case g2 = g4
3.7. Physical properties
3.7.1. Momentum distribution
3.7.2. Tunneling density of states
4. Renormalization group for interacting fermions
5. Single impurity in a 1D system: scattering theory for interacting fermions
5.1. First-order interaction correction to the transmission coefficient
5.1.1. Hartree interaction
5.1.2. Exchange
5.2. Renormalization group
5.3. Electrons with spins
5.4. Comparison of bulk and edge tunneling exponents
6. Bosonization solution
6.1. Spinless fermions
6.1.1. Bosonized Hamiltonian
6.1.2. Bosonization of fermionic operators
6.1.3. Attractive interaction
6.1.4. Lagrangian formulation
6.1.5. Correlation functions
6.2. Fermions with spin
6.2.1. Tunneling density of states
7. Transport in quantum wires
7.1. Conductivity and conductance
7.1.1. Galilean invariance
7.1.2. Kubo formula for conductivity
7.1.3. Drude conductivity
7.1.4. Landauer conductivity
7.2. Dissipation in a contactless measurement

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Dmitrii L. Maslov

7.3. Conductance of a wire attached to reservoirs
7.3.1. Inhomogeneous Luttinger-liquid model
7.3.2. Elastic-string analogy
7.3.3. Kubo formula for a wire attached to reservoirs
7.3.4. Experiment
7.4. Spin component of the conductance
7.5. Thermal conductance: Fabry-Perrot resonances of plasmons
8. Acknowledgments
Appendix A. Polarization bubble for small q in arbitrary dimensionality
Appendix B. Polarization bubble in 1D
Appendix B.1. Small q
Appendix B.2. q near 2kF
Appendix C. Some details of bosonization procedure
Appendix C.1. Anomalous commutators
Appendix C.2. Bosonic operators
Appendix C.2.1. Commutation relations for bosonic fields ϕ and ϑ
Appendix C.3. Problem with backscattering
References

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Abstract
Some aspects of physics on interacting fermions in 1D are discussed in a tutoriallike manner. We begin by showing that the non-analytic corrections to the Fermiliquid forms of thermodynamic quantities result from essentially 1D collisions embedded into a higher-dimensional phase space. The role of these collisions increases progressively as dimensionality is reduced until, finally, they lead to a
breakdown of the Fermi liquid in 1D. An exact solution of the Tomonaga-Luttinger
model, based on the Ward identities, is reviewed in the fermionic language. Tunneling in a 1D interacting systems is discussed first in terms of the scattering theory for
interacting fermions and then via bosonization. Universality of conductance quantization in disorder-free quantum wires is discussed along with the breakdown of
this universality in the spin-incoherent case. A difference between charge (universal) and thermal (non-universal) conductances is explained in terms of Fabry-Perrot
resonances of charge plasmons.

Fundamental aspects of electron correlations and quantum transport

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1. Introduction
The theory of interacting fermions in one dimension (1D) has survived several
metamorphoses. From what seemed to be a purely mathematical exercise up until
the 60s, it had evolved into a practical tool for predicting and describing phenomena in conducting polymers and organic compounds–which were the 1D systems
of the 70s. Beginning from the early 90s, when the progress in nanofabrication
led to creation of artificial 1D structures–quantum wires and carbon nanotubes,
the theory of 1D systems started its expansion into the domain of mesoscopics;
this trend promises to continue in the future. Given that there is already quite a
few excellent reviews and books on the subject [1]-[10] , I should probably begin with an explanation as to what makes this review different from the others.
First of all, it is not a review but–being almost a verbatim transcript of the lectures given at the 2004 Summer School in Les Houches–rather a tutorial on some
(and definitely not all) aspects of 1D physics. A typical review on the subject
starts with describing the Fermi Liquid (FL) in higher dimensions with an aim of
emphasizing the differences between the FL and its 1D counter-part –Luttinger
Liquid (LL). My goal–if defined only after the manuscript was written–was rather
to highlight the similarities between higher-D and 1D systems. The progress in
understanding of 1D systems has been facilitated tremendously and advanced to
a greater detail, as compared to higher dimensions, by the availability of exact or
asymptotically exact methods (Bethe Ansatz, bosonization, conformal field theory), which typically do not work too well above 1D. The downside part of this
progress is that 1D effects, being studied by specifically 1D methods, look somewhat special and not obviously related to higher dimensions. Actually, this is
not true. Many effects that are viewed as hallmarks of 1D physics, e.g., the suppression of the tunneling conductance by the electron-electron interaction and
the infrared catastrophe, do have higher-D counter-parts and stem from essentially the same physics. For example, scattering at Friedel oscillations caused by
tunneling barriers and impurities is responsible for zero-bias tunneling anomalies in all dimensions [11, 16]. The difference is in the magnitude of the effect
but not in its qualitative nature. Following the tradition, I also start with the FL
in Sec. 2, but the main message of this Section is that the difference between
D = 1 and D > 1 is not all that dramatic. In particular, it is shown that the
well-known non-analytic corrections to the FL forms of thermodynamic quantities (such as a venerable T 3 ln T -correction to the linear-in-T specific heat in
3D) stem from rare events of essentially 1D collisions embedded into a higherdimensional phase space. In this approach, the difference between D = 1 and
D > 1 is quantitative rather than qualitative: as the dimensionality goes down,
the phase space has difficulties suppressing the small-angle and 2kF −scattering
events, which are responsible for non-analyticities. The special point when these

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Dmitrii L. Maslov

events go out of control and start to dominate the physics happens to be in 1D.
This theme is continued in Sec.5, where scattering from a single impurity embedded into a 1D system is analyzed in the fermionic language, following the
work by Yue, Matveev, Glazman [11]. The drawback of this approach–the perturbative treatment of the interaction–is compensated by the clarity of underlying
physics. Another feature which makes these notes different from the rest of the
literature in the field is that the description goes in terms of the original fermions
for quite a while (Secs.2 through 5) , whereas the weapon of choice of all 1D
studies–bosonization– is invoked only at a later stage (Sec. 6 and beyond). The
rationale–again, a post-factum one–is two-fold. First, 1D systems in a mesoscopic environment–which are the main real-life application discussed here– are
invariably coupled to the outside world via leads, gates, etc. As the outside world
is inhabited by real fermions, it is sometimes easier to think of, e.g., both the
interior and exterior a quantum wire coupled to reservoirs in terms of the same
elementary quasi-particles. Second, after 40 years or so of bosonization, what
could have been studied within a model of fermions with linearized dispersion
and not too strong interaction–and this is when bosonization works–was probably studied. (As all statements of this kind, this is one is also an exaggeration.)
The last couple of years are characterized by a growing interest in either the effects that do not occur in a model with linearized dispersion, e.g., Coulomb drag
due to small-momentum transfers [17] and energy relaxation, or situations when
strong Coulomb repulsion does not permit linearization of the spectrum at any
energies [19, 20, 21]. Experiment seems to indicate that the Coulomb repulsion
is strong in most systems of interest, thus studies of a strongly-coupled regime
are quite timely. Once the assumption of the linear spectrum is abandoned, the
beauty of a bosonized description is by and large lost, and one might as well
come back to original fermions. Sec.6 is devoted to transport in quantum wires,
mostly in the absence of impurities. The universality of conductance quantization is explained in some detail, and is followed by a brief discussion of the recent
result by Matveev [19], who showed that incoherence in the spin sector leads to
a breakdown of the universality at higher temperatures (Sec. 7.4). Also, a difference in charge (universal) and thermal (non-universal) transport–emphasized by
Fazio, Hekking, and Khmelnitskii [22]– in addressed in Sec.7.5. What is missing
is a discussion of transport in a disordered (as opposed to a single-impurity) case.
However, this canonically difficult subject, which involves an interplay between
localization and interaction, is perhaps not ready for a tutorial-like discussion at
the moment. (For a recent development on this subject, see Ref.[18].)
Even a brief inspection of these notes shows that the choice between making
them comprehensive or self-contained was made for the latter with a focus on a
relatively small number of topics. It is quite easy to see what is missing: there
is no discussion of lattice effects, bosonization is introduced without the Klein

Fundamental aspects of electron correlations and quantum transport

7

factors, the sine-Gordon model is not treated in depth, chiral Luttinger liquids
are not discussed at all, and the list goes on. The discussion of the experiment
is scarce and perfunctory. However, the few subjects that are discussed are provided with quite a detailed–perhaps somewhat excessively detailed– treatment,
so that a reader may not feel a need to consult the reference list too often. For
the same reason, the notes also cover such canonical procedures as the perturbative renormalization group in the fermionic language (Sec. 4) and elementary
bosonization (Sec. 6), which are discussed in many other sources and a reader
already familiar with the subject is encouraged to skip them.
Also, a relatively small number of references (about one per page on average)
indicates once again that this is not a review. The choice of cited papers is subjective and the reference list in no way pretends to represent a comprehensive
bibliography to the field. My apologies in advance to those whose contributions
to the field I have failed to acknowledge here.
= kB = 1 through out the notes, unless specified otherwise.

2. Non-Fermi liquid features of Fermi liquids: 1D physics in higher dimensions

One often hears the statement that, by and large, a Fermi liquid (FL) is just a
Fermi gas of weakly interacting quasi-particles; the only difference being the
renormalization of the essential parameters (effective mass, g− factor) by the interactions. What is missing here is that the similarity between the FL and Fermi
gas holds only for leading terms in the expansion of the thermodynamic quantities (specific heat C(T ), spin susceptibility χs , etc.) in the energy (temperature) or spatial (momentum) scales. Next-to-leading terms, although subleading,
are singular (non-analytic) and, upon a closer inspection, reveal a rich physics
of essentially 1D scattering processes, embedded into a high-dimensional phase
space.
In this chapter, I will discuss the difference between “normal” processes which
lead to the leading, FL forms of thermodynamic quantities and “rare” 1D processes which are responsible for the non-analytic behavior. We will see that the
role of these rare processes increases as the dimensionality is reduced and, eventually, the rare processes become the norm in 1D, where the FL breaks down.
In a Fermi gas, thermodynamic quantities form regular, analytic series as function of either temperature, T, or the inverse spatial scale (bosonic momentum q)
of an inhomogeneous magnetic field. For T ≪ EF , where EF is the Fermi

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Dmitrii L. Maslov

energy, and q ≪ kF , where kF is the Fermi momentum, we have
C(T )/T =
χs (T, q = 0) =
χs (T = 0, q) =

γ + aT 2 + bT 4 + . . . ;
χ0 (0) + cT 2 + dT 4 + . . . ;
s
χ0 (0) + eq 2 + f q 4 + . . . ,
s

(2.1a)
(2.1b)
(2.1c)

where γ is the Sommerfeld constant, χ0 is the static, zero-temperature spin suss
ceptibility (which is finite in the Fermi gas), and a . . . f are some constants. Even
powers of T occur because of the approximate particle-hole symmetry of the
Fermi function around the Fermi energy and even powers of q arise because of
the analyticity requirement 1
Our knowledge of the interacting systems comes from two sources. For a system with repulsive interactions, one can assume that as long as the strength of the
interaction does not exceed some critical value, none of the symmetries (translational invariance, time-reversal, spin-rotation, etc.), inherent to the original Fermi
gas, are broken. In this range, the FL theory is supposed to work. However, the
FL theory is an asymptotically low-energy theory by construction, and it is really
suitable only for extracting the leading terms, corresponding to the first terms
in the Fermi-gas expressions (2.1a-2.1c). Indeed, the free energy of a FL as an
ensemble of quasi-particles interacting in a pair-wise manner can be written as
[25]

F − F0 =

k

(ǫk − µ) δnk +

1
2

fk,k′ δnk δnk′ + O δn3 ,
k

(2.2)

k,k′

where F0 is the ground state energy, δnk is the deviation of the fermion occupation number from its ground-state value, and fk,k′ is the Landau interaction
function. As δnk is of the order of T /EF , the free energy is at most quadratic
in T, and therefore the corresponding C(T ) is at most linear in T. Consequently,
the FL theory–at least, in the conventional formulation–claims only that
C ∗ (T )/T =
χ∗ (T, q) =
s
1 The

γ ∗;
χ∗ (0) ,
s

expressions presented above are valid in all dimensions, except for D = 2 with quadratic
dispersion. There, because the density of states (DoS) does not depend on energy, the leading correction to the γT − term in C(T ) is exponential in EF /T and χs does not depend on q for q ≤ 2kF .
However, this anomaly is removed as soon as we take into account a finite bandwidth of the electron
spectrum, upon which the universal (T 2n and q 2n ) behavior of the series is restored.

Fundamental aspects of electron correlations and quantum transport

9

U
symmetry breaking

Fermi−liquid theory

Microscopic models:
‘‘non−ideal Fermi gas"
high−density Coulomb gas
...

EF

T, Q, H ...

Fig. 1. Combined “diagram of knowledge". x-axis: energy scale (given by temperature T , bosonic
momentum Q, magnetic field H) in appropriate units. y-axis: interaction strength. Fermi liquid
works for not necessarily weak interactions (but smaller than the critical value for an instability of
the ground state denoted by the red dot) but at the lowest energy scales. Microscopic models work
for weak interactions but for arbitrary energies.

where γ ∗ and χ∗ (0) differ from the corresponding Fermi-gas values, and does
s
not say anything about higher-order terms 2 .
Higher-order terms in T or q can be obtained within microscopic models
which specify particular interaction and, if an exact solution is impossible–which
is always the case in higher dimensions– employ some kind of a perturbation theory. Such an approach is complementary to the FL: the former nominally works
for weak interactions 3 but at arbitrary temperatures, whereas FL works both
for weak and strong interactions, up to some critical value corresponding to an
instability of some kind, e.g., a ferromagnetic transition, but only in the lowtemperature limit. In the { temperature, interaction} plane, the validity regions
of these two approaches are two strips running along the two axes (cf. Fig. 1).
For weak interactions and at low temperatures, the regions should overlap.
Microscopic models (Fermi gas with weak repulsion, Coulomb gas in the
high-density limit, electron-phonon interaction, paramagnon model, etc.) show
that the higher-order terms in the specific heat and spin susceptibility are non2 Strictly speaking, non-analytic terms in C(T ) can be obtained from the free energy (2.2) by
taking into account the non-analytic terms in the quasi-particle spectrum, see Ref. [29]b.
3 Some results of the perturbation theory can be rigorously extended to an infinite order in the
interaction, and most of them can be guaranteed to hold even if the interactions are not weak.

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Dmitrii L. Maslov

analytic functions of T and q [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38].
For example,
C(T )/T
C(T )/T
χs (q)
χs (q)

=
=
=
=

γ3 − α3 T 2 ln T (3D);
γ2 − α2 T (2D) ;
χs (0) + β3 q 2 ln q −1 (3D);
χs (0) + β2 |q| (2D),

(2.3a)
(2.3b)
(2.3c)
(2.3d)

where all coefficients are positive for the case of repulsive electron-electron interaction 4
As seen from Eqs. (2.3a-2.3d), the non-analyticities become stronger as the
dimensionality is reduced. The strongest non-analyticity occurs in 1D, where–
as far as single-particle properties are concerned–the FL breaks down:
C(T )/T
χ (q)

= γ1 + α1 ln T (1D);
= χ0 + β1 ln |q| (1D).

It turns out that the evolution of the non-analytic behavior with the dimensionality reflects an increasing role of special, almost 1D scattering processes in
higher dimensions. Thus non-analyticities in higher dimensions can be viewed
as precursors of 1D physics for D > 1.
It is easier to start with the non-analytic behavior of a single-particle property, the self-energy, which can be related to the thermodynamic quantities via
standard means [23] (see also appendix A). Within the Fermi liquid,
ReΣR (ε, k) =
−ImΣR (ε, k) =

−Aε + Bξk + . . .
C(ε2 + π 2 T 2 ) + . . .

(2.4a)
(2.4b)

Expressions (2.4a) and (2.4b) are equivalent to two statements: i) quasi-particles
have a finite effective mass near the Fermi level
4 Notice that not only the functional forms but also the sign of the q− dependent term in the
spin susceptibility is different for free and interacting systems. “Wrong” sign of the q− dependent
corrections has far-reaching consequences for quantum critical phenomena. For example, it precludes
a possibility of a second-order, homogeneous quantum ferromagnetic phase transition in an itinerant
system [39]. What is possible is either a first-order transition or ordering at finite q, e.g. helical
structure. In 1D, a homogeneous ferromagnetic state is forbidden by the Lieb-Mattis theorem [46],
which states that the ground state of 1D fermions, interacting via spin-independent but otherwise
arbitrary forces, is non-magnetic. One could speculate that the non-analyticities in higher dimensions
indicate the existence of a higher-D version of the Lieb-Mattis theorem. Certainly, this does not
mean that ferromagnetism does not exist in higher dimensions (it is hard to deny the existence of,
e.g., iron). However, ferromagnetism may not exist in models dealing only with itinerant electrons in
continuum.

Fundamental aspects of electron correlations and quantum transport

11

q ,ω

k, ε

k q , ε

ω

k, ε

Fig. 2. Self-energy to first order in the interaction with a dynamic bosonic field.

m∗ = m0

A+1
,
B+1

and ii) damping of quasiparticles is weak: the level width is much smaller than
the typical quasi-particle energy

2

Γ = −2ImΣR (ε, k) ∝ max |ε| , T 2 ≪ |ε| , T.

Landau’s argument for the ε2 (or T 2 ) behavior of ImΣR relies on the Fermi
statistics of quasiparticles and on the assumption that the effective interaction is
screened at large distances [23]. It requires two conditions. One condition is obvious: the temperature has to be much smaller than the degeneracy temperature
∗
∗
TF = kF vF , where vF is the renormalized Fermi velocity. The other condition
is less obvious: it requires inter-particle scattering to be dominated by processes
with large (generically, of order kF ) momentum transfers. Once these two conditions are satisfied, the self-energy assumes a universal form, Eqs. (2.4a) and (
2.4b), regardless of a specific type of the interaction (e-e, e-ph) and dimensionality. To see this, let’s have a look at ImΣR (ε, k) due to the interaction with some
“boson” (Fig. 2).
The wavy line in Fig.2 can be, e.g., a dynamic Coulomb interaction, phonon

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Dmitrii L. Maslov

propagator, etc. On the mass shell (ε = ξk ) at T = 0 and for ε > 0, we have 5
ImΣR (ε) = −

ε

2
D+1

(2π)

dω
0

dd qImGR (ε − ω, k − q) ImV R (ω, q) .

(2.5)
The constraint on energy transfers (0 < ω < ε) is a direct manifestation of
the Pauli principle which limits the number of accessible energy levels. In real
space and time, V (r, t) is a propagator of some field which has a classical limit
(when the occupation numbers of all modes are large). Therefore, V (r, t) is a
real function, hence ImV is an odd function of ω. I will make this fact explicit
writing ImV as
ImV R (ω, q) = ωW (|ω| , q) .
Now, suppose that we integrate over q and the result does not depend on ω. Then
we immediately get
−ImΣR (ε) ∼ C

ε
0

dωω ∼ Cε2 ,

where C is the result of the q− integration which contains all the information
about the interaction. Once we got the ε2 -form for ImΣR (ε) , the ε- term in
ReΣR (ε) follows immediately from the Kramers-Kronig transformation, and we
have a Fermi-liquid form of the self-energy regardless of a particular interaction
and dimensionality. Thus a sufficient condition for the Fermi liquid is the separability of the frequency and momentum integrations, which can only happen if
the energy and momentum transfers are decoupled.
Now, what is the condition for separability? As a function of q, W has at
least two characteristic scales. One is provided by the internal structure of the
interaction (screening wavevector for the Coulomb potential, Debye wavevector
for electron-phonon interaction, etc.) or by kF , whichever is smaller. This scale
(let’s call it Q) does not depend on ω. Moreover, as |ω| is bounded from above
by ε, and we are interested in the limit ε → 0, one can safely assume that Q ≫
|ω| /vF . The role of Q is just to guarantee the convergence of the momentum
integral in the ultraviolet, that is, to ensure that for q ≫ Q the integrand falls
off rapidly enough. Any physical interaction will have this property as larger
momentum transfer will have smaller weight. The other scale is |ω| /vF . Now,
5 To get Eq. (2.5), one can start with the Matsubara form of diagram Fig. 2, convert the Matsubara
sums into the contour integrals, use the dispersion relation

D R (ε) =

1
π

∞
−∞

dε′

ImD R (ε′ )
,
ε′ − ε − i0+

which is valid for any retarded function, and take the limit T → 0.

Fundamental aspects of electron correlations and quantum transport

13

let’s summarize this by re-writing ImV in the following scaling form
ImV R (ω, q) = ω

|ω| q
,
vF Q Q

1
U
QD

,

where U is a dimensionless function and the factor Q−D was singled out to keep
the units right.
In the perturbation theory, the Green’s function in (2.5) is a free one. Assum2
ing the free-electron spectrum ξk = (k 2 − kF )/2m,
ImGR ε − ω, k − q = −πδ ε − ω − ξk + vk · q − q 2 /2m .
On the mass shell,
ImGR ε − ω, k − q |ε=ξk = −πδ ω − vk · q + q 2 /2m .
The argument of the delta-function simply expresses the energy and momentum
conservation for a process ε → ε − ω, k → k − q. The angular integral involves
only the delta-function. For any D, this integral gives
δ (. . . )

Ω

=

1
vF q

AD

ω + q 2 /2m
vF q

,

where vk was replaced by vF because all the action takes place near the Fermi
surface. For D = 3 and D = 2,
A3 (x)
A2 (x)

= 2θ(1 − |x|);
2θ (1 − |x|)
√
.
=
1 − x2

The constraint on the argument of AD is purely geometric: the magnitude of the
cosine of the angle between k and q has to be less then one. For power-counting
purposes, function AD has a dimensionality of 1. Therefore, its only role is to
provide a lower cut-off for the momentum integral. Then, by power counting
ImΣR (ε) ∼

1
vF QD

ε

dqq D−2 U

dωω
0

q≥|ω|/vF

|ω| q
,
vF q Q

.

(2.6)

Now, if the integral over q is dominated by q ∼ Q and is convergent in the infrared, one can put ω = 0 in this integral. After this step, the integrals over ω and
q decouple. The ω− integral gives ε2 regardless of the nature of the interaction

14

Dmitrii L. Maslov

p
b)

a)
p-q

k

k+q

k

p

k

c)

p+q

k+q

k

d)

k+q
k

p

k
k

k

p+q

k

d)

k
p

p-q

k

-k
-k

q

Fig. 3. a) and b) Non-trivial second order diagrams for the self-energy. c) Same diagrams as in a)
and b) re-drawn as a single “sunrise" diagram. d) Diagrams relevant for non-analytic terms in the
self-energy. e) Kinematics of scattering in a polarization bubble: the dynamic part Π ∝ ω/vF q
comes from the processes in which the internal fermionic momentum (p) is almost perpendicular to
the external bosonic one (q).

Fundamental aspects of electron correlations and quantum transport

15

and dimensionality whereas the q− integral supplies a prefactor which entails all
the details of the interaction
ImΣR (ε) = CD

ε2
.
vF Q

For example, for a screened Coulomb interaction in the weak-coupling (highdensity) limit Q = κ, where κ is the screening wavevector, we have in 3D
−ImΣR (ε) =

π 2 κ ε2
.
64 kF EF

Now we can formulate a sufficient (but not necessary) condition for the Fermiliquid behavior. It will occur whenever if kinematics of scattering is such that
the typical momentum transfers are determined by some internal and, what is
crucial, ω− independent scale, whereas the energy transfers are of order of the
quasi-particle energy (or temperature). Excluding special situations, such as
the high-density limit of the Coulomb interaction, Q is generically of order of
the ultraviolet range of the problem ∼ kF . In other words, isotropic scattering
guarantees a ε2 - behavior. Small-angle scattering with typical angles of order
ε/vF ≪ Q ≪ kF gives this behavior as well.
The ε2 − result seems to be quite general under the assumptions made. When
and why these assumptions are violated?
A long-range interaction, associated with small-angle scattering, is known to
destroy the FL. For example, transverse long-range (current-current [44] or gauge
[45]) interactions, which–unlike the Coulomb one–are not screened by electrons,
lead to the breakdown of the Fermi liquid. However, the current-current interaction is of the relativistic origin and hence does the trick only at relativistically
small energy scales, whereas the gauge interaction occurs only under special circumstances, such as near half-filling or for composite fermions. What about a
generic case when nothing of this kind happens? It turns out that even if the
bare interaction is of the most benign form, e.g., a delta-function in real space,
there are deviations from a (perceived) FL behavior. These deviations get amplified as the system dimensionality is lowered, and, eventually, lead to a complete
breakdown of the FL in 1D.
A formal reason for the deviation from the FL-behavior is that the argument
which led us to the ε2 -term is good only in the leading order in ω/qvF . Recall
that the angular integration gives us q −1 factors in all dimensions, and, to arrive
at the ε2 result we put ω = 0 in functions AD and U. If we want to get a next
term in ε, then we need to expand U and A in ω. Had such expansions generated
regular series, ImΣR would have also formed regular series in ε2 : ImΣR =
aε2 + bε4 + cε6 + . . . . However, each factor of ω comes with q −1 , so that no

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Dmitrii L. Maslov

matter how high the dimensionality is, at some order of ω/vF q we are bound to
have an infrared divergence.
2.1. Long-range effective interaction
Let’s look at the simplest case of a point-like interaction. A frequency dependence of the self-energy arises already at the second order. At this order, two
diagrams in Fig. 3 are of interest to us. For a contact interaction, diagram b) is
just -1/2 of a) (which can be seen by integrating over the fermionic momentum p
first), so we will lump them together. Two given fermions interact via polarizing
the medium consisting of other fermions. Hence, the effective interaction at the
second order is just proportional to the polarization bubble
ImV R (ω, q) = −U 2 ImΠR (ω, q).
Let’s focus on small angle-scattering first: q ≪ 2kF . It turns out that in all
three dimensions, the bubble has a similar form (see Appendix Appendix A for
an explicit derivation of this result)
−ImΠR (ω, q) = νD

ω
BD
vF q

ω
vF q

,

(2.7)

D−2
where νD = aD mkF
is the DoS in D dimensions [a3 = (2π)−2 , a2 =
−1
(2π) , a1 = 1/2π] and BD is a dimensionless function, whose main role is
to impose a constraint ω ≤ vF q in 2D and 3D and ω = vF q in 1D. Eq. (2.7)
entails the physics of Landau damping. The constraint arises because collective
excitations–charge- and spin-density waves– decay into particle-hole pairs. Decay occurs only if bosonic momentum and frequency (q and ω) are within the
particle-hole continuum (cf. Fig. 4). For D > 1, the boundary of the continuum
for small ω and q is ω = vF q, hence the decay takes place if ω < vF q. The
rest of Eq. (2.7) can be understood by dimensional analysis. Indeed, ΠR is the
retarded density-density correlation function; hence, by the same argument we
applied to ImV R , its imaginary part must be odd in ω. For q ≪ kF , the only
combination of units of frequency is vF q, and the frequency enters as ω/vF q.
Finally, a factor νD makes the overall units right. In 1D, the difference is that the
continuum shrinks to a single line ω = vF q, hence decay of collective excitations
is possible only on this line. In 3D, function B3 is simply a θ− function

ImΠR (ω, q) = −ν3

ω
θ (q − |ω/vF |) .
vF q

Next-to-leading term in the expansion of ImΣR in ε comes from retaining the

Fundamental aspects of electron correlations and quantum transport

ω

17

ω

ω = vFq

2kF

ω = vFq

q

2kF

(a)

q

(b)

Fig. 4. Particle-hole continua for D > 1 (left) and D = 1 (right). For the 1D case, only half of the
continuum (q > 0) is shown.

a)

q~Q

|ω|

q~
b)

|ω|

/vF

/vF

|q − 2k |~ |ω| /vF
F

c)

Fig. 5. Kinematics of scattering. a) “Any-angle” scattering. Momentum transfer q is of order of the
intrinsic scale of the interaction or kF , whichever is smaller, and is independent of the energy transfer,
ω, which is of order of the initial energy ǫ. This process contributes regular terms to the self-energy.
b) Dynamic forward scattering: q ∼ |ω|/vF . c) Dynamic backscattering: |q − 2kF | ∼ |ω|/vF .
Processes b) and c) are responsible for the non-analytic terms in the self-energy.

18

Dmitrii L. Maslov

lower limit in the momentum integral of Eq. (2.6), upon which we get
−ImΣR

∼

∼
∼

Q∼kF

ε

U 2 mkF

dω

U2

mkF
2
vF

2

ε
0
3

dqq 2

ω/vF

0

aε − b |ε| .





dωω  kF −

FL

1 ω
vF q vF q


ω
vF






beyond FL

The first term in the square brackets is the FL contribution that comes from
q ∼ Q. The second term is a correction to the FL coming from q ∼ ω/vF .
Thus, contrary to a naive expectation an expansion in ε is non-analytic. The fraction of phase space for small-angle scattering is small–most of the self-energy
comes from large-angle scattering events (q ∼ Q); but we already start to see the
importance of the small-angle processes. Applying Kramers-Kronig transforma3
tion to the non-analytic part (|ε| ) in ImΣR , we get a corresponding non-analytic
contribution to the real part as
ReΣR

non−an

∝ ε3 ln |ε| .

Correspondingly, specific heat which, by power counting, is obtained from ReΣR
by replacing each ε by T , also acquires a non-analytic term6
C(T ) = γ3 T + β3 T 3 ln T.
This is the familiar T 3 ln T term, observed both in He3 [40] and metals [41]
(mostly, heavy-fermion materials) 7 .
In 2D, the situation is more dramatic. The q− integral diverges now logarith6 One has to be careful with the argument, as a general relation between C(T ) and the singleparticle Green’s function [23] involves the self-energy on the mass shell. In 3D, the contribution
to Σ from forward scattering, as defined in Fig. 6, vanishes on the mass shell; hence there is no
contribution to C(T ) [50]. The non-analytic part of C(T ) is related to the backscattering part of the
self-energy (scattering of fermions with small total momentum), which remains finite on the mass
shell. That forward scattering does not contribute to non-analyticities in thermodynamics is a general
property of all dimensions, which can be understood on the basis of gauge-invariance [42].
7 The T 3 ln T -term in the specific heat coming from the electron-electron interactions is often referred to in the literature as to the “spin-fluctuation” or “paramagnon” contribution [27, 28]. Whereas
it is true that this term is enhanced in the vicinity of a ferromagnetic (Stoner) instability, it exists even
far way from any critical point and arises already at the second order in the interaction [29].

Fundamental aspects of electron correlations and quantum transport

19

mically in the infrared:
−ImΣR (ω) ∼
∼

U2
2 m
vF

ε

dωω
0

U2
EF
mε2 ln
.
vF
|ε|

∼kF
∼|ω|/vF

dq
q

Now, dynamic forward-scattering (with transfers q ∼ ω/vF ) is not a perturbation anymore: on the contrary, the ε dependence of ImΣR is dominated by forward scattering (the ε2 ln |ε|-term is larger than the “any-angle” ε2 -contribution
). Correspondingly, the real part acquires a non-analytic term ReΣ ∝ ε |ε|, and
the specific heat behaves as 8
C(T ) = γ2 T − β2 T 2 .
The non-analytic T 2 -term in the specific heat has been observed in recent experiments on monolayers of He3 adsorbed on a solid substrate [43]9 .
Finally, in 1D the same power-counting argument leads to ImΣR ∝ |ε| and
ReΣR ∝ ε ln |ε| 10 Correspondingly, the “correction” to the specific heat behaves
as T ln T and is larger than the leading, T − term. This is the ultimate case of
dynamic forward scattering, whose precursors we have already seen in higher
dimensions 11 .
8 again,

only processes with small total momentum contribute
a T 2 term in C(T ) does not fit your definition of non-analyticity, you have to recall that the
right quantity to look at is the ratio C(T )/T. Analytic behavior corresponds to series C(T )/T =
γ + δT 2 + σT 4 + . . . . whereas we have a T 2 ln T and T terms as the leading order corrections to
the Sommerfeld constant γ for D = 3 and D = 2, correspondingly.
10 Special care is required in 1D as in the perturbation theory one gets a strong divergence in the
self-energy corresponding to interactions of fermions of the same chirality (Fig. 8a,c). This point will
be discussed in more detail in Section 2.3 (along with a weaker but nonetheless singularity in 2D).
For now, let us focus on a regular part of the self-energy corresponding to the interaction of fermions
of opposite chirality (Fig. 8b).
11 Bosonization predicts that C(T ) of a fermionic system is the same as that of 1D bosons, which
scales as T for D = 1 [10]. This is true only for spinless fermions, in which case bosonisation
provides an asymptotically exact solution. For electrons with spins, the bosonized theory is of the
sine-Gordon type with the non-Gaussian (cosφ) term coming from the backscattering of fermions of
opposite spins. Even if this term is marginally irrelevant and flows down to zero at the lowest energies,
at intermediate energies it results in a multiplicative ln T factor in C(T ) and a ln max{q, T, H}
correction to the spin susceptibility (where H is the magnetic field, and units are such that q, T, and
H have the units of energy). The difference between the non-analyticities in D > 1 and D = 1 is
that the former occurs already at the second order in the interaction, whereas the latter start only at
third order. Naive power-counting breaks down in 1D because the coefficient in front of T ln T term
in C(T ) vanishes at the second order, and one has to go to third order. In the sine-Gordon model, the
third order in the interaction is quite natural: indeed, one has to calculate the correlation function of
the cosφ term, which already contains two coupling constants; the third one occurs by expanding the
exponent to leading (first) order. For more details, see [47],[48],[49].
9 If

20

Dmitrii L. Maslov

Even if the bare interaction is point-like, the effective one contains a longrange part at finite frequencies. Indeed, the non-analytic parts of Σ and C(T )
come from the region of small q, and hence large distances. Already to the sec˜
ond order in U , the effective interaction U = U 2 Π(ω, q) is proportional to the
dynamic polarization bubble of the electron gas, Π (ω, q). In all dimensions,
ImΠR is universal and singular in q for |ω| /vF ≪ q ≪ kF
ImΠR (ω, q) ∼ νD

ω
.
vF |q|

Although the effective interaction is indeed screened at q → 0 –and this is
why the FL survives even if the bare interaction has a long-range tail–it has a
˜
slowly decaying tail in the intermediate range of q. In real space, U (r) behaves
−1
D−1
as ω/r
at distances kF ≪ r ≪ vF /|ω|.
Thus, we have the same singular behavior of the bubble in all dimensions, and
the results for the self-energy differ only because the phase volume q D is more
effective in suppressing the singularity in higher dimensions than in lower ones.
There is one more special interval of q: q ≈ 2kF , i.e., Kohn anomaly. Usually, the Kohn anomaly is associated with the 2kF - non-analyticity of the static
bubble, and its most familiar manifestation is the Friedel oscillation in electron
density produced by a static impurity (discussed later on). Here, the static Kohn
anomaly is of no interest for us as we are dealing with dynamic processes. However, the dynamic bubble is also singular near 2kF . For example, in 2D,
ImΠR (q ≈ 2kF , ω) ∝

ω
θ (2kF − q) .
kF (2kF − q)

Because of the one-sided singularity in ImΠR near q = 2kF , the effective interaction oscillates and falls off as a power of r. By power counting, if a static
Friedel oscillation falls off as sin 2kF r/rD , then the dynamic one behaves as
ω sin 2kF r
˜
U ∝ (D−1)/2 .
r
Dynamic Kohn anomaly results in the same kind of non-analyticity in the selfenergy (and thermodynamics) as the forward scattering. The “dangerous” range
of q now is |q − 2kF | ∼ ω/vF –“dynamic backscattering”. It is remarkable that
the non-analytic term in the self-energy is sensitive only to strictly forward or
backscattering events, whereas processes with intermediate momentum transfers
contribute only to analytic part of the self-energy. To see this, we perform the
analysis of kinematics in the next section.

Fundamental aspects of electron correlations and quantum transport
k
k

1

2

k

1

α

α

β

β 2

k

a)

k

k

1

k

α

α
k

2 β

1

β 2

b)

k

k

1

21
k

2

β

α

1α

β

k

2

c)

Fig. 6. Scattering processes responsible for divergent and/or non-analytic corrections to the selfenergy in 2D. a) “Forward scattering”–an analog of the “g4 ”-process in 1D. All four fermionic
momenta are close to each other. b) Backscattering–an analog of the “g2 ”-process in 1D. The net
momentum before and after collision is small. Initial momenta are close to final ones. Although the
momentum transfer in such a process is small, we still refer to this process as “backscattering” (see
the discussion in the main text). c) 2kF − scattering.

2.2. 1D kinematics in higher dimensions
The similarity between non-FL behavior in 1D and non-analytic features in higher
dimensions occurs already at the level of kinematics. Namely, one can make a
rather strong statement: the non-analytic terms in the self-energy in higher dimensions result from essentially 1D scattering processes. Let’s come back to
self-energy diagram 3a. In general, integrations over fermionic momentum p and
bosonic q are independent of each other: one can first integrate over (p, ε), forming a bubble, and then integrate over (q, ω). Generically, p spans the entire Fermi
surface. However, the non-analytic features in Σ come not from generic but very
specific p which are close to either to k or to −k.
Let’s focus on the 2D case. The ε2 ln |ε| term results from the product of two
−1
q -singularities: one is from the angular average of ImG and the other one
from the dynamic, ω/vF q, part of the bubble. In Appendix Appendix A, it is
shown that the ω/vF q singularity in the bubble comes from the region where p is
almost perpendicular to q. Similarly, the angular averaging of ImG also pins the
angle between k and q to almost 90◦ .
ImGR (ε − ω, k − q) = −πδ (ε − ω − qvF cos θ′ ) →
ω
ε−ω
∼
≪ 1 → θ′ ≈ π/2.
cos θ′ =
vF q
vF q
As p and k are almost perpendicular to the same vector (q), they are either almost
parallel or anti-parallel to each other. In terms of a symmetrized (“sunrise”) selfenergy (cf. Fig. 3), it means that either all three internal momenta are parallel to
the external one or one of the internal one is parallel to the external whereas the
other two are anti-parallel 12 . Thus we have three almost 1D processes:
12 In

3D, conditions p ⊥ q and k ⊥ q mean only that p and k lie in the same plane. However,

22

Dmitrii L. Maslov

• all four momenta (two initial and two final) are almost parallel to each other;
• the total momentum of the fermionic pair is near zero, whereas the transferred
momentum is small;
• he total momentum of the fermionic pair is near zero, whereas the transferred
momentum is near 2kF .
These are precisely the same 1D processes we are going to deal with in the
next Section–the only difference is that in 2D, trajectories do have some angular
spread, which is of order |ω| /EF . The first one is known as “g4 ” (meaning: all
four momenta are in the same direction) and the other one as “g2 ” (meaning: two
out of four momenta are in the same direction). Both of these processes are of
the forward-scattering type as the transferred momentum is small. In 1D, these
processes correspond to scattering of fermions of same (g4 ) or opposite chirality
(g2 ). The last (2kF ) process is known “g1 ” in 1D.
It turns out that of these two processes, the g2 - and 2kF - ones, are directly
responsible for the ε2 ln ε behavior. The g4 -process leads to a mass-shell singularity in the self-energy both in 1D and 2D, discussed in the next section, but does
not affect the thermodynamics, so we will leave it for now.
What about 2kF − scattering? Suppose electron k scatters into −k emitting
an electron-hole pair of momentum 2k. In general, 2k of the e-h pair may consist
of any two fermionic momenta which differ by 2k : p and p + 2k. But since
2k ≈ 2kF , the components of the e-h pair will be on the Fermi surface only if
p ≈ −k and p + 2k ≈ k. Only in this case does the effective interaction (bubble)
have a non-analytic form at finite frequency. Thus 2kF - scattering is also of the
1D nature for D > 1.
What we have said above, can be summarized in the following pictorial way.
Suppose we follow the trajectories of two fermions, as shown in Fig. 7. There
are several types of scattering processes. First, there is “any-angle” scattering
which, in our particular example, occurs at a third fermion whose trajectory is not
shown. This scattering contributes regular, FL terms both to the self-energy and
thermodynamics. Second, there are dynamic forward-scattering events, when
q ∼ |ω| /vF . These are not 1D processes, as fermionic trajectories enter the
interaction region at an arbitrary angle to each other. In 3D, a third order in
such processes results in the non-analytic behavior of C(T )–this is the origin
it is still possible to show that for a closed diagram, e.g., thermodynamic potential, p and k are
either parallel or anti-parallel to each other. Hence, the non-analytic term in C(T ) also comes from
the 1D processes. In addition, there are dynamic forward scattering events (marked with a star in
Fig. 7) which, although not being 1D in nature, do lead to a non-analyticity in 3D. Thus, the T 3 ln T
anomaly in C(T ) comes from both 1D and non-1D processes [50] . The difference is that the former
start already at the second order in the interaction whereas the latter occur only at the third order. In
2D, the entire T − term in C(T ) comes from the 1D processes.

Fundamental aspects of electron correlations and quantum transport

g1
g

23

g2

4

dynamic forward scattering
1D forward or backscattering
‘‘any−angle" scattering −−> regular (FL) contribution

Fig. 7. Typical trajectories of two interacting fermions. Explosion: “any-angle” scattering at a third
fermion (not shown) which leads to a regular (FL) contribution. Five-corner star: dynamic forward
scattering q ∼ |ω|/vF . This process contributes to non-analyticity in 3D (to third order in the interaction) but not in 2D. Four-corner star: 1D dynamic forward and backscattering events, contributing
to non-analyticities both in 3D and 2D.

of the “paramagnon” anomaly in C(T ). In 2D, dynamic forward scattering does
not lead to non-analyticity. Finally, there are processes, marked by“g1 ”, “g2 ”,
and “g4 ", when electrons conspire to align their initial momenta so that they are
either parallel or antiparallel to each other. These processes determine the nonanalytic parts of Σ and thermodynamics in 2D (and also, formally, for D < 2.)
A crossover between D > 1 and D = 1 occurs when all other processes but g1 ,
g2 , and g4 are eliminated by a geometrical constraint.
We see that for non-analytic terms in the self-energy (and thermodynamics),
large-angle scattering does not matter. Everything is determined by essentially
1D processes. As a result, if the bare interaction has some q dependence, only
two Fourier components matter: U (0) and U (2kF ). For example, in 2D
ImΣR (ε) ∝

ReΣR (ω) ∝
C(T )/T

=

χs (Q, T ) =

U 2 (0) + U 2 (2kF ) − U (0)U (2kF ) ε2 ln |ε| ;
U 2 (0) + U 2 (2kF ) − U (0)U (2kF ) ε |ε| ;

γ ∗ − a U 2 (0) + U 2 (2kF ) − U (0)U (2kF ) T ;
χ∗ (0) + bU 2 (2kF ) max {vF Q, T } ;
s

where a and b are coefficients. These perturbative results can be generalized for
the Fermi-liquid case, when the interaction is not necessarily weak. Then the
leading, analytic parts of C(T ) and χs are determined by the angular harmonics

24

Dmitrii L. Maslov

of the Landau interaction function
ˆ
ˆ
F (p, p′ ) = Fs (θ) I + Fa (θ) σ · σ ′ ,
where θ is the angle between p and p′ . In particular,
γ∗

=

χ∗ (0) =
s

γ0 (1 + cos θFs ) ;
1 + cos θFs
,
χ0
s
1 + Fa

where γ0 and χ0 are the corresponding quantities for the Fermi gas. Because of
s
the angular averaging, the FL part is rather insensitive to the details of the interaction. As generically Fs and Fa are regular functions of θ, the whole Fermi
surface contributes to the FL renormalizations. Vertices U (0) and U (2kF ), occurring in the perturbative expressions, are replaced by scattering amplitudes at
angle θ = π
ˆ
ˆ
A (p, p′ ) = As (θ) I + Aa (θ) σ · σ ′ ,
Beyond the perturbation theory [37],
= γ ∗ − a A2 (π) + 3A2 (π) T ;
¯ s
a
∗
¯ 2 (π) max {vF Q, T } .
χs (Q, T ) = χs (0) + bAa
C(T )/T

Non-analytic parts are not subject to angular averaging and are sensitive to a
detailed behavior of As,a near θ = π 13 .
2.3. Infrared catastrophe
2.3.1. 1D
By now, it is well-known that the FL breaks down in 1D and an attempt to apply the perturbation theory to 1D problem results in singularities. Let’s see what
precisely goes wrong in 1D. I begin with considering the interaction of fermions
of opposite chirality, as in diagram Fig. 8b. Physically, a right-moving fermion
emits (and then re-absorbs) left-moving quanta of density excitations (same for
left-moving fermion emitting/absorbing right-moving quanta). Now, instead of
the order-of-magnitude estimate (2.7), which is good in all dimensions but only
for power-counting purposes, I am going to use an exact expression for the bubble, Eq. (B. 4), formed by left-moving fermions. On the Fermi surface (k = kF ,
13 The renormalization of the scattering amplitudes by the Cooper channel of the interaction results
in additional ln T -dependences of Ac,s (π)

Fundamental aspects of electron correlations and quantum transport

25

-

+
b)

a)

-

+

+

+

+

+

+

+

c)

+

+

+

+

+

Fig. 8. Self-energy in 1D. ± refer to right (left)-moving fermions.

we have
ε

−ImΣR (ε) ∼
+−

U 2 ν1

dω
dω

0
2

∼

U ν1

∼

dqImGR (ε − ω, k − q) ImΠR
+
−
dqδ (ε − ω + vF q) (ω/vF ) δ (ω + vF q)

ε

g 2 |ε| ,

0

where g ≡ U/vF is the dimensionless coupling constant. The corresponding real
part behaves as ε ln |ε| . What we got is bad, as ImΣR scales with ε in the same
way as the energy of a free excitation above the Fermi level and ReΣR increases
faster than ε (which means that the effective mass depends on ε as ln |ε|), but not
too bad because, as long as g ≪ 1, the breakdown of the quasi-particle picture
occurs only at exponentially small energy scales: ε
EF exp(−1/g 2 ). Now,
let’s look at scattering of fermions of the same chirality. This time, I choose to
be away from the mass shell.
−ImΣR
++

ε

∝

dω
0

dq δ (ε − ω − vF (k − q)) ωδ (ω − vF q)
ImGR
+

=

ε2 δ (ε − vF k) .

(2.8)

ImΠR
+

(2.9)

26

Dmitrii L. Maslov

It is not difficult to see that the full (complex) self-energy is simply
ΣR ∝ −
++

ε2
.
ε − vF k + i0+

(2.10)

On the mass shell (ε = vF k) we have a strong–delta-function–singularity. This
anomaly was discovered by Bychkov, Gor’kov, and Dzyaloshinskii back in the
60s [52], who called it the “infrared catastrophe”. Indeed, it is similar to an
infrared catastrophe in QED, where an electron can emit an infinite number of
soft photons. Likewise, since we have linearized the spectrum, a 1D fermion
can emit an infinite number of soft bosons: quanta of charge- and spin-density
excitations. The point is that in 1D there is a perfect match between momentum
and energy conservations for a process of emission (or absorption) of a boson
with energy and momentum related by ω = vF q :
k′
ε′

=
=

k−q
ε − ω = vF k − vF q = vF (k − q) .

On the mass-shell, the energy and momentum conservations are equivalent. Imagine that you want to find a probability of certain scattering process using a Fermi
Golden rule. Then you have a product of two δ− functions: one reflecting the
momentum and other energy conservation. But if the arguments of the deltafunctions are the same, you have an essential singularity: a square of the deltafunction. As a result, the corresponding probability diverges.
A pole in the self-energy [Eq. (2.10)] indicates the non-perturbative and specifically 1D effect: spin-charge separation. Indeed, substituting Eq. (2.10) we get
two poles corresponding to excitations propagating with velocities vF (1 ± g)
(recall that g ≪ 1). This peculiar feature is confirmed by an exact solution (see
Section 3): already the g4 −interaction leads to a spin-charge separation (but not
to anomalous scaling). What we did not get quite right is that the velocities of
both–spin- and charge-modes–are modified by the interactions. In fact,the exact
solution shows that the velocity of the spin-mode remains equal to vF , whereas
the velocity of the charge mode is modified.
Obviously, there is no spin-charge separation for spinless electrons. Indeed, in
this case diagram Fig. 8a does not have an additional factor of two as compared to
Fig. 8c (but is still of opposite sign), so that the forward-scattering parts of these
two diagrams cancel each other. As a result, there is no infrared catastrophe for
spinless fermions.
2.3.2. 2D
What we considered in the previous section sounds like an essentially 1D effect.
However, a similar effect exists also in 2D (more generally, for 1 ≤ D ≤ 2).

Fundamental aspects of electron correlations and quantum transport

27

This emphasizes once again that the difference between D = 1 and D > 1 is not
as dramatic as it seems.
In 2D, the self-energy also diverges on the mass shell, if one linearizes the
electron’s spectrum, albeit the divergence is weaker than in 1D–to second order,
it is logarithmic 14 . The origin of the divergence can be traced back to the form
of the polarization bubble at small momentum transfer, Eq. (A. 1). Integrating
over the angle in 2D, we get
ImΠR (ω, q) = −

m
2π

ω
2

(vF q) − ω 2

θ (vF q − |ω|) .

(2.11)

ImΠR (ω, q) has a square-root singularity at the boundary of the particle-hole
continuum, i.e., at ω = vF q. (This is a threshold singularity of the van Hove
type: the band of soft electron-hole pairs is terminated at ω = vF q, but the
spectral weight of the pairs is peaked at the band edge). On the other hand,
expanding ǫk+q in GR (ε + ω, k+q) as ξk+q = ξk + vF q cos θ and integrating
over θ, we obtain another square-root singularity
2

dθImGR = −2π (vF q) − (ε + ω − ξk )

2

−1/2

.

(2.12)

On the mass shell (ω = ξk ), the arguments of the square roots in Eqs. (2.11) and
(2.12) coincide, and the integral over q diverges logarithmically. The resulting
contribution to ImΣR diverges on the mass shell (ε = ξk ) [53, 54, 55],[34],[37]
ImΣR (ε, k) = −
g4

EF
u 2 ε2
ln
,
8π EF
|ε − ξk |

where ∆ ≡ ε − ξk and u ≡ mU/2π. The process responsible for the logsingularity is the “g4 ” process in Fig. 6. On the other hand, g2 and g1 processes
give a contribution which is finite on the mass shell
ImΣR +g2 (ε, k) = −
g1

u 2 ε2
EF
ln
.
4π EF
|ε + ξk |

(The divergence at ε = −ξk is spurious and is removed by going beyond the logaccuracy [34],[37].) We see therefore that the familiar form of the self-energy in
2D [ε2 ln |ε|, see Ref. [56]] is valid only on the Fermi surface (ξk = 0) . The logarithmic singularity in ImΣR on the mass shell is eliminated by retaining the finite
curvature of single-particle spectrum (which amounts to keeping the q 2 /2m term
in ξk+q ). This brings in a new scale ε2 /EF . The emerging singularity in (2.3.2)
14 In

3D, there is no mass-shell singularity to any order of the perturbation theory.

28

Dmitrii L. Maslov

is regularized at |ε − ξk | ∼ ε2 /EF and the ε2 ln |ε| behavior is restored. However, higher orders diverge as power-laws and finite curvature does not help to
regularize them. This means that–in contrast to 3D–the perturbation theory must
be re-summed even for an infinitesimally weak interaction. Once this is done, the
singularities are removed. Re-summation also helps to understand the reason for
the problems in the perturbation theory. In fact, what we were trying to do was
to take into account a non-perturbative effect–an interaction with the zero-sound
mode–via a perturbation theory. Once all orders are re-summed, the zero-sound
mode splits off the continuum boundary–now it is a propagating mode with velocity c > vF . This splitting is what regularizes the divergences. The resulting
state is essentially a FL: the leading term in Σ behaves as ε2 ln |ε| . However,
some non-perturbative features remain: for example, the spectral function exhibits a second peak away from the mass shell corresponding to the emission of
the zero-sound waves by fermions. A two-peak structure of the spectral function
is reminiscent of the spin-charge separation, although we do not really have a
spin-charge separation here: in contrast to the 1D case, the spin-density collective mode lies within the continuum and is damped by the particle-hole pairs.

3. Dzyaloshinskii-Larkin solution of the Tomonaga-Luttinger model
3.1. Hamiltonian, anomalous commutators, and conservation laws
In the Tomonaga-Luttinger model [57],[58] one considers a system of 1D spin1/2 fermions with a linearized dispersion. Only forward scattering of left- and
right-moving fermions is taken into account (g2 and g4 − processes), whereas
backscattering is neglected. This last assumption means that the interaction potential is of sufficiently long-range, so that U (2kF ) ≪ U (0) . [We will come
back to this condition later.] Coupling between fermions of the same chirality
(g4 ) is assumed to be different from coupling between fermions of different chirality (g2 ). If the original Hamiltonian contains only density-density interaction,
then g2 = g4 . A difference between g2 and g4 leads to an unphysical (within
this model) current-current interaction. We will keep g2 = g4 , however, at the
intermediate steps of the calculations as it helps to elucidate certain points. At
the end, one can make g2 equal to g4 without any penalty. In addition, in some
physical situations, g2 = g4 . 15 In what follows I will follow the original paper
by Dzyaloshinskii and Larkin (DL) [59] and a paper by Metzner and di Castro
15 For example, Coulomb interaction between the electrons at the edges of a finite-width Hall bar
(in the Integer Quantum Hall Effect regime) has this feature: electrons of the same chirality are
situated on the same edge, whereas electrons of different chirality are on opposite edges; hence the
matrix elements for the g2 − and g4 - interactions are different.

Fundamental aspects of electron correlations and quantum transport

29

[60], where the Ward identity used by Dzyaloshinskii and Larkin is derived in a
detailed way.
The Hamiltonian of the model is written as
H
Hint

=
≡

H0 + Hint ;
H 2 + H4 ,

where
H0 = vF
k,σ

k a† (k)a+,σ (k) − a† (k)a−,σ (k)
+,σ
−,σ

is the Hamiltonian of free fermions (± denote right/left moving fermions and σ
is the spin projection) and
H2
H4

=
=

g2
L

q

g4
2L

q

σ,σ′

ρ+,σ (q) ρ−,σ′ (−q) ;

σ,σ′

ρ+,σ (q) ρ+,σ′ (−q) + ρ−,σ (q) ρ−,σ′ (−q) ,

with
ρ±,σ =

k

a† (k + q) a±,σ (k) .
±,σ

To avoid additional complications, I assume that the interaction is spin-independent.
To simplify the notations and to emphasize the similarity between this model and
QED, I will set vF to unity in this section.
Introducing the chiral charge- and spin densities as
ρc
±
ρs
±

=
=

ρ±,↑ + ρ±,↓ ;
ρ±,↑ − ρ±,↓ ,

and total charge density and current as
ρc
jc

=
=

ρc + ρc ;
+
−
ρc − ρc ,
+
−

the interaction part of the Hamiltonian reduces to
Hint =
q

1
1
(g2 + g4 ) ρc (q) ρc (−q) + (g4 − g2 ) j c (q) j c (−q) .
2
2

(3.1)

As we have already said, for g2 = g4 , the interaction is of a pure density-density
type. Notice also that the spin density and current drop out of the Hamiltonian–
this is to be expected for a spin-invariant interaction. To make a link with QED,

30

Dmitrii L. Maslov

let us introduce Minkowski current j µ with µ = 0, 1 so that j 0 = ρc (=j0 ) and
j 1 = j c (=-j1 ). Then the interaction can be written as a 4-product of Minkowski
currents in a Lorentz-invariant form
gµν jν j ν ,

Hint =
q

where
g00

=

g11

=

g01

=

1
(g2 + g4 ) ;
2
1
(g4 − g2 ) ;
2
g10 = 0.

(3.2)

In what follows, we will need the following anomalous commutators
[ρ±,σ (q) , H0 ] =
[ρ±,σ (q) , H2 ] =
[ρ±,σ (q) , H4 ] =

±qρ±,σ (q) ;
g2
± qρ∓,σ (q) ;
2π
g4
± qρ±,σ (q) .
2π

The derivation of these commutation relations can be found in a number of standard sources [61, 10] and I will not present it here. Adding up the commutators,
we get
[ρ±,σ , H] =
=

[ρ±,σ , H0 + H2 + H4 ]
g4 c
g2 c
qρ ±
qρ
±qρ±,σ ±
2π ∓ 2π ±

Adding up equations for spin-up and -down fermions, we obtain
ρc , H = ±qρc ±
±,
±

g4
g2 c
qρ∓ ± qρc .
π
π ±

Finally, adding up the ± components yields
i∂t ρc = [ρc , H] = vc qj c ,
where

(3.3)

g4 − g2
π
(recall that vF = 1). Eq. (3.3) is a continuity equation reflecting charge conservation. As if we did not have enough new notations, here is another one
vc ≡ 1 +

Qµ = (ω, q)

Fundamental aspects of electron correlations and quantum transport

=

K =

+

+

+

+

31

+

...

(a)

Λ

=

=

+

+

+

+

+

+

+

+

...

(b)

Fig. 9. a) Three-leg correlator K. b) Vertex part Λ.

and
Qµ = (ω, vc q) .
In these notations and after a Fourier transform, the continuity equation can be
written as
Qµ j µ = 0.
The same relation for free particles reads
Qµ j µ = 0.
3.2. Reducible and irreducible vertices
Now, construct a mixed (fermion-boson) correlator
µ
K±,σ (k, q|t, t1 , t′ ) = − T j µ (q, t) a±,σ (k, t1 ) a† (k + q, t1 ) ,
1
±,σ

where µ = 0, 1 and
j0
j1

= ρc + ρc ;
+
−

= ρc − ρc .
+
−

(3.4)

32

Dmitrii L. Maslov
µν

Α

Γ
Λ

g µν
=

+

Fig. 10. Relation between vertices Λ and Γ.

K µ is an analog of the three-leg vertex in QED , except that in QED the “boson”
is the µ− the component of the photon field
QED : K µ = − T Aµ a¯ .
a
A diagrammatic representation of K µ is a three-particle (one boson and two
fermions) diagram (cf. Fig. 9a).
The diagrams with self-energy insertions to solid lines simply renormalize the
Green’s functions. Absorbing these renormalizations, we single out the vertex
part, re-writing K µ as
K µ = G2 Λµ .
(3.5)
Notice that there are as many vertex parts as there are bosonic degrees of freedom. In (3+1) QED, Λ0 is a scalar vertex and Λµ=1,2,3 are the components of
the vector vertex. Diagrams representing Λµ are shown in Fig. 9b. These series can be re-arranged further by separating the photon-irreducible vertex part,
Γµ . A photon-irreducible part is obtained by separating the corrections to the
bosonic line, i.e., taking into account polarization. Vertices Λµ and Γµ are related via a kind of Dyson equation, which is simpler than the usual Dyson in
a sense that there is no Λµ on the right-hand-side. Diagrammatically, this relation is represented by Fig.10 where a shaded bubble is an exact (renormalized)
current-current correlation function
Aµν (q, t) = −

i µ
j (q, t) j µ (−q, 0) .
V

Algebraically, equation in Fig.10 says
Λµ = Γµ + Aµν gµλ Γλ .
±,σ
±,σ
±,σ

(3.6)

(We remind the reader that indices ±, σ simply specify the fermionic flavor which
is not mixed in our approximation of forward-scattering and spin-independent

Fundamental aspects of electron correlations and quantum transport

33

forces, so all relations are applicable to each individual flavor). The coupling
constants g µν relate currents to densities. According to Eqs. (3.1) and (3.2), densities couple to densities and currents to currents with no cross terms. Opening
the matrix product in Eq. (3.6), we obtain
Λµ = Γµ + Aµ0 g00 Γ0 + Aµ1 g11 Γ1 .
i
i

(3.7)

3.3. Ward identities
A Ward identity for vertex Λµ is obtained by applying i∂t to K µ in Eq. (3.4)
and using the continuity equation (3.4).16 Performing this operations and Fourier
transforming in time, we obtain
µ
Qµ Ki (K, Q) = Gi (K) − Gi (K + Q) ,

where i denotes the branch
i ≡ ±, σ.

Recalling Eq. (3.5), we see that the Ward identity becomes
−1
Qµ Λµ (K, Q) = G−1 (K + Q) − Gi (K) ,
i
i

(3.8)

which is identical to a corresponding identity in QED. For those who like to see
things not masked by fancy notations, here is Eq. (3.8) in an explicit form
ωΛ0 (ε, k; ω, q) − vc qΛ1 (ε, k; ω, q) = G−1 (ε + ω, k + q) − G−1 (ε, k) . (3.9)
i
i
i
i
Notice that Eqs.(3.8,3.9) contain renormalized velocity vc . In what follows, we
will actually need a Ward identity not for Λµ but for the photon-irreducible vertex Γµ . This one is obtained by deriving the continuity equation for 4-current
correlation function Aµν . To this end, one applies i∂t to A0ν and uses continuity
equation (3.3), which yields 17
Qµ Aµν =

2
qδν,1 .
π

(3.11)

16 When

differentiating, recall that the T − product can be represented by step-functions in time
which, upon differentiating, yield delta-functions
17 To get this result, recall the form of the anomalous density-density commutator
2
[j µ (q) , j ν (−q)] = ǫµν qL,
π
where ǫ00 = ǫ11 = 0, ǫ01 = −ǫ10 = 1. Now open the T − product in A0ν and apply i∂t
i∂t A0ν (q, t)

=
=

i
(i∂t ) θ (t) j 0 (q, t) j ν (−q, 0) + θ (−t) j ν (−q, 0) j 0 (q, t)
V
1
q
δ (t) [j 0 (q, 0) , j ν (−q, 0)] + va qA1ν = 2 δν,1 + va qA1ν . (3.10)
V
π

−

In 4-notations, (3.10) is equivalent to (3.11).

34

Dmitrii L. Maslov

Now, we form a scalar product between Qµ and Eq. (3.7), using continuity
equation for Aµν (3.11). This brings us to
Qµ Λµ = Qµ Aµ ,
i
where
Qµ Λ µ
i

= Qµ Γµ + Aµ0 g00 Γ0 + Aµ1 g11 Γ1
i
i
i

= Qµ Γµ + Qµ Aµ0 g00 Γ0 + Qµ Aµ1 g11 Γ1
i
i
i
=0

= ωΓ0 −
i
=

ωΓ0
i

=2q/π

qΓ1 +
i

vc

2q 1 g4 − g2 1
Γi
π 2 π

=1+(g4 −g2 )/π

− qΓ1 = Qµ Γµ .
i

Finally, the Ward identity for photon-irreducible vertex is
Qµ Γµ = G−1 (K + Q) − G−1 (K) .
i
i

(3.12)

It is remarkable that the left-hand-side of Eq. (3.12) contains the bare Fermi
velocity (= 1) instead of the renormalized one. This is true even if we allowed
for spin-dependent interaction in the Hamiltonian.
It seems that we have not achieved much, as the conservation law was simply
cast into a different form. However, in our 1D problem with a linearized spectrum a further progress can be made because the current and density (for given
chirality) are just the same quantity (up to an overall factor of the Fermi velocity):
Γ1 = ±Γ0
±,σ
±,σ
Therefore, we have a closed relation between just one vertex and Green’s functions. Suppressing the 4-vector index µ, we get the Ward identity for the density
vertex
G−1 (K + Q) − G−1 (K)
±,σ
±,σ
.
(3.13)
Γ0 (K, Q) =
±,σ
ω∓q
This is the identity that we need to proceed further with the DzyaloshinskiiLarkin solution of the Tomonaga-Luttinger problem. Notice that (3.13) contains
fully interacting Green’s functions.

Fundamental aspects of electron correlations and quantum transport
Π+
g4

V++

g4

=

+

Π−

111
000
1
0
111
000
1
0
111
000
1
0
111
000
1
0
111
000
1
0
111
000
1
0
111
000
1
0
111
000
1
0
1
0
111
000
1
0
1
0
1
0
111
000
1
0
1
0
111
000
1
0
1
0

V++

V
1111
0000 =V -+
+1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000

g2

+

Π−
V
+-

g2

=

g2
−

+

35

Π+
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000

V =V
-++

1
0
1
0
111
000
1
0
111
000
+
1
0
111
000
1
0
111
000
111
000
111
000
111
000
111
000
11
00
1
0
1
0
111
000
1
0+
11
00
1
0
111
000
11
00
1
0
1
0
111
000
1
0

g4

+

V+-

Fig. 11. Dyson equation for the effective interaction. Solid line: Green’s function of a right-moving
fermion. Dashed line: Green’s function of a left-moving fermion. Single wavy line: bare interaction
of fermions of the same chirality; spiral line: same for the fermions of opposite chirality. Double
wavy and spiral lines represent the renormalized interactions.

3.4. Effective interaction
Effective interaction is obtained by collecting polarization corrections to the bare
one. Diagrammatically, this procedure is described by the Dyson equation, represented in Fig.11. The interaction and polarization bubble are matrices with
components
ˆ
V =

V++
V+−

V+−
V++

ˆ
, V0 =

g4
g2

g2
g4

ˆ
,Π =

Π+
0

0
Π−

,

where we used an obvious symmetry V++ = V−− , V+− = V−+ . The Dyson
equation in the matrix form reads
ˆ
ˆ
ˆ ˆˆ
V = V0 + V0 ΠV ,
or, in components,
V++
V+−

= g4 + g4 Π+ V++ + g2 Π− V+− ;
= g2 + g2 Π− V++ + g4 Π− V+− .

(3.14)

The bubbles in these equations are fully renormalized ones, i.e., they are built on
exact Green’s functions and contain a vertex (hatched corner):
Π± (ω, q) = −2i

dkdε

2 G±

(2π)

(ε + ω, k + q) G± (ε, k) Γ0 (ε, k; ω, q) .
±

36

Dmitrii L. Maslov

Now we use the Ward identity for Γ0 (3.13) to get 18
±
Π± (ω, q) = −2i

1
ω∓q

dkdε
2

(2π)

[G± (ε, k) − G± (ε + ω, k + q)] . (3.15)

Eq. (3.15) looks exactly the same as a free bubble [cf. Eq. (B. 1)] except that
it contains exact rather than free Green’s functions. Because we managed to
transform the product of two Green’s functions into a difference, frequency integration in Eq. (3.15) can be performed term by term yielding exact momentum
distribution functions n± (k) and n± (k + q) :
Π± (ω, q) =

1
ω∓q

dk
[n± (k) − n± (k + q)] .
π

(3.16)

At the first glance, it seems that we have not achieved much so far. Indeed, we
traded one unknown quantity (Π± ) for another (n± ). Both of them include the
interaction to all orders and without any further simplification we are stuck. In
fact, we have already made an important simplification: when specifying the
model, we assumed only forward scattering. This means that the interaction
is sufficiently long-range in real space so that backscattering can be neglected.
Equivalently, in the momentum space it means that our interaction operates only
in a narrow window of width q0 near the Fermi points, ±kF . Thus the states far
away from the Fermi points are not affected by the interaction. The momentum
integral in (3.16) comes from regions far away from the Fermi surface where
unknown functions n± can be approximated by free Fermi steps. This approximation is good as long as q0 ≪ kF . The solution is going to be exact only in
a sense that there will be no constraints on the amplitude of the interaction (parameters g2 and g4 ) but not its range.19Now we understand better why the title of
the paper by Dzyaloshinskii and Larkin [59] is “Correlation functions for a onedimensional Fermi system with long-range interaction (Tomonaga model)”20.
18 I skipped over a subtlety related to the infinitesimal imaginary parts i0+ in the denominator.
Works the same way. If you are unhappy with this, imagine that we work with Matsubara frequencies.
Then there are no i0+ s whatsoever.
19 In higher dimensions, we have a familiar problem of the Coulomb potential. Because it’s a
power-law potential, one cannot separate it into “amplitude” and “range”. There is in fact a single dimensionless parameter, rs , which must be small for the perturbation theory–Random Phase
Approximation–to work. Once rs ≪ 1, we have two things: the screened potential is simultaneously
weak and long-ranged. The Tomonaga-Luttinger model unties these two things: the interaction is
assumed to be long-ranged but not necessarily weak.
20 What seemed to be just a matter of mathematical convenience in the 70s, turns out to be quite a
realistic case these days. If a wire of width a is located at distance d to the metallic gate, the Coulomb
potential between electrons in the wire is screened by their images in the gate. Typically, d ≫ a. A
simple exercise in electrostatics shows that in this case U (0) is larger than U (2kF ) by large factor
ln (d/a) [62].

Fundamental aspects of electron correlations and quantum transport

37

With this simplification, the momentum integration proceeds in the same way
as for free fermions (see Appendix Appendix B) with the result that the fully
interacting bubbles are the same as free ones
Π± (ω, q) = Π0 (ω, q) = ±
±

q
1
.
π ω − q + i0+ sgnω

(3.17)

This is a truly remarkable result which is a cornerstone for the DL solution 21 .
Because our bubbles were effectively “liberated” from the interaction effects,
system (3.14) is equivalent to what we would have obtained from the Random
Phase Approximation (RPA). It turns out that RPA is asymptotically exact in 1D
in the limit q0 /kF → 0. Solving the 2 by 2 system, we obtain for the effective
interaction
V++ (ω, q) = (ω − q)

2
2
g4 (ω + q) + g4 − g2 q/π
,
ω 2 − u2 q 2 + i0+

where 22
u=

1+

2
2g4
g 2 − g2
+ 4
.
π
π

For g4 = g2 ≡ g,
V++ (ω, q) = g

ω2

ω2 − q2
.
− u2 q 2 + i0+

(3.18)

3.5. Dyson equation for the Green’s function
The Dyson equation for right-moving fermions reads
d2 Q

Σ+ (P ) = i

2 G+

(2π)

(P − Q) V++ (Q) Γ0 (P, Q) .
+

Diagrammatically, this equation is shown in Fig. 12. For linear dispersion,
Σ± (ε, p) = ε ∓ p − G−1 (ε, p)
±
Substituting this relation back into the Dyson equations, we obtain
(ε − p) G+ (ε, p) = 1+i
21 In

dωdq
(2π)

2 G+

(ε, p) G+ (ε − ω, p − q) V++ (ω, q) Γ0 (ε, p; ω, q) .
+

QED, this statement is known as Furry theorem (W. H. Furry, 1937)
that as long as g4 = g2 , the left-right symmetry is broken, i.e., the potential is not
symmetric with respect to q → −q.
22 Notice

38

Dmitrii L. Maslov

V++ ,V-11111
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
00000

Σ+ =
−

Γ−
+

G+
−
Fig. 12. Dyson equation for the self-energy.

Using the Ward identity (3.13) , we get
dωdq

(ε − p + Σ0 ) G+ (ε, p) = 1 + i

(2π)2

G+ (ε, p) G+ (ε − ω, p − q)

V++ (ω, q) −1
G+ (ε, p) − G−1 (ε − ω, p − q)
+
ω−q
dωdq
V++ (ω, q)
+ G+ (ε, p) × const,
2 G+ (ε − ω, p − q)
ω−q
(2π)
×

=1+i
where

dωdq V++ (ω, q)
(2π)2 ω − q

const =i

. A constant term can always be absorbed into Σ, which simply results in a shift
of the chemical potential. We are free to choose this shift in such a way that
const=0, so that the Dyson equation reduces to
(ε − p) G+ (ε, p) = 1 + i

dωdq

2 G+

(2π)

(ε − ω, p − q)

V++ (ω, q)
. (3.19)
ω−q

Notice that Eq. (3.19) is an integral equation with a difference kernel, which
can be reduced to a differential equation for G. Before we demonstrate how it is
done, let’s have a brief look at a case when there is no coupling between left- and
right-moving fermions: g2 = 0. In this case,
V++ (ω, q) = π

(w − 1) (ω − q)
,
ω − wq + i0+

where
w = 1 + g4 /π.

Fundamental aspects of electron correlations and quantum transport

39

Eq. (3.19) takes the form
dωdq G+ (ε − ω, p − q)
.
4π
ω − wq + i0+

(ε − p) G+ (ε, p) = 1 + i (w − 1)

This equation is satisfied by the following function
G+ (ε, p) =

1
ε − p + i0+

ε − wp + i0+

.

(3.20)

This is an example of a non-Fermi-liquid behavior: the pole of a free G splits into
the product of two branch cuts, one peaked on the mass shell of free fermions
(ε = p) and another one at the renormalized mass shell (ε = wp). As left- and
right movers are totally decoupled in this problem, the same result would have
been obtained for two separate subsystems of left- and right movers. For example, Eq. (3.20) predicts that an edge state of an integer quantum Hall system is
not a Fermi liquid, if spins are not yet polarized by the magnetic field [63]. The
same procedure for a spinless system would give us a pole-like G with a renormalized Fermi velocity. The non-Fermi-liquid behavior described by Eq. (3.20)
is rather subtle: it exists only if both ε and p are finite. In the limiting case of
p = 0 (tunneling DoS) we are back to a free-fermion behavior G (ε, 0) = ε−1 .
Also, the ε− integral of Eq. (3.19) gives a step-like distribution function in momentum space. The spectral function, however, is characteristically non-FL-like:
instead of delta-function peak we have a whole region |p| < |ε| < w |p| in which
ImG is finite. At the edges of this interval ImG has square-root singularities.
3.6. Solution for the case g2 = g4
Substituting the effective interaction (3.18) into Dyson equation (3.19), we obtain
(ε − p) G+ (ε, p) = 1 + i

ω+q
dωdq
G (ε − ω, p − q) g (q) 2
,
2
4π
ω − u2 q 2

where
u=

1 + 2g/π.

Notice that the constant g is replaced by a momentum-dependent interaction,
g (q) . The reason is that without such a replacement the integral diverges at the
upper limit. Here, the assumption of a cut-off in the interaction becomes important again. Transforming back to real time and space
G (x, t) =

dεdp

2 G (ε, p) e

(2π)

i(px−εt)

,

40

Dmitrii L. Maslov

we obtain the Dyson equation in a differential form
∂
∂
+
∂t ∂x

G (x, t) = P (x, t) G (x, t) − iδ (x)δ(t) ,

(3.21)

ω+q
.
ω 2 − u2 q 2 + i0+

(3.22)

where
P (x, t) =

1
4π 2

dωdqei(qx−ωt) g (q)

The integral for P diverges if g is constant. To ensure convergence, we will
approximate g (q) = ge−|q|/q0 . An actual form of the cut-off function is not
important as long as we are interested in such times and spatial intervals such
−1
that x, t ≫ q0 . The integral over ω is solved by closing the contour around the
poles of the denominator ω = ±u |q| (1 + i0+ ) . For t > 0, we need to choose
the one with Imω < 0. Doing so, we obtain
sgnq + u i(qx−u|qt|)
dω
··· = i
e
.
2π
2u
Solving the remaining q− integral, we obtain for P (x, t)
P (x, t) =

g
4πu

u+1
u−1
−
x − ut + i/q0
x + ut + i/q0

.

For t < 0, one needs to change q0 → −q0 in the last formula.
The delta-function term can be viewed as a boundary condition
G (x, 0+) − G (x, 0−) = −iδ (x) .

(3.23)

Once the function P (x, t) is known, Eq. (3.21) is trivially solved in terms of new
variables r = x − t, s = x + t. For example, for t > 0
s

G+ (r, t > 0) = G0 (x, t) f> (r) exp i

ds′ P (r, s′ ) ,

(3.24)

r

where function f> (r) is determined by the analytic properties of G as a function
of ε. Substituting result for P (x, t) into Eq. (3.24), we get
G+ (x, t > 0) =

1
G0 (x, t) f> (x − t)
2π

where

x − t + i/q0
x − ut + i/q0

α+1/2

x − t − i/q0
x + ut − i/q0

2

α=

(u − 1)
.
8u

(3.25)

α

.

Fundamental aspects of electron correlations and quantum transport

41

Formula for t < 0 is obtained by choosing another function f< and replacing
q0 → −q0 . Functions f>,< are determined from the analytic properties. First of
all, recall that
1
.
G0 (x, t) =
x − t + isgnt0+

We see that although G0 is not an analytic function of t for any t, it is analytic
for Ret > 0 in the right lower quadrant (Imt < 0) and for Ret < 0 in the upper
left quadrant (Imt > 0). The interaction cannot change analytic properties of a
Green’s function hence we should expect the same properties to hold for full G.
23

From the boundary condition (3.23), it follows that
f> (x) = f< (x)
and f (0) = 0.
Analyzing different factors in the formula for G, we see that only the term
α
(x − t ∓ i/q0 ) does not satisfy the required analyticity property. This term is
eliminated by choosing function f (x) as
2
f (x) = q0 x2 + 1

−α

.

Finally, the result for G takes the form
1
x − t + iγ
1
+
2π x − t + isgnt0
x − ut + iγ
1
× 2
α,
[q0 (x − ut + iγ) (x + ut − iγ)]

G+ (x, t) =

1/2

where γ = sgnt/q0 . It seems somewhat redundant to keep two different damping terms (isgnt0+ and γ) in the same equation. However, these terms contain
different physical scales. Indeed, isgnt0+ enters a free Green’s function and 0+
there has to be understood as the limit of the inverse system size. On the other
hand, γ contains a cut-off of the interaction. Obviously, |γ| ≫ 1/L → 0+ for
a realistic situation. The difference between the two cutoffs becomes important
23 Indeed,

G (x, t)

this property follows immediately from the Lehmann representation for G
=

|Mν0 |2 eipν x e−i(Eν −E0 )t , for t > 0;

−i
ν

=

|Mν0 |2 e−ipν x ei(Eν −E0 )t , for t < 0,

i
ν

where Mν0 are the matrix elements between the ground state and state ν with energy Eν > E0 . The
required property simply follows from the condition for convergence of the sum.

42

Dmitrii L. Maslov

for the momentum distribution function and tunneling DoS, discussed in the next
Section.
3.7. Physical properties
3.7.1. Momentum distribution
Having an exact form of the Green’s function, we can now calculate the momentum distribution of, e.g., right-moving fermions:
n+ (p) =
=
=
=

−i

∞
−∞

dxe−ipx G+ x, t → 0+

∞
1
e−ipx
i
dx
−
2
2π −∞ x + i0+ [q0 x2 + 1]α
∞
1
1
i
dxe−ipx P − iπδ (x)
−
α
2
2π −∞
x
[q0 x2 + 1]
∞
1
1
1
sin |p| x
,
− sgnp
dx
2 x2 + 1]α
2 π
x
[q0
0

We are interested in the behavior at p → 0 (which means |p| ≪ q0 ). The final
result for n+ (p) depends on whether α is larger or smaller than 1/2 [59, 64].
• For α < 1/2 (“weak interaction”), one cannot expand sin px in x because the
resulting integral diverges at x = ∞. Instead, rescale px → y
n+ (p) =

1
1
−
2 π

∞
0

dy

1
sin y
y (q /p)2 y 2 + 1
0

α

and neglect 1 in the denominator. This gives
n+ (p) =
where

1
+ C1
2

|p|
q0

2α

sgnp

(3.26)

sin πα
Γ (−2α) .
π
Notice that n+ (p) is finite (= 1/2) at p = 0, although its derivative is singular.
We should be able to recover the Fermi-gas step at p = 0 by setting α = 0 in
(3.26). Indeed,
1
1
lim C1 = α
=−
α→0
−2α
2
and
1 − sgnp
,
n (p) =
2
C1 =

Fundamental aspects of electron correlations and quantum transport

43

which is just the Fermi-gas result. Notice also that there is nothing special about
the limit α → 0 24 . Indeed, constant C1 has a regular expansion in α
1
C1 = − − γα + . . . ,
2
2α

where γ = 0.577 . . . and factor (|p| /q0 ) can be expanded for finite p and small
α as
2α
(|p| /q0 ) = 1 + 2α ln |p| /q0 .
To leading order in α, we obtain
n+ (p) =

1
1
− sgnp [1 + 2α ln |p| /q0 ] = n0 (p) − αsgnp ln |p| /q0 ,
2
2

which is a perfectly regular in α (but logarithmically divergent at p → 0) behavior. Once again, it is not surprising: despite the fact that the results for a 1D
system differ dramatically from that for the Fermi gas, they are still perturbative,
i.e., analytic, in the coupling constant.
• For α > 1/2 (“strong interaction”), it is safe to expand sin px and the result
is
1
n+ (p) = − C2 p/q0 ,
2
where
1 Γ (α − 1/2)
C1 = √
.
Γ (α)
2 π
In this case, no remains of a jump at the Fermi point is present in n+ (p) which
is a regular, linear function near p = 0.
• Finally, α = 1/2 is a special case, where expansion in p results in a logdivergent integral. To log-accuracy
n+ (p) =

q0
1 p
1
ln .
−
2 π q0 |p|

In general, n (p) is some hypergeometric function of p/q0 which decays rapidly
for p ≫ q0 and approaches 1 for p ≪ −q0 . A posteriori, this justifies the replacement of exact n (p) by its free form in the Dyson equation.
3.7.2. Tunneling density of states
Now we turn to the tunneling DoS
1
N (ε) = − ImGR (ε, x = 0) .
π
24 contrary

to some statements in the literature.

44

Dmitrii L. Maslov

Recalling that [23]
GR (ε) =
=

G (ε) , for ε > 0;
G∗ (ε) , for ε < 0,

we see that
ImGR (ε, 0) = sgnεImG (ε, 0)
and
N (ε) =
=

1
1
− sgnεImG (0, ε) = − sgnε
dteiεt G (0, t) −
π
π
1
dteiεt {G (0, t) − G∗ (0, −t)}
− sgnε
π

For t → ∞,
G (0, t) =

dte−iεt G∗ (0, t)

const
(−t)1+2α

and
G (0, t) − G∗ (−t)
is an odd function of t. Thus
N (ε) =
=

1
1
1
dteiεt G (0, t) − dte−iεt G∗ (0, t)
− sgnεImG (0, ε) = − sgnε
π
π
2i
∞
∞
1
sin εt
− sgnε
dt sin εt {G (0, t) − G∗ (0, −t)} ∝ sgnε
dt 1+2α .
π
t
0
0

The integral is obviously convergent for α < 1/2. In this case,
2α

Ns (ε) ∝ |ε|

,

which means that the local tunneling DoS is suppressed at the Fermi level. Actually, the exponent for α > 1/2 is the same, however, the prefactor is a different
function of α [65].
The DoS in Eq. (3.7.2) with exponent 2α, where α is given by Eq. (3.25)
corresponds to tunneling into the “bulk” of a 1D system, i.e., when the tunneling
contact (with a tip of an STM or another carbon nanotube crossing the first one)
is far away from its ends. In the next Section, we will analyze tunneling into an
edge of a 1D conductor, which is characterized by a different exponent, α′ .

Fundamental aspects of electron correlations and quantum transport

45

4. Renormalization group for interacting fermions

The Tomonaga-Luttinger model can be solved exactly as it was done in the previous Section– only in the absence of backscattering. Backscattering can be treated
via the Renormalization Group (RG) procedure. This treatment is standard by
now and discussed in a number of sources [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. For the
sake of completeness, I present here a short derivation of the RG equations. A
reader familiar with the procedure can skip this Section and go directly to Sec. 5,
where these equations will be used in the context of a single-impurity problem.
An exact solution of the previous Section is parameterized by two coupling
constants, g2 and g4 , which are equal to their bare values. In the RG language,
it means that these couplings do not flow. Let’s see if this is indeed the case. In
what follows, I will neglect the g4 − processes, as their effect on the flow of other
couplings is trivial, and, for the sake of simplicity, consider a spin-independent
interaction. To second order, the renormalization of the g2 − coupling is accounted for by two diagrams: diagrams a) and b) of Fig. 13.
Diagram a) is a correction to g2 in the particle-particle channel. The correction to g2 is given by
(2)

g2

a

=

2
g2

dq

2

(2π)

dωG+ (iε1 + iω, k1 + q) G− (iε2 − iω, k2 − q) .

Without a loss of generality, one can choose all momenta to be on the Fermi
“surface”: k1 = k2 = k3 = k4 = 0. Choose q > 0 (the other choice q < 0 will
simply double the result)
(2)

g2

a

=
=
=

2
g2

dq

(2π)2

dωG+ (iε1 + iω, q) G− (iε2 − iω, −q)

Λ/2

2
g2
2

(2π)

2
2πig2
2

(2π)

dq

dω

0
Λ/2

dq
0

1
1
i (ε1 + ω) − q i (ε2 − ω) − q

g2
iΛ
1
= 2 ln
ε1 + ε2 + ω + 2iq
4π ε1 + ε2

Adding the result up with the (identical) q < 0 contribution, we find
(2)

g2

a

=

2
g2
iΛ
.
ln
2π ε1 + ε2

46

Dmitrii L. Maslov

Diagram b) is a correction to g2 in the particle-hole channel:
(2)

g2

=

b

=

2
g2

dq

2

(2π)

Λ/2

2
g2
2

(2π)

= −

dωG+ (iε1 + iω, q) G− (iε4 + iω, q)
dq

dω

0
Λ/2d

2πi

2
2 g2
(2π)

q
0

1
1
i (ε1 + ω) − q i (ε4 + ω) + q

g2
iΛ
1
.
= − 2 ln
ε1 − ε4 + ω + 2iq
4π ε1 − ε4

As in the previous case, the final result is:
(2)

g2

b

=−

2
g2
iΛ
.
ln
2π ε1 − ε4

If we sum only the Cooper ladders, adding up more vertical interaction lines
to diagram a), the full vertex becomes
Γpp =

g2
.
1 + g2 ln ε1iΛ 2
+ε

(To keep track of the signs, one needs to recall that in Matsubara frequencies each
interaction line comes with the minus sign from the expansion of the S− matrix).
The resulting vertex blows up for attractive interaction (g2 < 0) as ε1 + ε2 → 0,
which is nothing more than a Cooper instability.
Likewise, untwisting the crossed lines in diagram b) and adding more interaction lines, we get the particle-hole vertex
Γph =

g2
.
1 − g2 ln ε1iΛ 4
−ε

This vertex has an instability for repulsive interaction (g2 > 0). In fact, none of
these instabilities occur. To see this, add up the results of diagrams a) and b)
(2)

g2

a+b

=

2
g2
iΛ
g2
iΛ
ε1 − ε4
ln
= 2 ln
− ln
.
2π
ε1 + ε2
ε1 − ε4
2π ε1 + ε2

In the RG, one changes the cut-off and follow the corresponding evolution of the
(2)
,
couplings. As the cut-off dependence cancelled out in the result for g2
coupling g2 remains invariant under the RG flow.
Backscattering generates additional diagrams: diagrams c)-f) in Fig.13.

a+b

Fundamental aspects of electron correlations and quantum transport

ε1

ε 1+ω

ε3

ε2

ε 2−ω

ε4

ε 1+ω

ε4 +ω

c)

b)

a)

ε+ε 1−ε

d2)

d1)

f1)

47

3

e)

f2)

Fig. 13. Second order diagrams for couplings g2 (solid wavy line) and g1 (dashed wavy line). Straight
solid and dashed lines correspond to Green’s functions of right- and left moving fermions, correspondingly.

48

Dmitrii L. Maslov

Diagram c) describes repeated backscattering in the particle-particle channel,
which is equivalent to forward scattering. Therefore, this diagram gives a correc∗
tion to g2 − coupling. Using the relation between G± , i.e., G± = − (G∓ ) , we
find
(2)

g2

c

2
g1

=

dq

2

(2π)
2
g1

=

dωG− (iε1 + iω, q) G+ (iε4 + iω, q)

dq

2

(2π)

dωG+ (iε1 + iω, q) G− (iε4 + iω, q)
(2)

The last integral is the same as for g2
(2)

g2

=

c

2
g1
(1)
2 dg2
g2

a
∗

∗

.

. Thus,
=

2
g1
−iΛ
.
ln
2π ε1 + ε2

The rest of the diagrams provide corrections to g1 .
Diagram d1) is the same as diagram a) except for the prefactor being equal
to g1 g2 :
iΛ
g1 g2
(2)
ln
=
g1
2π
ε1 + ε2
d1
Diagram d2) is a complex-conjugate of diagram d1). The sum of diagrams
d1) and d2) is equal to
(2)

g1

g1 g2
g1 g2
iΛ
−iΛ
+
ln
ln
2π
ε1 + ε2
2π
ε1 + ε2
g1 g2
Λ
.
ln
π
ε1 + ε2

=

d1+d2

=

Diagrams e) is a polarization correction to the bare g1 −coupling:
(2)

g1

e

=

2
g1 Π2kF (ω = ε1 − ε2 , q = 0) .

−
fermionic loop

Using Eq. (B. 8), we obtain
(2)

g1

e

=

Ns 2
Λ
g1 ln
,
2π
|ε1 − ε2 |

where Ns is the degeneracy factor (=2 for spin 1/2 fermions, occupying a single
valley in the momentum space).

Fundamental aspects of electron correlations and quantum transport

49

Diagram f1) is the same as the bubble insertion, except for no minus sign, no
degeneracy factor (Ns ) factor, and the overall coefficient is g1 g2:
(2)

g1

f1

=−

Λ
1
g1 g2 ln
.
2π
|ε1 − ε2 |

Diagram f2) is equal to f1). Their sum
(2)

g1

f 1+f 2

1
Λ
= − g1 g2 ln
π
|ε1 − ε2 |

Collecting all contributions together, we obtain
=

−g2 + g2

Γ2

=

g2 −

(2)

(2)

(2)

−Γ2

a

+ g2

b

+ g2

c

;

2
g2
g2
iΛ
iΛ
−iΛ
g2
+ 2 ln
;
ln
− 1 ln
2π ε1 + ε2
2π ε1 − ε4 2π ε1 + ε2
cancel out in the RG sense

−Γ1

=

−g1 +

Γ1

=

g1 −

(2)
g2

(2)

(2)

d

+ g2

e

+ g2

f

;

g1 g2
Ns 2
Λ
Λ
1
Λ
−
ln
g ln
+ g1 g2 ln
.
π
ε1 + ε2
2π 1 |ε1 − ε2 | π
|ε1 − ε2 |

Second and fourth terms in Γ1 also cancel out in the RG sense. Changing the
cut-off from Λ to Λ + d Λ, we obtain two differential equations
dΓ2
dl
dΓ1
dl

Γ2
1
;
2π
Γ2
= −Ns 1 ,
2π
= −

(4.1)
(4.2)

where l = ln Λ. We see that a quantity
1
1
¯
Γ = Γ2 −
Γ1 = const = g2 −
g1 .
Ns
Ns

(4.3)

is invariant under RG flow, therefore its value can be obtained by substituting the
bare values of the coupling constants (g2 and g1 ) into (4.3). The RG-invariant
combination is then
1
¯
g1 .
(4.4)
Γ = g2 −
Ns
For spinless electrons (Ns = 1),
¯
Γ = Γ2 − Γ1 = U (0) − U (2kF ) .

50

Dmitrii L. Maslov

This last result can be understood just in terms of the Pauli principle. Indeed,
the anti-symmetrized vertex for spinless electrons is obtained by switching the
outgoing legs of the diagram (p1 , p2 → p3 , p4 ). To first order,
Γ (p1 , p2 ; p3 , p4 ) = U (p1 − p3 ) − U (p1 − p4 ) .
Choosing p3 = p1 − q and p4 = p2 + q, we obtain [recall that U (q) = U (−q)]
Γ (p1 , p2 |q) = U (q) − U (p1 − p2 − q).
One of the incoming fermions is a right mover (p1 = pF ) and the other one is a
left mover (p2 = −pF ). As q is small compared to pF , we obtain
Γ (p1 , p2 |q) = U (0) − U (2kF ).
In fact, for spinless electrons g2 and g1 processes are indistinguishable25 as we
do not know whether the right-moving electron in the final state is a right-mover
of the initial state, which experienced forward scattering, or the left-mover of the
initial state, which experienced backscattering. A proper way to treat the case of
spinless fermions is to include backscattering into Dzyaloshinskii-Larkin scheme
from the very beginning, re-write the Hamiltonian in terms of forward scattering
¯
with invariant coupling Γ, and proceed with the solution. All the results will then
¯
be expressed in terms of Γ rather than of g2 .
Solving the equation for Γ1 , gives on scale ε
Γ1 =

1
−1

(g1 )

+

Ns
2π

ln Λ/ε

.

(4.5)

At low energies, Γ1 renormalizes to zero (Γ∗ = Γ1 (l = ∞) = 0), if the inter1
action is repulsive, and blows up at ε = Λ exp (−1/|g1|), if the interaction is
attractive. Coupling Γ2 also flows to a new value which can be read off from
Eq. (4.4)
1
Γ∗ = g 2 −
g1 .
2
Ns
Roughly speaking, g1 is not important for repulsive interaction as the effective
low-energy theory will look like a theory with forward scattering only. This does
25 That does not mean that backscattering is unimportant! It comes with a different scattering
amplitude U (2kF ) . In fact, it is only backscattering which guarantees that the Pauli principle is
satisfied, namely, for a contact interaction, when U (0) = U (2kF ) , we must get back to a Fermi
gas as fermions are not allowed to occupy the same position in space and hence they cannot interact
via contact forced. Our invariant combination U (0) − U (2kF ) obviously satisfies this criterion. We
will see that bosonization does have a problem with respecting the Pauli principle, and it takes some
effort to recover it.

Fundamental aspects of electron correlations and quantum transport

51

not really mean, however, that one can consider a fixed point as a new problem
in which backscattering is absent, and apply our exact solution to this problem.
Instead, one should calculate observables, derive the RG equations for flows, and
use current values of coupling constants in these RG equations. An example of
this procedure will be given in the next Section, where we will see that the flow
of Γ1 provides additional renormalization of the transmission coefficient in an
interacting system.
Assigning different coupling constants to the interaction of fermions of parallel (g1|| ) and anti-parallel (g1⊥ ) spins, one could see that the coupling which
diverges for attractive interaction is in fact g1⊥ . This clarifies the nature of the
gap that RG hints at (in fact, a perturbative RG can at most just give a hint): it is a
spin gap. This becomes obvious in the bosonization technique, as the instability
occurs in the spin-sector of the theory. An exact solution by Luther and Emery
[66] for a special case of attractive interaction confirms this prediction.

5. Single impurity in a 1D system: scattering theory for interacting fermions
A single impurity or tunneling barrier placed in a 1D Fermi gas reduces the conductance from its universal value–e2/h per spin orientation–to
G = Ns

e2
|t0 |2 ,
h

(5.1)

where t0 is the transmission amplitude. The interaction renormalizes the bare
transmission amplitude. As a result, the conductance depends on the characteristic energy scale (temperature or applied bias), which is observed as a zerobias anomaly in tunneling. This effect is not really a unique property of 1D
: in higher dimensions, zero-bias anomalies in both dirty and clean (ballistic)
regimes [67, 12, 13, 14] as well as the interaction correction to the conductivity [67, 15], stem from the same physics, namely, scattering of electrons from
Friedel oscillations produced by tunneling barriers or impurities. 1D is special
in the magnitude of the effect: the conductance varies significantly already on
the energy scale comparable to the Fermi energy, whereas in higher dimensions
the effect of the interaction is either small at all energies or becomes significant
only at low energies (below some scale which is much smaller than EF as long
as the parameter kF l, where l is the elastic mean free path, is large. The 1D
zero-bias anomaly is described quite simply in a bosonized language [68], which
does not require the interaction to be weak. We will use this description in Sec.6.
However, in this Section I will choose another description–via the scattering theory for fermions rather than bosons–developed by Matveev, Yue, and Glazman
[11]. Although this approach is perturbative in the interaction, it elucidates the

52

Dmitrii L. Maslov

underlying mechanism of the zero-bias anomaly and allows for an extension to
higher-dimensional case (which was done for the case of tunneling in Ref.[12]
and transport in Ref.[15]).
5.1. First-order interaction correction to the transmission coefficient
In this section we consider a 1D system of spinless fermions with a tunneling
barrier located at x = 0 [11]. For the sake of simplicity, I assume that the barrier
is symmetric, so that transmission and reflection amplitude for the waves coming
from the left and right are the same. Also, I assume that e-e interaction is present
only to the right of the barrier, whereas to the left we have a Fermi gas. Such
a situation models a setup when a tunneling contact separates a 1D interacting
system (quantum wire or carbon nanotube) and a “good metal”, where one can
be neglect the interaction. We also assume that the interaction potential U (x)
is sufficiently short-ranged, so that U (0) is finite and one can neglect over-thebarrier interaction. However, U (0) = U (2kF ) (otherwise, spinless electrons do
not interact at all26 ).
The wave function of the free problem for a right-moving state is:
0
ψk (x)

=
=

1
√ eikx + r0 e−ikx , x < 0;
L
1
√ t0 eikx , x > 0.
L

(5.2)

For a left-moving state:
0
ψ−k (x)

=
=

1
√ e−ikx + r0 eikx , x > 0;
L
1
√ t0 e−ikx , x < 0.
L

(5.3)

√
Here k = 2mE > 0. To begin with, we consider a high barrier: |t0 | ≪ 1, r0 ≈
−1. Then the free wavefunction reduces to
0
ψk (x)

=
=

2i
√ sin kx, x < 0 (incoming from the left+reflected);
L
1
√ t0 eikx , x > 0 (transmitted left → right);
L

(5.4)
(5.5)

26 For a contact potential [which leads to U (0) = U (2k )], the four-fermion interaction for the
F
2
2
spinless case reduces to Ψ† (0) Ψ2 (0) . By Pauli principle, Ψ† (0) = Ψ2 (0) = 0, so that
the interaction is absent.

Fundamental aspects of electron correlations and quantum transport

a)

53

b)
x2

x

x1

x’

x

x1

x2

x’

Fig. 14. Correction to the Green’s function: exact with respect to the barrier and first order in the
interaction.

0
ψ−k (x)

=
=

1
√ t0 e−ikx , x < 0 (transmitted right → left);
(5.6)
L
2i
− √ sin kx, x > 0 (incoming from the right+reflected).(5.7)
L

The barrier causes the Friedel oscillation in the electron density on both sides of
the barrier. The interaction is treated perturbatively, via finding the corrections to
the transmission coefficient due to additional scattering at the potential produced
by the Friedel oscillation. Diagrammatically, the corrections to the Green’s function are described by the diagrams in Fig.14, where a) represents the Hartree and
b) the exchange (Fock) contributions, correspondingly. Compared to the textbook case, though, the solid lines in these diagrams are the Green’s functions
composed of the exact eigenstates in the presence of the barrier (but no interaction). Because the barrier breaks translational invariance, these Green’s functions
are not translationally invariant as well. I emphasized this fact by drawing the diagrams in real space, as opposed to the momentum -space representation. Notice
also that the Hartree diagram is usually discarded in textbooks because the bubble there corresponds to the total charge density (density of electrons minus that
of ions), which is equal to zero in a translationally invariant and neutral system.
However, what we have in our case is the local density of electrons at some distance from the barrier. Friedel oscillation is a relatively short-range phenomenon
(the period of the oscillation is comparable to the electron wavelength), and it is
possible to violate the charge neutrality locally on such a scale. As a result, the
Hartree correction is not zero.
To first-order in the interaction, an equivalent way of solving the problem is
to find a correction to the wave-function, rather than the Green’s function, in
the Hartree-Fock method. The electron wave-function which includes both the

54

Dmitrii L. Maslov

barrier potential and the electron-electron interaction is
ψk (x)

=

∞

0
ψk (x) +

×

∞
0

0

dx′ G> (x,x′ , E)
0

′′

0
dx [VH (x′′ )δ(x′ − x′′ ) + Vex (x′ , x′′ )]ψk (x′′ ) ,

(5.8)

where G> is the Green’s function of free electrons on the right semi-line, E
0
is the full energy of an electron, VH and Vex are the Hartree and the exchange
potentials. The Hartree potential is
dx′ U (x − x′ )δn(x′ ),

VH (x) =

(5.9)

where δn(x) = n(x) − n0 is the deviation of the electron density from its uniform value (in the absence of the potential) and U (x) is the interaction potential.
Hartree interaction is a direct interaction with the modulation of the electron density by the Friedel oscillation. For a high barrier, which is essentially equivalent
to a hard-wall boundary condition, the electron density is
kF

n(x)

=

δn (x)

=

sin 2kF x
dk
sin2 (kx) = n0 1 −
2π
2kF x
0
− sin (2kF x) /2πx,
4

→

(5.10)
(5.11)

where n0 = kF /π is the density of electrons. Then,
VH (x) = −

1
2π

∞
0

dx′ U (x − x′ )

sin 2kF x′
.
x′

(5.12)

Notice that although the bare interaction is short-range, the effective interaction
has a slowly-decaying tail due to the Friedel oscillation. (The integral goes over
only for positive values of x′ because electrons interact only there.)
The exchange potential is equal to
Vex (x, x)

=

−U (x − x′ )
kF

+
0

kF
0

dk 0 ′
ψ (x )
2π k

dk 0
ψ (x′ )
2π −k

∗

∗

0
ψ−k (x) .

0
ψk (x)

(5.13)

Since we assumed that electrons interact only if they are located to the right of
the barrier, the integral in (5.8) runs only over x, x′ > 0 and the Green’s function
is a Green’s function on a semi-line. The wave-function in (5.8) needs to be
evaluated at x → ∞, which means that we will only need an asymptotic form of

Fundamental aspects of electron correlations and quantum transport

55

the Green’s function far away from the barrier. This form is constructed by the
method of images
G> (x, x′ , E) = G0 (x, x′ , E) − G0 (x, −x′ , E),
0

(5.14)

where

1 ik|x−x′ |
e
ivk
√
is the free Green’s function on a line with k = 2mE and vk = k/m. Coordinate x′ is confined to the barrier, whereas x → ∞, thus x > x′ and
G0 (x, x′ , E) =

G> (x, x′ , E) = −
0

2
sin(kx′ )eikx .
vk

5.1.1. Hartree interaction
Our goal is to present the correction to the wave-function for electrons going
from x < 0 to x > 0 in the form
1
0
ψk − ψk = √ δteikx ,
L

(5.15)

where δt is the interaction correction to the transmission coefficient. Substituting
(5.15) into (5.8), we obtain for the Hartree contribution to t
2
δtH
=−
t0
vF

∞

dx sin kxeikx VH (x),

0

where one can replace vk → vF in all non-oscillatory factors. For a deltafunction potential, U (x) = U δ (x)
VH (x) = −

U sin 2kF x
.
2π
x

(5.16)

However, the δ− function potential is not good enough for us, because the Hartree
and exchange contributions cancel each other for this case. Friedel oscillation
arises due to backscattering. With a little more effort, one can show that U in the
last formula is replaced by U (2kF ) :27
VH (x) = −
27 Notice

U (2kF ) sin 2kF x
.
2π
x

that the sign of the Hartree interaction is attractive near the barrier (assuming the sign of
the e-e interaction is repulsive at 2kF ): for x → 0, VH (x) → −U (2kF ) kF /π. The reason is that
the depletion of electron density near the barrier means that the positive background is uncompensated. As a result, electrons are attracted to the barrier and transmission is enhanced by the Hartree
interaction.

56

Dmitrii L. Maslov

Substituting this into δt/t yields
δtH
t0

=
=
=

=
=

U (2kF )
πvF
U (2kF )
πvF
U (2kF )
πvF

∞

dx sin (kx) eikx

0
∞

sin 2kF x
x

sin 2kF x
1
e2ikx − 1 regular correction to Imt
2i
x
0
∞
sin 2kF x
U (2kF )
1
=
dx e2ikx
2i
x
2πvF
0
∞
sin 2kF x
×
dx sin 2kx + i−1 cos 2kx yet another regular correction
x
0
∞
U (2kF )
sin 2kF x
dx sin 2kx
2πvF
x
0
k + kF
kF
U (2kF )
ln
≈ α′ F ln
,
2k
4πvF
|k − kF |
|k − kF |
dx

where
α′ F =
2k

g1
,
4πvF

and g1 = U (2kF ) . In deriving the final result, all terms regular in the limit
k → kF were discarded.
5.1.2. Exchange
Now both x and x′ > 0. We need to select the largest wave-function, i.e., such
that does not involve a small transmitted component. Obviously, this is only
0
possible for k < 0 (second term in (5.13)) and ψ−k , given by (5.7). Substituting
the free wave-functions into the equation for the exchange interaction, we get
Vex (x, x′ ) = −U (x − x′ )ρ(x, x′ ),

(5.17)

where the 1D density-matrix is
kF

ρ(x, x′ ) = 4
0

dk
sin(kx) sin(kx′ )
2π

(5.18)

kF

dkx
[cos k(x − x′ ) − cos k(x + x′ )]
2π
0
sin kF (x + x′ )
= ··· −
,
π(x + x′ )
= 2

(5.19)
(5.20)

Fundamental aspects of electron correlations and quantum transport

57

where . . . stand for the term which depends on x − x′ . This term does not lead to
the log-divergence in δt and will be dropped 28 . For x = x′ , we get the correction
to the density δn (x) , as we should.
Correction to the transmission coefficient
∞

∞

sin kF (x′ + x′′ )
.
x′ + x′′
0
0
(5.21)
After a little manipulation with trigonometric functions, which involves dropping
of the terms depending only on x − x′ , we arrive at
δtex /t0 = −

δtex
t0

2
πvF

dx′

dx′′ U (x′ − x′′ ) sin kx′ eikx

′′

∞
+∞
1
dx+
dq
U (q)
2v
4π F 0
q
x+
0
×{sin 2(k − kF + q)x+ − sin 2(k − kF − q)x+ },

= −

where

x′ + x′′
.
2
provides a lower cut-off for the q− integral
x+ =

Integral over x+

(5.22)
(5.23)

(5.24)

+∞

=
=

dx+
{sin 2(k − kF + q)x+ − sin 2(k − kF − q)x+ }
x+
0
π
π
sgn(q + k − kF ) + sgn(q − k + kF )
2
2
πθ(q − |k − kF |).

Now
1
δtex
=−
t0
4πvF

+∞
|k−kF |

dq
U (q).
q

(5.25)
(5.26)
(5.27)

(5.28)

As U (q) is regular at q → 0 29 , one can take U (q) out of the integral at q = 0
(denoting U (0) = g2 )
δtex
1
≈−
g2
t0
4πvF
28 Notice

q0
|k−kF |

q0
dq
= −α′ ln
.
0
q
|k − kF |

that the important part of the exchange potential is repulsive near the barrier. This means
that electrons are repelled from the barrier and transmission is suppressed.
29 If U (q) has a strong dependence on q for q → 0 (which is the case for a bare Coulomb potential
U (q) ∝ ln q), this dependence affects the resulting dependence of the transmission coefficient on
energy |k − kF | , i.e., on the temperature and/or bias. Instead of a familiar power-law scaling of the
tunneling conductance for the short-range interaction, the conductance falls off with energy faster
than any power law for the bare Coulomb potential.

58

Dmitrii L. Maslov

Combining the exchange and Hartree corrections together (in doing so, we choose
the smallest upper cut-off for the log which we assume to be the inverse interaction range, q0 )) we get
q0
,
(5.29)
δt = −t0 α′ ln
|k − kF |
where

α′ = α′ − α′ F =
0
2k

g2 − g1
; asymmetric geometry.
4πvF

(5.30)

It can be shown in a similar manner that if we had interacting regions on both
sides of the barrier, the result for α′ would be double of that in Eq. (5.30).
α′ = α′ − α′ F =
0
2k

g2 − g1
; symmetric geometry.
2πvF

(5.31)

The sign of the correction to t depends on the sign of g2 −g1 = U (0)−U (2kF ) .
Notice that transmission is enhanced, if U (2kF ) > U (0). Usually, this behavior
is associated with attraction. We see, however, that even if the interaction is
repulsive at all q but U (2kF ) > U (0), it works effectively as an attraction.
The case U (2kF ) > U (0) is not a very realistic one, at least not in a situation
when electrons interact only among themselves. Other degrees of freedom, e.g.,
phonons, must be involved to give a preference to 2kF − scattering.
5.2. Renormalization group
It is tempting to think that the first-order in interaction correction to t0 in Eq. (5.30)
′
is just an expansion of the scaling form t ∝ |k − kF |α . A poor-man RG indeed
shows that this is the case. Near the Fermi level, k − kF = (E − EF ) /vF =
ε/vF so that the first-order correction to t is
t1 = t0 1 − α′ ln

W0
|ε|

,

where W0 = q0 vF is the effective bandwidth. The meaning of this bandwidth
is that the states at ±W0 from the Fermi level (=0) are not affected by the interaction. For |ε| = W0 , t1 = t0 . Suppose that we want to reduce to bandwidth
W0 → W1 < W0 and find t at |ε| = W1
t1 = t0 1 − α′ ln

W0
W1

.

It is of crucial importance here that coefficient α′ (which will become the tunneling exponent in the scaling form we are about to get) is proportional to the
RG-invariant combination U (0) − U (2kF ) = g2 − g1 for spinless electrons.

Fundamental aspects of electron correlations and quantum transport

59

This means that α′ is to be treated as a constant under the RG flow. Repeating
this procedure using t, found at the previous stage instead of a bare t0 , n times,
we get
Wn
.
tn+1 = tn 1 − α′ ln
Wn+1
The renormalization process is to be stopped when the bandwidth coincides with
the physical energy |ε| , at which t is measured. In the continuum limit (tn+1 −
tn = dt; Wn+1 = Wn − dW ), this equation reduces to a differential one
dW
dt
= α′
t
W
Integrating from t (ε) to t0 (and, correspondingly, from W = |ε| to W = W0 ),
we obtain
α′
t (ε) = t0 (|ε| /W0 ) .
5.3. Electrons with spins
Now let’s introduce the spin. The effect will be more interesting than just multiplying the result for the tunneling conductance by a factor of two (which is
all what happens for non-interacting electrons.) To keep things general, I will
assume an arbitrary “spin” (which may involve other degrees of freedom) degeneracy Ns and put Ns = 2 at the end. In this section we will exploit the result of
Sec.4 stating the backscattering amplitude flows under RG. This flow affects the
renormalization of the transmission coefficient at low energies.
Repeating the steps for the first-order correction to t for the case of electrons
with spin is straightforward: one just has to recall that the Hartree correction is
multiplied by Ns (as the polarization bubble involves summation over all isospin
components, it is simply multiplied by a factor of Ns ). On the contrary, the exchange interaction is possible only between electrons of the same spin, so there
are Ns identical exchange potentials for every spin component. I am going to
discuss the strong barrier case first in the symmetric geometry. Then, taking into
account what we have just said about the factor of Ns , we can replace the result
for spinless electrons (5.30) by
α′ → α′ = α′ − Ns α′ F =
0
2k

g2 − Ns g1
.
4πvF

(5.32)

(and similarly for the symmetric geometry of the tunneling experiment). Correspondingly, the correction to the transmission coefficient (for a given spin projection) changes to
tσ = t0 (1 − α′ ln W/ |ε|) .

60

Dmitrii L. Maslov

The tunneling conductance is found from the Landauer formula
G=

e2
h

Ns
2

σ=1

|tσ | ,

where, as the barrier is spin-invariant, the sum simply amounts to multiplying the
result for a given spin component by Ns . Now, the result in Eq. (5.32) seems to be
interesting, as the 2kF contribution gets a boost. If Ns U (2kF ) > U (0), we have
in increase of the barrier transparency. It does not seem too hard to satisfy this
condition. For example, it is satisfied already for the delta-function potential 30
and Ns = 2. However, as opposed to the spinless case, α′ is not an RG-invariant
but flows under renormalizations. Let’s split α′ into an RG-invariant part (4.4)
and the rest
α′

=
=

2
1
1 Ns − 1
1
U (0) −
U (2kF ) −
U (2kF )
4πvF
Ns
4πvF Ns
2
1 Ns − 1
g1 ,
α′ −
s
4πvF Ns

where
α′ =
s

1
4πvF

g2 −

1
g1
Ns

=

¯
Γ
.
4πvF

(5.33)

The condition for the tunneling exponent to be negative is more restrictive that it
seemed to be: g1 > Ns g2 . It is not hard to see that the RG equation for tσ now
changes to
2
dt
1 Ns − 1
Γ1 (l) ,
(5.34)
= −t α′ −
s
dl
4πvF Ns
where Γ1 (l) is given by
Γ1 =

1
(g1 )

−1

+

Ns
2π l

.

Integrating (5.34), we find
tσ = t0 1 +

Ns g1
W
ln
4πvF
|ε|

where
βs =
30 as

βs

α′
s

(|ε| /W )

,

2
Ns − 1
.
2
Ns

now electrons have spins, they are allowed to be at the same point in space and interact.

Fundamental aspects of electron correlations and quantum transport

61

In particular, for Ns = 2, we get
W
g1
ln
2πvF
|ε|

3/4

tσ = t0 1 +

g1
W
ln
2πvF
|ε|

3/2

G = G0 1 +

α′
s

(|ε| /W )

.

and conductance
′

(|ε| /W )2αs ,

(5.35)

where G0 is the conductance for the free case. Thus the flow of the backscattering amplitude results in a multiplicative log-renormalization of the transmission
coefficient. One can check the first-order result is reproduced if expand the RG
result to the first log.
An interesting feature of this result is that it predicts three possible types of
behavior of the conductance as function of energy.
1. weak backscattering:
α′ > 0 → g1 < g2 /Ns .

In this regime already the first-order correction corresponds to suppression of
the conductance, which decreases monotonically as the energy goes down.
2. intermediate backscattering:
α′ > 0 but α′ < 0 → g2 /Ns < g1 < Ns g2 .
s
In this regime, the first-order correction enhances the transparency, but the RG
result shows that the for ε → 0, the transmission goes to zero. It means that
at higher energies, when the RG has not set in yet, the conductance increases
as the energy goes down, but at lower energies the conductance decreases.
The dependence of G (ε) on ε is non-monotonic–there is a maximum at the
intermediate energies.
3. strong backscattering:
α′ < 0 → g1 > Ns g2 .
s

In this regime, tunneling exponent α′ is negative and the conductance ins
creases as the energy goes down.
5.4. Comparison of bulk and edge tunneling exponents
Tunneling into the bulk of a 1D system is described by the density of states obtained, e.g., in the DL solution of the Tomonaga-Luttinger model (no backscattering). The “bulk” tunneling exponent is equal to
2

2α =

(u − 1)
,
4u

62

Dmitrii L. Maslov

where
u=

1 + 2g2 /πvF .

In this Section, we considered tunneling into the edge for weak interaction and
found that the conductance scales with exponent 2α′ (5.33). To compare the two
s
exponents, we need to expand the DL exponent for weak interaction
2α =

2
g2
.
4π 2 vF

For g1 = 0, the edge exponent is
2α′ =
s

1
g2 .
2πvF

We see that for weak coupling tunneling into the edge is stronger affected by the
interaction than tunneling into the bulk: the former effect starts at the first order
in the interaction whereas the latter starts at the second order. This difference has
a simple physical reason which is general for all dimensions. In a translationally
invariant system, the shape of the Green’s function is modified in a non-trivial
way only starting at the second order. For example, the imaginary part of the
self-energy (decay of quasi-particles) occur only at the second order. The firstorder corrections lead only to a shift in the chemical potential and, if the potential
is of a finite-range, to a renormalization of the effective mass. If the translational
invariance is broken, non-trivial changes in the Green’s function occur already
at the first order in the interaction. That tunneling into the bulk and edge are
characterized by different exponents is also true in the strong-coupling case (cf.
Sec.6). As the relation between the bulk and edge exponents is known for an
arbitrary coupling, one can eliminate the unknown strength of interaction and
express one exponent via the other. Knowing one exponent from the experiment,
one can check if the observed value of the second exponent agrees with the data.
This cross-check was cleverly used in the interpretation of the experiments on
single-wall carbon nanotubes [71, 72].

6. Bosonization solution
Bosonization procedure in described in a number of books and reviews [1]-[10].
Without repeating all standard manipulations, I will only emphasize the main
steps in this Section, focusing on a couple of subtle points not usually discussed
in the literature. A reader familiar with bosonization may safely skip the first
part of this Section, and go directly to Secs. 6.1.5 and 6.2.1, where tunneling
exponents are calculated. Some technical details of the bosonization procedure

Fundamental aspects of electron correlations and quantum transport

63

are presented in Appendix Appendix C. As in Sec.3, vF = 1 in this Section,
unless specified otherwise.
6.1. Spinless fermions
6.1.1. Bosonized Hamiltonian
We start from a Hamiltonian of interacting fermions without spin
H=

1
L

ξk a† ak +
k
p,k,q

1
2L2

Vq a† a† ak ap .
p−q k+q
p,k,q

The interacting part of the Hamiltonian can be re-written using chiral densities
ρ± (q) =

a†
p−q/2 ap+q/2
p≷0

as
Hint = g2

1
L

q

ρ+ (q) ρ− (−q) +

g4 1
2 L

q

ρ+ (q) ρ+ (−q) + ρ− (q) ρ− (−q) .

The interacting part is in the already bosonized form. For a linearized dispersion
ξk = |k| − kF ,
it is also possible to express the free part via densities. To check this, let’s assume
that H0 can indeed be written as
H0 =

1
L

q

Aq [ρ+ (q) ρ+ (−q) + ρ− (q) ρ− (−q)] ,

where Aq is some unknown function. Commuting ρ+ with H0 , and making use
of the anomalous commutator [ρ+ (q) , ρ+ (−q)] = qL/2π, we obtain
[ρ+ (q) , H0 ] =
=

1
L

Aq′ [ρ+ (q) , ρ+ (q ′ ) ρ+ (−q ′ )]
q′

Aq ρ+ (q)

q
.
π

On the other hand, the same commutator can be calculated directly in a model
with the linearized spectrum, using only the fermionic anticommutation relations
[69]. This gives
[ρ+ (q) , H0 ] = qρ+ (q) .

64

Dmitrii L. Maslov

Comparing the two results, we see that
Aq = π
and thus
H0 = π

1
L

q

ρ+ (q) ρ+ (−q) + ρ− (q) ρ− (−q) .

Combining the free and interacting parts of the Hamiltonian, we obtain
H=π

g4
1
1+
L
2π

q

1
{ρ+ (q) ρ+ (−q) + ρ− (q) ρ− (−q)}+π g2
L

q

ρ+ (q) ρ− (−q) .

Notice that if only the g4 − interaction is present, the system remains free but the
Fermi velocity changes.
It is convenient to expand the density operators over the normal modes
ρ+ (x)

=
q>0

ρ− (x)

=
q<0

|q|
bq eiqx + b† e−iqx ;
q
2πL
|q|
bq eiqx + b† e−iqx .
q
2πL

One can readily make sure that density operators defined in this way reproduce
the correct commutation relations, given that bq , b†′ = δq,q′ . In terms of these
q
operators, the Hamiltonian reduces to
H =π 1+

g4
2π

q>0

q b† bq + b† b−q + πg2
q
−q

q

q b† b† + bq b−q .
q −q

Introducing new bosons via a Bogoliubov transformation
c†
q
c†
−q

= cosh θq bq + sinh θq b† ;
−q
= cosh θq b† + sinh θq bq
−q

and choosing θq so that the Hamiltonian becomes diagonal, i.e.,
tanh 2θq =
we obtain
H=

1
L

g2 /2π
,
1 + g4 /2π
ω q c† cq ,
q
q

Fundamental aspects of electron correlations and quantum transport

65

where
ωq = u |q| ,
and 31
u=

1+

g4
2π

2

−

2 1/2

g2
2π

.

(6.2)

For the spinless case, backscattering can be absorbed into forward scattering.
The resulting expression for the renormalized velocity for the case g2 = g4 = g1
is (cf. Appendix Appendix C.3 )
u=

1+

g2 − g1
.
2π

6.1.2. Bosonization of fermionic operators
The Ψ− operators of right/left movers can be represented as
1
e±2πi
Ψ± (x) = √
2πa

x
−∞

ρ± (x′ )dx′

,

(6.3)

where a is the ultraviolet cut-off in real space. Using the commutation relations
for ρ± , one can show that the (anti) commutation relations
Ψ± (x) , Ψ† (x′ ) = δ (x − x′ )
±
are satisfied.
The argument of the exponential can be re-written as two bosonic fields
Ψ± (x)
ϕ± (x)

√
1
e±i πϕ± (x) ;
2πa
= ϕ (x) ∓ ϑ (x) .

=

√

(6.4)

31 Let’s now introduce backscattering. Since for spinless fermions backscattering is just an exchange process to forward scattering of fermions of opposite chirality (think of a diagram where
you have right and left lines coming in and then interchange them at the exit), the only effect of
backscattering is to replace g2 → g2 − g1 . (cf. discussion in Sec. 4). Instead of (6.2), we then have

u=

1+

g4
2π

2

−

g2 − g1
2π

2 1/2

.

(6.1)

Now, consider a delta-function interaction, when g1 = g2 = g4 . Pauli principle says that we
should get back to the Fermi gas in this case. However, u still differs from unity (Fermi velocity)
and thus our result violates the Pauli principle. See Appendix Appendix C.3 for a resolution of the
paradox.

66

Dmitrii L. Maslov

Equating exponents in (6.3) and (6.4), we obtain
x

√
π [ϕ (x) − ϑ (x)] =

2π

=

2π

√
π [ϕ (x) + ϑ (x)] =

q>0
x

2π
−∞

=

dx′ ρ+ (x′ )

(6.5)

−∞

|q| 1
bq eiqx − b† e−iqx ;
q
2πL iq
dx′ ρ− (x′ )

(6.6)

|q| 1
bq eiqx − b† e−iqx .
q
2πL iq

2π
q<0

Solving for ϕ (x) and ϑ (x) gives
ϕ (x) =
ϑ (x) =

−i

1
−∞<q<∞

i
−∞<q<∞

sgnq bq eiqx − b† e−iqx ;
q

(6.7)

eiqx bq − b† e−iqx .
q

(6.8)

2 |q| L

1

2 |q| L

Using (6.7) and (6.8), one can prove that ϕ (x) and ∂x ϑ (x) satisfy canonical
commutation relations between coordinate and momentum (cf. Appenidx Appendix C.2.1)
(6.9)
[ϕ (x) , ∂x′ ϑ (x′ )] = iδ (x − x′ ) .
Using Eqs. (6.5) and (6.6), we obtain the density and current as the gradients of
the bosonic fields
x
x
√
√
π
dx′ (ρ+ (x′ ) + ρ− (x′ )) = π
dx′ ρ (x′ ) →
ϕ (x) =
−∞

ρ (x) =
ϑ (x) =

−∞

1
√ ∂x ϕ;
π
x
√
√
dx′ (ρ+ (x′ ) + ρ− (x′ )) = − π
− π
−∞

j (x) =

1
− √ ∂x ϑ (x) .
π

The continuity equation,
∂t ρ + ∂x j = 0
relates the Heisenberg fields ϕ (x, t) and ϑ (x, t)
∂t ϕ = ∂x ϑ.

x
−∞

dx′ j (x′ ) →

Fundamental aspects of electron correlations and quantum transport

67

The current can be also found as
1
j (x, t) = − √ ∂t ϕ (x, t) .
π
We will use this relation later. Expressing H in the real space via ρ±
dx ρ2 + ρ2 +
+
−

H=π

g4
2

dx ρ2 + ρ2 + g2
+
−

dxρ+ ρ−

and using the relations
1
ρ± = √ (∂x ϕ ∓ ∂x ϑ) ,
2 π
we obtain a canonical form of H in terms of the bosonic fields
H

=
=

1
2
1
2

2

dx (∂x ϕ) + (∂x ϑ)

2

+

g2 + g4
4π

2

dx (∂x ϕ) +

g2 − g4
4π

u
2
2
(∂x ϕ) + uK (∂x ϑ) ,
K

dx

(6.10)

where 32
1+

u=

g4
2π

2

−

g2
2π

For g4 = g2 ≡ g, we have
u=

1 + g/π; K =

2

; K=

1
.
1 + g/π

1+
1+

g4 −g2
2π
g4 +g2
2π

.

(6.11)

(6.12)

Notice that in this case uK = 1. This is important: in the next Section, we will
see that this product renormalizes the Drude weight (and the persistent current).
Neither of these quantities are supposed to be affected by the interactions, as the
Galilean invariance remains intact. We see that it is indeed the case in our model.
If backscattering is present (but g2 = g4 ), the parameters change to (cf. Appendix Appendix C.3)
u

=

K

=

1+

g2 − g1
;
π
1

1 + (g2 − g1 ) /π

(6.13)
.

2

dx (∂x ϑ)

(6.14)

32 As we have already seen in Sec.3, a difference between g and g leads to the current-current
2
4
interaction in the Hamiltonian. In the bosonized form, this interaction is the (g2 − g4 ) (∂x ϑ)2 term
in the first line of Eq. (6.10).

68

Dmitrii L. Maslov

Had we started with another microscopic model, e.g., with fermions on a lattice
but away from half-filling, the effective low-energy theory would have also been
described by Hamiltonian (6.10) albeit with different–and, in general, unknown–
parameters u and K. The term “Luttinger liquid” (LL) [70] refers to a universal
Hamiltonian of type (6.10), which describes the low-energy properties of many
seemingly different systems. In that sense, the LL is a 1D analog of higherdimensional Fermi liquids, which describe the low-energy properties of a large
class of fermionic systems, while encoding the quantitative differences in their
high-energy properties by a relatively small set of parameters.
6.1.3. Attractive interaction
What happens for the case of an attractive interaction, g < 0? Formally, for
g < −π (or g2 − g1 < π), u2 in Eqs.(6.12,6.13) is negative, which seems to
suggest some kind of an instability. Actually, this is not the case [59], as a 1D
system of spinless fermions does not have any phase transitions even at T = 0.
All it means that the interacting system is a liquid rather than a gas, i.e., it does
not require external pressure to mantain its volume. An equilibrium value of the
density is fixed by given ambient pressure. To see this, restore the Fermi velocity
vF = πn/m, where n is the density
2
u2 = vF

1+

g
πvF

=

πn
m

2

+

ng
m

(6.15)

and recall the thermodynamic relation
u2 = m−1 ∂P/∂n,

(6.16)

where P is the pressure. Integrating (6.16) with the boundary condition P (n = 0) =
0, we obtain the constituency relation
P =

2

π
m

n3
g 2
+
n .
3
2m

For g < 0, there is a metastable region of negative pressure. This means that if
the ambient pressure is equal to zero, the thermodynamically stable value of the
density is given by the non-zero root of the equation P (n) = 0:
n∗ =

3
|g| m.
2π 2

The square of the sound velocity at this density is positive:
2

(u∗ ) =

3 2
g .
4π 2

Fundamental aspects of electron correlations and quantum transport

69

The Fermi velocity at n = n∗ is
∗
vF = πn∗ /m =

and parameter K

3
|g|
2π 2

√
∗
vF
3
K= ∗ =
u
π

is a universal number, independent of the interaction.
6.1.4. Lagrangian formulation
In what follows, it will be more convenient to work in the Lagrangian rather the
Hamiltonian formulation (and also in complex time). A switch from the Hamiltonian to Lagrangian formulation is done via the usual canonical transformation
S=

dx

dt (qp − H (p, q)) ,
˙

(6.17)

where H is the Hamiltonian density defined such that
H=

dxH

and q and p are the canonical coordinate and momentum, correspondingly. According to commutation relation (6.9),
q = ϕ p = ∂x ϑ.

(6.18)

Performing a Wick rotation, t → −iτ, we reduce the quantum-mechanical problem into a statistical-mechanics one with the partition function
Z=

Dϕ

Dϑe−SE ,

where the Euclidian action
SE =

dτ

dx

1
1 u
2
2
(∂x ϕ) + uK (∂x ϑ) − i∂τ ϕ∂x ϑ .
2K
2

In a Fourier-transformed form
SE =

d2 k

1 u 2
1
q ϕk ϕ−k + uKϑk ϑ−k + iqωϕk ϑ−k ,
2K
2

70

Dmitrii L. Maslov

where k ≡ (q, ω). If one needs only an average composed of fields of one type
(ϕ or ϑ), then the other field can be integrated out. This leads to two equivalent
forms of the action
Sϕ

=
=

Sϑ

=
=

1
2K
1
2K
K
2
K
2

1 2
ω + uq 2 ϕk ϕ−k
u
1
2
2
(∂τ ϕ) + (∂x ϕ) ;
dx dτ
u
1 2
ω + uq 2 ϑk ϑ−k
d2 k
u
1
2
2
(∂τ ϑ) + (∂x ϑ) .
dx dτ
u
d2 k

(6.19a)
(6.19b)
(6.19c)
(6.19d)

In calculating certain correlation functions, e.g., the fermionic Green’s function, one also needs a cross-correlator ϕϑ . This one is computed by keeping
both ϕ and ϑ in the action.
It is convenient to re-write the action in the matrix form
1
SE =
d2 k η † D−1 ηk ,
ˆ ˆ ˆ
k
2
where

ϕk
ϑk

ηk =
ˆ
and the inverse matrix of propagators
ˆ
D−1 =

q 2 uK
iqω

iqω
u
q2 K

.

Inverting the matrix, we obtain
ˆ
D=

1
u2 q 2 + ω 2

uK
−iω/q

−iω/q
u
K

.

The space-time propagators can be found by performing the Fourier transforms
ˆ
of D. For diagonal terms, one really does not need to do it, as it is obvious from
(6.19a) and (6.19c) that these propagators just coincide with the Green’s function
of a 2D Laplace’s equations. Recalling that the potential of a line charge is a
log-function of the distance, we obtain
Φ (z) =
Θ (z) =

a2
K
,
ln
4π x2 + (u |τ | + a)2
a2
1
ln
ϑ (z) ϑ (0) − ϑ2 (0) =
4πK x2 + (u |τ | + a)2

ϕ (z) ϕ (0) − ϕ2 (0) =

Fundamental aspects of electron correlations and quantum transport

71

where a is “lattice constant”, z ≡ (x, τ ) , and x2 + u2 τ 2 ≫ a2 . 33 These are
the two correlation functions we will need the most. In addition, there is also an
off-diagonal propagator
Ξ (z) =

ϕ (z) ϑ (0) − ϕ (0) ϑ (0) = ϑ (z) ϕ (0) − ϑ (0) ϕ (0)
d2 k eik·z − 1

=

ϕk ϑ−k = −i

d2 k eik·z − 1

ω/q
.
u2 q 2 + ω 2

Ξ (z) depends only on the ratio x/uτ and thus does not change the powercounting. To see this, introduce polar coordinates q = k cos α/u, ω = sin α, x =
(z/u) cos β, and τ = z sin β. Then
Ξ (x, τ )

=

d2 k

−i

=

−i

ei(qx−ωτ ) − 1

2

(2π)

∞

1
2

(2π)

0

dk
k

2π
0

ω/q
u2 q 2 + ω 2

dα eik cos(α+β) − 1 tan α.

The resulting integral is a function of only β = tan−1 (x/uτ ).
6.1.5. Correlation functions
Now we can calculate various correlation functions, including the Green’s function.
Non-time-ordered Green’s function for right movers:
G+ (x, τ )

=
=

†
B
− Tτ ψ+ (x, τ ) ψ+ (0, 0)
√
√
1
B
Tτ ei π(ϕ(1)−ϑ(1)) e−i π(ϕ(0)−ϑ(0)) ,
2πa

(6.20)

B
where (1) ≡ (x, τ ) and (0) ≡ (x = 0, τ = 0) , and where Tτ is a bosonic timeordering operator. 34 I will use the well-known result, valid for an average of the
product of the exponentials of gaussian fields (see the books by Tsvelik [6] or
Giamarchi [10] for a derivation)

Tτ Πj eiAj γ(zj ) = δ

j

Aj ,0

× e−

k>j

Aj Ak Tτ γ(zj )γ(zk )

1

e− 2

k

A2 γ 2 (zj )
k

.
(6.21)

33 A non-symmetric appearance of the cut-off with respect to time and space coordinates reflect an
asymmetric way the sums over bosonic momenta and frequencies were cut. We adopted a standard
procedure in which the sum of over q is regularized by exp (− |k| a) , whereas the frequency sum is
unlimited. Other choices of regularization are possible.
34 Surely, it is not a conventional definition of the Green’s function, but it is easier to work with this
one for now, and restore the fermionic Tτ product at the end.

72

Dmitrii L. Maslov

[Eq. (6.21) is essentially a field-theoretical analog of the probability theory result
for the average of eiAγ , where γ is a Gaussian random variable.] For example, in
the average
√
√
Av (z) = Tτ ei πϕ(z) e−i πϕ(0)
√
√
A1 = π, A2 = − π and
Av (z) = eπ

Tτ [ϕ(z)γ(0)−ϕ2 (0)]

1/4K

a2

=

x2 + (u |τ | + a)

2

.

Similarly, with the help of (6.21), Eq. (6.20) reduces to
G+ (x, τ )

=
=

1 π ϕ(1)ϕ(0)−ϕ2 (0) τ π ϑ(1)ϑ(0)−ϑ2 (0) τ −2
e
e
e
2πa
1 πΦ(x,τ ) πΘ(x,τ ) −2Ξ(x,τ )
e
e
e
2πa

ϕ(1)ϑ(0)−ϕ(0)ϑ(0)

(6.22)

K+K −1

=

1
2πa

4

a2

eif (x/uτ ) ,

x2 + (u |τ | + a)2

where . . . τ stands for a time-ordered product and where it was used that in a
translationally invariant and equilibrium system ϕ2 (0) = ϕ2 (1) (same for
ϑ). Function f (x/uτ ) is a phase factor which does not effect the power-counting.
Bulk tunneling DoS For x = 0,
G (0, τ ) ∝ τ −

K+K −1
2

.

By power-counting,
ν (ε) ∝ |ε|

K+K −1
2

−1

= |ε|

(K−1)2
2K

.

(6.23)

This is an analog of the DL result for the spinless case.
Edge tunneling DoS In a tunneling experiment, one effectively measures the
local DoS at the sample’s surface. In a correlated electron system, the boundary
condition affects the wavefunction over a long (exceeding the electron wavelength) distance from the surface. Therefore, the surface DoS differs significantly from the “bulk” one. If a tunneling barrier is high, then–to leading order
in transmission– the DoS can be found via imposing a hard-wall boundary condition. The presence of the surface (boundary) can be taken into account by
imposing the boundary conditions on the number current
1
j (x = 0, τ ) = − √ ∂τ ϕ = 0.
i π

τ

(6.24)

Fundamental aspects of electron correlations and quantum transport

73

at x = 0. This means that ϕ is pinned at the boundary, i.e., it takes some timeindependent value. In the gradient-invariant theory, we can always choose this
constant to be zero. Thus,
ϕ (0, τ ) = 0.
This suggests that the local correlator Φ(0, τ ) = 0, and the long-time behavior
of the Green’s function in Eq. (6.22) is determined only by the correlator of the
ϑ fields. Had the boundary not affected this correlator, we would have arrived at
1
.
(wrong)
G (x = x′ = 0, τ ) ∝ exp (πΘ (x = 0, τ )) ∝
1/2K
|τ |

But then we have a problem, as Eq. (wrong) does not reproduce the free-fermion
1
behavior for K = 1. Consequently, the DoS at the edge νe (ε) ∝ |ε| 2K −1 would
have not reproduced the free behavior either. What went wrong is that we pinned
one field but forgot the other one is canonical conjugate to the first one. By
the uncertainty principle, fixing the “coordinate” (ϕ) increases the uncertainty
in the “momentum” (ϑ)–and vice versa. Thus, fluctuations of ϑ fields should
increase. A rigorous solution to this problem is to change the fermionic basis
from the plane waves to the solutions of the Schrodinger equation with the hardwall boundary condition and to bosonize in this basis. This was done by Eggert
and Affleck [75] and Fabrizio and Gogolin [76] , [9]. Here I will give an heuristic
argument based on a simple image construction, which leads to the same result.
Eq. (6.24) translates into the boundary conditions for the bosonic propagators:
Φe (x, x′ , τ ) = 0; ∂x,x′ Θe (x, x′ , ω) = 0,

(6.25)

′

for x, x = 0, where subindex e denotes the correlators in a semi-infinite system.
Since Φe and Θe satisfy the Laplace’s equation, we can view these propagators
as potentials produced by some fictitious charges. Then, Φe and Θe can be constructed from the propagators of an infinite sample by the method of images:
Φe (x, x′ , τ )
Θe (x, x′ , τ )
For x = x′ ,

= Φ(x − x′ , τ ) − Φ(x + x′ , τ );
= Θ (x − x′ , τ ) + Θ (x + x′ , τ ) .

Φe (0, 0, τ ) = 0; Θe (0, 0, τ ) = 2Θ (0, τ ) .

(6.26)

Hence, pinning the ϕ field enhances the rms fluctuations of the ϑ field by a factor
of two. This leads us to
G+ (0, 0, τ ) ∝ exp (πΦe (0, 0, τ )) exp (πΘe (0, 0, τ ))
a
2π
ln
= exp (2πΘe (0, τ )) ∝ exp
2πK |τ |

∝ |τ |

−1/K

.

74

Dmitrii L. Maslov

Consequently, the DOS becomes
K −1 −1

νe (ε) ∝ |ε|

.

(6.27)

This result by Kane and Fisher [68] initiated the new (and still continuing) surge
of interest to 1D systems (in terms of the impurity scattering time, this result was
obtained earlier in Refs. [73, 74]). For tunneling from a contact with energyindependent DoS (“Fermi liquid”) into a 1D system, the tunneling conductance
scales as νe (ε)
K −1 −1

G(ε) ∝ νe (ε) ∝ |ε|

.

Now we see that the free-fermion behavior is correctly reproduced for K = 1.
Expanding the tunneling exponent K −1 −1 with parameter K from Eq. (6.14)
for the weak-coupling case gives
K −1 − 1 ≈

g2 − g1
.
2πvF

This is the same result as the weak-coupling tunneling exponent (5.30) obtained
in Sec.5 via the scattering theory for interacting fermions.
Where do the “bulk" and “edge" forms of DoS match? Consider an object
G(x = x′ , ε). At the boundary, the DoS is of the “edge” form (6.27). Far
away from the boundary, the Green’s function does not depend on x and ν(ε)
acquires a “bulk" form (6.23). As a function of x, G(x = x′ , ε) varies on the
scale ≃ u/|ε| and the crossover between two limiting forms of ν occurs on this
scale. Choosing the energy in a tunneling experiment, i.e., temperature or bias–
whichever is larger, determines how far from the boundary one should go in order
to see a change in the scaling behavior.

6.2. Fermions with spin
For fermions with spin, each component of the fermionic operator is bosonized
separately
√
1
exp ±i π (ϕσ ∓ ϑσ ) .
ψ±,σ = √
2πa
Index σ of the bosonic field does not mean that bosons acquired spin. We simply
have more bosonic fields. Charge and spin densities and currents are related to

Fundamental aspects of electron correlations and quantum transport

75

the derivatives of the bosonic fields
1
1
√ (ϕ′ ∓ ϑ′ ) ρσ = ρ+,σ + ρ−,σ = √ ϕ′ ;
ρ±,σ =
σ
2 π σ
π σ
1
jσ = ρ+,σ − ρ−,σ = √ ϑ′ ;
π σ
ρc

1
= ρ↑ + ρ↓ = √ ϕ′ + ϕ′ =
↓
π ↑

2 ′
ϕ
π c

ρs

1
= ρ↑ − ρ↓ = √ ϕ′ − ϕ′ =
↓
π ↑

2 ′
ϕ ;
π s

jc

1
= j↑ + j↓ = √ ϑ′ + ϑ′ =
↓
π ↑

2 ′
ϑ ;
π c

js

1
= j↑ − j↓ = √ ϑ′ − ϑ′ =
↓
π ↑

2 ′
ϑ ,
π s

where ′ denotes ∂x and where the charge and spin bosons are defined as
ϕc,s =

ϕ↑ ± ϕ↓
ϑ↑ ± ϑ↓
√
; ϑc,s = √
.
2
2

(6.28)

I assume that the interaction is spin-invariant, i.e., couplings of ↑↑ and ↑↓ fermions
are the same. Substituting the relations between charge- and spin-densities into
the Hamiltonian, one arrives at the familiar bosonized Hamiltonian which consists of totally independent charge and spin parts
H
Hc
Hs

= Hc + Hs ;
1
uc
(∂x φc )2 + uc Kc (∂x θc )2 ;
=
dx
2
Kc
us
1
2
2
(∂x φs ) + us Ks (∂x θs )
dx
=
2
Ks
√
2g1
+
8πφσ .
dx cos
2
(2πa)

(6.29)

Parameters of the Gaussian parts are related to the microscopic parameters of the
original Hamiltonian

uc

=

1+

us

=

1−

g1
2π
g1
2π

1/2

2

1+

4g2 − g1
2π

1/2

; Ks =

1/2

; Kc =

1 + g1 /2π
1 − g1 /2π

1/2

1 + g1 /2π
1 + (4g2 − g1 ) /2π
.

1/2

;

76

Dmitrii L. Maslov

Notice that Kc < 1 for g1 < 2g2 (“repulsion”) and Kc > 1 for g1 > 2g2 (“attraction”). The boundaries for “repulsive” and “attractive” behaviors coincide with
those obtained when studying tunneling of interacting electrons. The velocity of
the charge part for g1 = 0 coincides with that found in the DL solution (Sec. 3)
uc =

1+

2g2
π

1/2

.

Scaling dimension of the backscattering term in the spin part can be read off
from the correlation function
1
1 i√8πφσ (z) −i√8πφs
= 4 exp
e
e
a4
a

8πKs a2
ln 2
4π
z

=

1
a4

4Ks

a
|z|

∝ a4(Ks −1) .

If we allowed for different coupling constants between electrons of different spin
orientations, then the coefficient in front of the cos term would have been g1⊥ .
For Ks > 1, the operator scales down to zero as a → 0, whereas for Ks < 1, it
blows up signaling an instability: a spin-gap phase.
The RG-flow of the spin-part is described by the Berezinskii-Kosterlitz-Thouless
∗
∗
phase diagram. The fixed-point value of g1 = 0 for Ks > 1. In the weak coupling limit, the RG reduces to a single equation for g1 , which we have derived in
the fermionic language in Sec.4
dg1
1
2
= −g1 → g1 =
,
0 )−1 + l
dl
(g1

(6.30)

6.2.1. Tunneling density of states
The procedure of finding the scaling behavior for the DoS reduces to a simple
recipe.
• Take the free Green’s function and split it formally into the spin and charge
parts
1
1
1
=
.
G (x, t) =
1/2
1/2
x−t
(x − t)
(x − t)

• In an interacting system, the exponent of 1/2 in the charge part is replaced by
−1
−1
Kc + Kc /4 and in the spin-part by Ks + Ks /4. If the spin-rotational
invariance is preserved, then the spin exponent remains equal to 1/2.
• For x = 0,
1
G (t) ∝
−1
(Kc +Kc )/4+1/2
t
and the DoS behaves as
2

2

(Kc −1)
(uc −1)
−1
ν (ε) ∝ |ε|(Kc +Kc )/4−1/2 = |ε| 4Kc = |ε| 4uc .

Fundamental aspects of electron correlations and quantum transport

77

Comparing this result for g1 = 0 with that by DL (Sec. 3), we see that the
bosonization solution gives the same result as the fermionic one.
• For tunneling into the edge, remove Kc , which comes from the correlator
−1
ϕϕ pinned by the boundary, and multiply Kc , which comes from ϑϑ , by a
factor of 2. This gives
1
Ge (t) ∝ −1
tKc /2+1/2
and

−1
Kc −1)

G ∝ νe (ε) ∝ |ε| 2 (
1

.

Expanding Kc back in the interaction
Kc

=

1 + g1 /2π
1 + (4g2 − g1 ) /2π

1/2

≈1−

g2 − (1/2)g1
,
π

we obtain the weak-coupling limit for the tunneling exponent
(1/2) 1/Kc − 1 ≈

g2 − (1/2)g1
2π

.

This coincides with the result obtained in the fermionic language (Sec.5). What
was missed in a bosonization solution is a multiplicative log-renormalization,
present in Eq. (5.35). This is because we evaluated G at the fixed point, where
∗
g1 = 0, rather then derived an independent RG equation for the flow of the
conductance. This procedure should bring in the log-factors (cf. Ref.[74] where
these factors were obtained for the impurity scattering time).

7. Transport in quantum wires
7.1. Conductivity and conductance
7.1.1. Galilean invariance
Interactions between electrons cannot change the response to an electric field in
a Galilean-invariant system–the electric field couples only to the center-of-mass
whose motion is not affected by the inter-electron interaction. This property is
reproduced by the bosonized theory provided that the product uK = 1 (= vF in
dimensional form.) To see this, combine the Heisenberg equation of motion for
density ρ (spinless fermions) with the continuity equation:
∂t ρ = i[H, ρ] = −∂x j.

(7.1)

78

Dmitrii L. Maslov

Calculating the commutator in Eq. (7.1) with the help of Eq. (6.9), we identify
the current operator as
uK
uK 2
√ ∂x ϑ = −∂x − √ ∂x ϑ
π
π
uK
= − √ ∂x ϑ .
π

∂t ρ

=

j

→

The current is not affected by the interaction as long as uK = 1.
7.1.2. Kubo formula for conductivity
The Kubo formula relates the conductivity, a response function to an electric field
at finite ω and q, to the current-current correlation function
σ (ω, q) =

e2
1
− + JJ
iω
π

R
qω

,

(7.2)

where it was used that n = kF /π and kF /m = vF = 1 in our units.
Electric current for electrons (e > 0)
e
J = −ej = √ ∂x ϑ.
π
In complex time,
R
x,τ

JJ

R
q,ωm

e
√
π

=
=

JJ

2
2
−∂x

ϑϑ

x,τ

→

e2 2
q ϑϑ q,ωm
π
2
e2
ωm
e2
e2
˜˜
=
−
+ J J q,ωm .
2
π
π ωm + u 2 q 2
π

=

(7.3)

The first term in (7.3) cancels the diamagnetic response in (7.2). Continuing
analytically to real frequencies, we find
σ (ω, q) =

−ω 2
e2 1
1 ˜˜
J J q,ωm→−iω+δ = −
iω
π iω − (ω + iδ)2 + u2 q 2
m

= i

ω
e2
.
π ω 2 − u2 q 2 + isgnωδ

(7.4)

Fundamental aspects of electron correlations and quantum transport

79

Consequently, the dissipative conductivity is equal to
Reσ (ω, q) =
=

−

1
e2
ωIm 2
π
ω − u2 q 2 + isgnωδ

e2
[δ (ω − uq) + δ (ω + uq)] .
2

(7.5)

7.1.3. Drude conductivity
In a macroscopic system, one is accustomed to take the limit q → 0 first: this corresponds to applying a spatially uniform but time-dependent electric field [61].
(For the lack of a better name, I will refer to the conductivity obtained in this way
as to the Drude conductivity). The Drude conductivity in our case is the same as
for the Fermi gas as the charge velocity drops out from the result
Reσ (ω, 0) = e2 δ (ω)
or, restoring the units,
Reσ (ω, 0) =

e2 vF

δ (ω) .

All it means that when a static electric field is applied to a continuous system of
either free or interacting electrons, the center-of-mass moves with an acceleration
and there is no linear response, as there is no “friction” that can balance the
electric force.
For electrons with spins, the electrical current is related only to the charge
component of the ϑ− field:
Jc = −ejc = e

2
∂x ϑc ,
π

√
where again uc Kc = 1. Because of the 2 factor in the current, the conductivity
is by a factor of two different from that in the spinless case
Reσ (ω, 0) =

2e2 vF

δ (ω) .

(Notice, however, that at fixed density vF is by a factor of 2 smaller for electrons
with spin, so that the conductivity is the same.)

80

Dmitrii L. Maslov

7.1.4. Landauer conductivity
Let’s consider now the opposite order of limits, corresponding to a situation when
a static electric field is applied over a part of the infinite wire. (Again, for the lack
of a better name, I will refer to this conductivity as to the Landauer conductivity.)
The electric field might as well be non-uniform; the only constraint we are going
to use is that the integral
dxE (x) ,
equal to the applied voltage, is finite. The induced current (which in 1D coincides
with the current density) is given by
=

dx′

dt′ σ (t − t′ ; x, x′ ) E (t′ , x′ )

=

J (t, x)

dx′

dω −iωt
e
σ(ω; x, x′ )E (ω, x′ ) .
2π

In linear response, the conductivity is defined in the absence of the field. As such,
it is still a property of a translationally invariant system and depends thus only on
x − x′ . This allows one to switch to Fourier transforms
J (t, x) =

dx′

dω
2π

dq iq(x−x′ ) −iωt
e
e
σ (ω, q) E (ω, x′ ) .
2π

(7.6)

Now use the fact that the applied field is static: E (x, ω) = 2πδ (ω) E0 (x) (upon
which the t-dependence of the current disappears, as it should be in the steady
state)
dq iq(x−x′ )
e
σ (0, q) E0 (x′ ) .
(7.7)
J(x) = dx′
2π
From (7.5),
σ (0, q) =

1 2
e δ (q) = Ke2 δ (q) ,
u

(7.8)

where uK = 1 was used again. Substituting (7.8) into (7.7), we see that the x−
dependence of the current also disappears
J=

Ke2
2π

dx′ E0 (x′ ) =

Conductance G = J/V is given by
G=

Ke2
,
2π

Ke2
V.
2π

Fundamental aspects of electron correlations and quantum transport

81

or, restoring the units,
e2
.
h
For electrons with spin, a similar consideration gives
G=K

G = Kc

2e2
.
h

(7.9)

(7.10)

We see that the conductance is renormalized by the interactions from it universal
value given by the Landauer formula for an ideal wire [68].
7.2. Dissipation in a contactless measurement
What kind of an experiment Eqs. (7.9) and (7.10) correspond to?
Suppose that we connect a wire of length L to an external resistor and place
the whole circuit into a resonator [79]. Now, we apply an ac electric field E (x, t)
of frequency ω0 and parallel to a segment of the wire of length LE ≪ L, and
measure the losses in the resonator. The external resistor takes care of energy
dissipation: as the wire is ballistic (also in a sense that electrons travel through
the wire without emitting phonons), the Joule heat can be generated only outside
the wire. Dissipated energy, averaged over many periods of the field, is given by
˙
Q=−

dx J (x, t) E (x, t) .

For a monochromatic field, E (x, t) = E0 (x) cos ω0 t and after averaging over
many periods of oscillations, we obtain
˙
Q=−

dx

dx′ Reσ (ω0 ; x, x′ ) E0 (x) E0 (x′ ) .

Now, choose the frequency in such a way that
LE ≪

u
≪ L,
ω0

(7.11)

where u is the velocity of the charge mode in the wire. Because the wavelength
of the charge excitations at frequency ω0 –acoustic plasmons– is much shorter
than the distance to contacts (L), the conductivity is essentially the same as for
an infinite wire and depends only on x − x′ . Performing partial Fourier transform
in Eq. (7.5), we find
Reσ (ω, x) =

e2
e2
cos (ωx/u) =
K cos (ωx/u) ,
2πu
2π

(7.12)

82

Dmitrii L. Maslov

so that
1 e2
˙
K
Q=−
2 2π

dx

dx′ cos [ω0 (x − x′ ) /u] E0 (x) E0 (x′ ) .

On the other hand, because |x, x′ | ≤ LE ≪ u/ω0 , the cosine can be replaced by
unity, and
e2
˙
Q = − KV 2 ≡ −GV 2 ,
2π
or
e2
G = K.
2π
Therefore, dissipation in a contactless measurements under the conditions specified by Eq. (7.11) corresponds to a renormalized conductance. To the best of
my knowledge, this experiment has not been performed. A typical (two-probe)
transport measurement is done by applying the current and measuring the voltage
drop between the reservoirs. In this case, the measured conductance does not correspond to Eqs.(7.9,7.10) but is rather given simply by e2 /h per spin projection–
regardless of the interaction in the wire [80],[81],[82]. This result is discussed in
the next Section.
7.3. Conductance of a wire attached to reservoirs
The reason why the two-terminal conductance is not renormalized by the interactions within the wire is very simple. For the Fermi-gas case, the conductance
of e2 /h per channel is actually not the conductance of wire itself–a disorder-free
wire by itself does not provide any resistance to the current. In a four-probe measurement, when the voltage and current are applied to and measured in different
contacts, the conductance of a disorder-free wire is, in fact, infinite. However, in
a two-probe measurement, the voltage and current contacts are the same. Finite
resistance comes only from scattering of electrons from the boundary regions,
connecting wide reservoirs to the narrow wire [83, 84], as shown in Fig.15a).
The universal value of e2 /h is approached in the limit of an adiabatic (smooth on
the scale of the electron wavelength) connection between the reservoirs and the
wire [85] 35 . As the resistance comes from the regions exterior to the wire, the
interaction within the wire is not going to modify the e2 /h− result. Another way
to think about it is to notice that the renormalized conductance (7.9,7.10) can be
35 Accidentally, the actual constraint on the adiabaticity of the connection is rather soft–it is enough
to require the radius of curvature of the transition region be just comparable to, rather than much larger
than, the electron wavelength [85].

Fundamental aspects of electron correlations and quantum transport

(a)

83

LL

FL

FL
(b)
LL

FL

*

*
*

FL

Fig. 15. a) A Luttinger-liquid (LL) wire attached to Fermi-liquid (FL) reservoirs. b) same for a single
impurity within the wire.

√ √
interpreted as a manifestation of the fractional charge e∗ = K( Kc ), associated with the excitations in a 1D system. However, the current coming from, e.g.,
the left reservoir is carried by integer charges, and as all these charges get eventually transmitted through the wire, the current collected in the right reservoir
is carried again by integer charges. Fractional charges is a transient phenomena
which, in principle, can be observed in an ac conductance or noise measurements
but not in a dc experiment. In the rest of this Section, these arguments will be
substantiated with some simple calculations.
7.3.1. Inhomogeneous Luttinger-liquid model
An actual system consists of two Fermi-liquid reservoirs connected via a Luttingerliquid (LL) wire and , due to the presence of the reservoirs, is not one-dimensional.
In the inhomogeneous Luttinger-liquid model, the actual system is replaced by an
effective 1D system, which is an infinite LL with a position-dependent interaction parameter K(x) (cf. Fig. 16). The actual reservoirs are higher (D = 2 or
3) systems, where the effect of the interaction can be disregarded. Consequently,
the reservoirs are modelled by one-dimensional free conductors with KL = 1.
In between, K (x) goes through some variation. Similarly, the charge velocity
is equal to the Fermi one in the reservoirs and varies with x in the middle of the
system. The potential difference applied to the system produces some distribution of the electric field along the wire. The shape of this distribution is irrelevant

84

Dmitrii L. Maslov

K(x)
(a)

K=1
L

v(x)
(b)

vL
E(x)
(c)

L

x

Fig. 16. Inhomogeneous Luttinger-liquid model.

in the dc linear response.
7.3.2. Elastic-string analogy
The (real-time) bosonic action for a spinless LL is
S=

1
2

d2 x

1
K(x)

u(x)(∂x ϕ)2 −

1
(∂t ϕ)2 .
u(x)

(7.13)

The density of the electrons (minus the background density) and the (number)
current are given by
√
√
j = −∂t ϕ/ π.
(7.14)
ρ = ∂x ϕ/ π,
The interaction with an external electromagnetic field Aµ is described by
e
Sint = √
2 π

d2 x {A0 ∂x ϕ − A1 ∂t ϕ} ,

(7.15)

so that the equation of motion for ϕ is
∂t

1
u
e
∂t ϕ − ∂x
∂x ϕ = √ E(x, t),
Ku
K
π

(7.16)

where E = −∂x A0 + ∂t A1 is the electric field. We assume that the electric field
is switched on at t = 0, so that E(x, t) = 0 for t < 0 and E(x, t) = E(x)

Fundamental aspects of electron correlations and quantum transport

85

( a)

φ( x,t )

IV

I

III

II

V

( b)

φ( x,t )

vt
L

L

L

x

Fig. 17. a) Solution of the wave equation in the homogeneous case for t = 5L/u. b) Schematic
solution in the inhomogeneous case for t ≫ L/u.

for t ≥ 0. The problem reduces now to determining the profile of an infinite
elastic string under the external force. In this language, ϕ(x, t) is the transverse
displacement of the string at point x and at time t, while the number current
√
j = −∂t ϕ/ π is proportional to the transverse velocity of the string.
To develop some intuition into the solution of Eq. (7.16), we first solve it in the
homogeneous case, when K =const, u =const, and E(x) =const for |x| ≤ L/2,
and is equal to zero otherwise. In this case, the solution of Eq. (7.16) for t > L/u,
is

2
2
t − x +L /4 for |x| ≤ L/2;


Lu

KeV
t − |x|/u for L/2 ≤ |x| ≤ ut − L/2
ϕ(x, t) = √ ×
 u t − |x|−L/2 2 for ut − L/2 ≤ |x| ≤ ut + L/2
2 π
 2L
u

0, for |x| ≥ ut + L/2,
where V = EL is the total voltage drop. This solution is depicted in Fig. 17a.
The profile of the string consists of two segments (I and II in Fig. 17a) whose
widths, equal to (ut− L), grow with time, and of three segments (III, IV and V in
Fig. 17a) whose widths are constant in time and equal to L. In segments I and II,
the profile of the string ϕ(x, t) is linear in x, and therefore, being the solution of
the wave equation, also in t; in segments III-V, the profile is parabolic. Outside
segments IV and V, the string is not perturbed yet, and ϕ(x, t) = 0. As time
goes on, the larger and larger part of the profile becomes linear. For late times,
the pulse produced by the force spreads outwards with velocity u, involving the
yet unperturbed parts of the string in motion; simultaneously, in all but narrow

86

Dmitrii L. Maslov

segments in the middle and at the leading edges of the pulse, the√
string moves
upwards with the t- and x-independent “velocity” ∂t ϕ = KeV /2 π. In terms
of the original transport problem, it means that the charge current J = −ej
is constant outside the wire (but not too close to the edges of the regions of
where the electron density is not yet perturbed by the electric field) and given by
J = Ke2 V /h. Therefore, the conductance (per spin orientation) is G = Ke2 /h.
We now turn to the inhomogeneous case. As in the previous case, the profile
consists of several characteristic segments (cf. Fig. 17b). In segments III-V, the
profile is affected by the inhomogeneities in K(x), u(x), and E(x) and depends
on the particular choice of the x-dependences in all these quantities. In segments
I and II however, the profile, being the solution of the free wave equation, is
again linear in x (and in t). Requiring the slopes of the string be equal and
opposite in segments I and II (which is consistent with the condition of the current
conservation), the solution in these regions can be written as ϕ(x, t) = A(t −
|x|/uL ). The constant A can be found by integrating Eq. (7.16) between two
symmetric points ±a , chosen outside the wire
+a

−

dx∂x
−a

e
u
∂x ϕ = √
K
π

+a
−a

eV
dxE(x) = √ .
π

(7.17)

√
Outside the wire, K(x) = KL and u(x) = uL , thus A = KL eV /2 π . Calculating the current, we get G = KL e2 /h and, recalling that KL = 1, we finally
arrive at G = e2 /h. Thus, the conductance is not renormalized by the interactions
in the wire.
7.3.3. Kubo formula for a wire attached to reservoirs
The Kubo formula for a translationally non-invariant system can be written as
σ (ω; x, x′ )

e2
δ (x − x′ )
iπω
e2
d (τ − τ ′ ) eiωm τ Tτ ∂τ ϕ (x, τ ) ∂τ ′ ϕ (x′ , τ ′ )
iπω

= −
+

|iωm →ω+iδ .

The diamagnetic contribution is cancelled by a delta-function term, which is obtained when integrating by parts in the time-ordered product [86, 10]. Having
this in mind, I will re-write the conductivity via the Fourier transform of the ϕϕcorrelator without the T − product
σ (ω; x, x′ ) = i

e2 2
ω Φω (x, x′ ) |iωm →ω+iδ .
πω m m

For a translationally invariant case, this reduces back to Eq. (7.4). Now, K (x)
and u (x) depend on position. The propagator of the ϕ fields satisfy the wave

Fundamental aspects of electron correlations and quantum transport

87

equation (or a Laplace’s equation as we are dealing with the imaginary time)
2
ωm
− ∂x
u (x) K (x)

u (x)
∂x
K (x)

Φωm (x, x′ ) = δ (x − x′ ) .

(7.18)

In a model of step-like variation of K (x) and u (x) (K = KW and u = uW
within the wire and K = KL = 1 and u = uL = 1 outside the wire), Eq.
(7.18) is complemented by the following boundary conditions: 1) Φωm (x, x′ )
is continuous at x = ±L/2, 2) u (x) K (x) ∂x Φωm (x, x′ ) is continuous at x =
±L/2; but 3) undergoes a jump of unit height at x = x′ . Solution of this problem
is totally equivalent to finding a potential of a point charge located somewhere
in a sandwich-like system, consisting of three insulators with different dielectric
constants. Two of these layers are semi-infinite and the third one (in the middle)
is of finite thickness. “Potential” Φωm (x, x′ ) can be found in a general form for
arbitrary x, x′ . In the expression for the current
J (t, x) =

dx′

dω
σ (ω; x, x′ ) E (ω, x′ ) ,
2π

x′ is within the wire; hence we need to know Φωm (x, x′ ) only for −L/2 ≤ x′ ≤
L/2. In a steady-state regime, one is free to measure the current through any
cross-section; let’s choose x also within the wire. As we are interested in the
limit ω → 0, when the plasmon wavelength is larger than the wire length, we can
put x = x′ . In the interval −L/2 < x = x′ < L/2 the solution of the Laplace’s
equation is
Φωm (x, x) =

KW
KW κ2 e−L/Lω + κ+ κ− cosh (2x/Lω )
−
+
,
2 |ωm | 2 |ωm |
eL/Lω κ2 − e−L/Lω κ2
−
+

(7.19)

where Lω = uW / |ωm | , uW is the charge velocity within the wire, and
−1
−1
κ± = K W − K L .

Letting ωm in (7.19) to zero (and thus Lω to ∞), we find that

KL
1
=
,
2iω
2iω
as by our assumption KL = 1. This result is true for any x, x′ within the wire for
ω→0
1
, for − L/2 < x, x′ < L/2.
Φω (x, x′ ) =
2iω
The Luttinger-liquid parameters of the wire drop out from the answer. The dc
conductivity reduces to its free value
Φω (x, x) =

σ (ω → 0; x, x′ ) =

e2
,
2π

88

Dmitrii L. Maslov

and, consequently, the conductance
G=

e2
h

is not renormalized by the interaction. The same consideration for electrons with
spins gives
G=

2e2
h

(7.20)

7.3.4. Experiment
Most of the experiments on quantum wires indeed show that the conductance is
quantizated in units of 2e2 /h at relatively high temperatures. 36 37 At lower
temperatures, the conductance decreases beyond the universal value and also the
quantization plateaux exhibit some structure as a function of the gate voltage
[90]. This can be interpreted as the effect of residual disorder: as was discussed
in Secs. 5,6 transmission decreases at lower energy scales. An effect of single
impurity in a quantum wire will be largely insensitive to the presence of reservoirs: as long as the transmission coefficient for an impurity is much smaller than
one, the largest voltage drops occurs near the impurity rather than at the contacts
to the wire. One can show that the scaling of the conductance with energy is determined by the interaction parameter K inside the wire [91], in a contrast to the
disorder-free case when only K outside the wire matters. Also, the mesoscopic
conductance fluctuations increase as the temperature goes down (the theory predicts that this effect is enhanced by the interaction [92]). As one is dealing here
with a crossover regime from scattering at a single impurity to that at many impurities, a quantitative analysis of the temperature dependences is difficult; another
complication arises from the finite length of the wire which cuts off scalings with
temperature and voltage. In addition, at higher temperatures the first quantization
plateau exhibits a well-defined step at about 0.7 × 2e2 /h [93, 94, 95, 96, 97, 98].
This “0.7” feature is not likely to result from spurious impurity scattering but
rather reveals some interesting physics beyond what has been discussed so far in
this review. Although the “0.7” feature deserves a review on its own, I will come
back to this subject briefly in the next Section.
36 However,

it has been observed recently that the conductance of carbon nanotubes is quantized in
units of e2 /h –as opposed to 4e2 /h, predicted by the non-interacting theory for this case [87].
37 A special case of a non-universal conductance quantization is very long wires grown by cleavededge overgrowth technique [88] can be attributed to a non-trivial coupling between the wire and 2D
reservoirs [89], characteristic for these systems.

Fundamental aspects of electron correlations and quantum transport

89

7.4. Spin component of the conductance
As we have shown in the previous sections, the Luttinger-liquid models predicts
that conductance of a disorder-free wire is given by e2 /h per channel at any temperature. Also, thanks to spin-charge separation, spin degrees of freedom do not
play an essential role in charge transport except for giving an overall factor of
two to the conductance. These two results hold as long as the Luttinger-liquid
model is a good description for interacting electrons in the wire. When does this
model break down? If the interaction is strong, electrons form almost a periodic
1D structure: quasi-Wigner crystal. The exchange energy of almost localized
electrons is exponentially small and, correspondingly, the spin velocity is small
too: us ≪ uc . The Luttinger-liquid model should work for energies (temperatures) much smaller than the smallest of the two (spin and charge) bandwidth
T ≪ us kF ≪ uc kF , when both spin- and charge degrees of freedom are coherent. The spin- incoherent regime, i.e., us kF ≪ T ≪ uc kF , has attracted considerable interest recently [19, 20, 21], and was shown to spoil the conductance
quantization in integer multiples of 2e2 /h [19] at temperatures larger than the
spin bandwidth (us kF ). In what follows, I present a short summary of Ref.[19].
In a quasi-Wigner-crystal regime, a reasonable starting point for describing
the spin sector is the Heisenberg model
Hs = Jex
l

Sl · Sl+1 ,

where spins are localized at “lattice sites” corresponding to positions of electrons.
Because the Lieb-Mattis theorem [46] forbids ferromagnetic ordering in 1D, the
sign of the exchange interaction must be antiferromagnetic: Jex > 0. Assuming
that electrons are well localized at distances a = 1/n from each other, Jex can be
√
estimated in the WKB approximation: Jex ∼ EF exp −c/ aB n , where c ∼ 1
and aB is the Bohr radius. A spin-1/2 chain is then mapped onto a Hubbard 1/2filled model of spinless fermions via the Jordan-Wigner transformation
Sz (l) = a† al − 1/2;
l
Sx (l) + iSy (l) =

a†
l

l−1

a† ex aJex
J

exp iπ
Jex =1

with the result
Jex
Hs = −
c† cl + H.c. + Jex
: c† cl − 1/2 :: c† cl − 1/2 : .
l+1
l
l+1
2
The spinless Hubbard model can be bosonized
√
aJex
u
1
2
2
(7.21)
dx (∂x φ) + uK (∂x θ) +
Hs =
16πφ .
2 cos
2
K
(2πa)

90

Dmitrii L. Maslov

A comparison to the Bethe Ansatz solution of the Hubbard model at half-filling
enables one to identify the parameters of the spinless LL with the microscopic
parameters of the spin-1/2 chain. In particular, for an isotropic spin chain (Jx =
Jy = Jz )
π
(7.22)
u = Jex a; K = 1/2.
2
Comparing the spin-part of the Hamiltonian of the original LL model, Eq. (6.29)
with that of the spinless LL, Eq. (7.21), one notices that they the same upon the
following mapping
√
us
u
1
=
; us Ks = 2uK,
φ = √ φs ; θ = 2θs ;
Ks
2K
2

(7.23)

or

π
Jex a; Ks = 2K = 1.
(7.24)
2
As Jex is exponentially small, so is the spin bandwidth. Therefore, the Luttingerliquid description is valid only at very low temperatures.
A translationally invariant LL still possesses spin-charge separation. However,
this is no longer true for a wire connected to non-interacting leads. To understand
this point, let’s come back to the inhomogeneous LL model (cf. Sec. 7.3.1),
where the electron density changes from a higher value in the leads to a lower
value within the wire. Because Jex depends on the local density, it is modulated
along the wire, and its minimum value is at the middle of the wire. In the leads,
we have a non-interacting system, where Jex ∼ EF ≫ T. However, in the middle
min
of the wire spins are incoherent, if Jex ≪ T. Thus a spin part of the electron
incoming from the lead at energy T above the Fermi level cannot propagate freely
through the wire because the spin band narrows down: it works as if there is a
barrier for spin excitations in the wire. Although charge plasmons propagate
freely, backscattering of spin plasmons leads to additional dissipation, and thus
to additional resistance. The total resistance of the wire consists now of two parts
us = u =

R = Rc + Rs .
The charge part, Rc , is due to propagation of charge plasmons. Since the charge
part is still described by the LL model, our previous result for universal conmin
ductance, Eq. (7.20), still holds and Rc = G −1 = h/2e2. For T ≪ Jex , only
athermal spin plasmons, with energies exceeding the width of the spin band, contribute to Rs . The number of such plasmons is exponentially small, hence
min
Rs ∝ exp −Jex /T ,

and total conductance G = (Rc + Rs )−1 is exponentially close to 2e2 /h. At high
min
temperatures T ≫ Jex , almost all spin plasmons are reflected by the wire.

Fundamental aspects of electron correlations and quantum transport

91

Then Rs ∼ Rc ∼ h/e2 , and the conductance differ substantially from its universal value. This qualitative picture can be confirmed in a particular simple case of
the XY − model for the spin-chain. In this case, Rs can be calculated explicitly
with the result [19]
1
h
Rs = 2
min /T ) + 1
2e exp (Jex
and, consequently, the conductance is equal to
G=

2e2 /h

−1 .

min
1 + [exp (Jex /T ) + 1]

min
For T → 0, G approaches the universal value of 2e2 /h. For T ≫ Jex ,G ap2
proaches another T -independent limit, equal to (2/3)2e /h. The actual number
in the high-temperature limit of the conductance is model-dependent (it is different, for example, for an isotropic spin-chain), but the main result, i.e., the
non-universality of conductance quantization at higher temperatures, survives.
As it was mentioned in Sec. 7.3.4, the experiment shows that there is a shoulder in the conductance preceding the first quantization plateau at a fractional
value of about 0.7×2e2/h. Surprisingly, this “0.7 feature” is more pronounced
at higher temperatures, and the T - dependence of this feature was reported to be
of an activated type [95]. The magnetic field transforms the “0.7 feature” into a
fully developed quantization plateau at e2 /h, which is to be expected in a fully
polarized, and thus spinless, regime. The sensitivity to the magnetic field hints
at the spin origin of the effect, and a significant theoretical effort was invested
in understanding how spins can explain the observed phenomena. Although the
effect, described in this Section, does have all qualitative characteristics of the
observed “0.7 feature”, it is not clear at the moment whether this feature indeed
corresponds to the spin-incoherent regime. Other explanations of the “0.7 feature” have been suggested (most prominently, the Kondo physics is believed to
be involved [99, 98]), but a further discussion of this point goes beyond the scope
of these notes.

7.5. Thermal conductance: Fabry-Perrot resonances of plasmons
There is an important difference between charge and thermal (electronic) conductances [22]. As we have just shown, the charge conductance is equal to e2 /h
regardless of interaction in the wire. This means that the transmission coefficient
of electrons is equal to unity. The effect of the temperature on the charge conductance is the same as for a non-interacting, perfectly transmitting wire: at finite
temperature, not only the lowest but also higher subbands of transverse quantization are populated, and the quantization plateaux are smeared. However, this

92

Dmitrii L. Maslov

effect is exponentially small for temperatures smaller than either the Fermi energy, EF , or the difference between the Fermi energy and the threshold of the
next subband of transverse quantization, ∆; whichever is smaller.
Thermal current is carried not by electrons but bosonic excitations: acoustic plasmons. In contrast to electrons, plasmons get reflected at the boundary
between the reservoirs and the wire due to the mismatch of charge velocities
(this reflection happens even for an adiabatically smooth transition). From the
plasmon’s point-of-view, a wire coupled to reservoirs represents a Fabry-Perrot
interferometer. Interference of plasmon waves scattered off the opposite ends of
the wire results in an oscillatory dependence of the transmission coefficient on
the frequency with a period given by the travel time of plasmons through the wire
2πωL = L/uW .
As long as λF ≪ L, this period is long: ωL ≪ EF . The difference between
charge and heat transport is that the chemical potential of plasmons is equal to
zero, and thus the characteristic scale for frequency is set by T . Therefore, the
thermal conductance varies with the temperature on a scale T ≃ ωL .
Suppose that a small temperature difference δT is maintained between the
reservoirs, connected by a quantum wire. As the Hamiltonian of an interacting
system is diagonalized of terms of plasmons, plasmons modes are decoupled and
contribute to the energy flux independently. Then the thermally averaged energy
current, i.e., the thermal current can be found via a Landauer-like argument
∞

JT =
0

dω
ω |t (ω)|2 nL
2π

ω
T + δT

− nR

ω
T

,

where nL,R (ω/T ) are the Bose distribution functions in the reservoirs. Expanding in small δT, we obtain the thermal conductance
GT =

1
JT
=
δT
8πT 2

∞
0

dω

ω2
|t (ω)|2 .
sinh ω/2T
2

(7.25)

For a free system, |t (ω)|2 = 1 and GT = πT /6. The charge and thermal conductances of a free system obey the Wiedemann-Franz law
GT
π2
= L0 = 2 ,
TG
3e

(7.26)

where L0 is the Lorentz number. This means that charge and energy are carried
by the same excitations. This is not so in a Luttinger liquid.
For an interacting system, relation (7.26) holds in the limit of T → 0. The
characteristic scale for frequencies in integral (7.25) is determined by T. For

Fundamental aspects of electron correlations and quantum transport

93

2

T ≪ ωL , one can substitute ω = 0 into |t (ω)| . Regardless of the interaction
2
strength, |t (0)| = 1 : a Fabry-Perrot interferometer becomes transparent in the
long wavelength limit. For T ωL , the result for GT depends on how the charge
velocity varies along the wire, and is thus non-universal. On the other hand, the
charge conductance is universal. Therefore, their ratio is non-universal and the
Wiedemann-Franz law is violated.
In a step-like model of Sec. 7.3.1, the transmission coefficient of plasmons is
equal to
1
2
.
|t (ω)| =
(K 2 −1)2
ω
1 + 4K 2 sin2 ωL
2

Obviously, |t (0)| = 1 regardless of K, an agreement with what was said above.
For T ≪ ωL , the Lorentz number is close to the universal value of π 2 /3e2 . For
2
2
T ≫ ωL , the oscillations of |t (ω)| become very fast, so that |t (ω)| can be
replaced by its averaged value
2

|t (ω)|

ωL

=
0

2K
dω
2
|t (ω)| = 2
.
ωL
K +1

The thermal conductance increases linearly with T, so that L0 approaches a constant but a non-universal value
L|T ≫ωL =

2K
L0 < L0 .
K2 + 1

(7.27)

As the Lorentz number varies with temperature in between two limits specified
by Eqs.(7.26) and (7.27), the Wiedemann-Franz law is violated.

8. Acknowledgments
Late R. Landauer and H. J. Schulz helped me–either directly or indirectly, through
their writings–to form my way of thinking about interactions and transport, and
my gratitude goes to their memories. I benefited tremendously from discussions
with B. L. Altshuler, C. Biagini, C. Pèpin, A. V. Chubukov, V. V. Cheianov, E.
Fradkin, F. Essler, S. Gangadharaiah, Y. Gefen, L. I. Glazman, P. M. Goldbart, I.
V. Gornyi, Y. B. Levinson, D. Loss, K. A. Matveev, B. N. Narozhny, A. A. Nersesyan, D. G. Polyakov, C. Pépin, M. Yu. Reizer, I. Safi, R. Saha, B. I. Shklovskii,
M. Stone, O. A. Starykh, S. Tarucha, S.-W. Tsai, A. Yacoby, O. M. Yevtushenko,
and M. B. Zvonarev on various subjects discussed in these notes. I am also grateful to the organizers of the 2004 Summer School in Les Houches–H. Bouchiat,
Y. Gefen, S. Guéron, and G. Montambaux–for assembling a very interesting program and providing a stimulating environment, as well as to the all participants of

94

Dmitrii L. Maslov

the School for their questions, attention, and patience. S. Gangadharaiah, L. Merrill, and R. Saha kindly helped me with producing the images. S. Gangadharaiah,
P. Kumar, C. Pépin, R. Saha, and S.-W. Tsai proofread parts of the manuscript
(which does not make them responsible for the typos). I acknowledge the hospitality of the Abdus Salam International Centre for Theoretical Physics (Trieste,
Italy), where a part of this manuscript was written. Financial support from NSF
DMR-0308377 is greatly appreciated.
Appendix A. Polarization bubble for small q in arbitrary dimensionality
The polarization bubble in Matsubara frequencies and at T = 0 is given by
Π (iω, q) =
=
=

Ns
(2π)D+1
Ns
D+1

(2π)
Ns

D+1

(2π)

dD pdεG (iε + iω, p + q) G (iε, p)
dD pdε
dD p

1
[G (iε + iω, p + q) − G (iε, p)]
iω − ξp+q + ξp

f (|p + q|) − f (p)
,
iω − ξp+q + ξp

where f is the Fermi function. Expanding in q, and switching from the integration over dD p to dξ, we obtain
Π (iω, q) = −Ns νD 1 −

dΩ
iω
ΩD iω − vF q cos θ

.

where ΩD = 4π (3D), = 2π (2D) , = 2 (1D) and νD is the DoS in D dimensions
per one of the Ns isospin components. For D=1, the integral over Ω is understood
as a sum of terms with cos θ = ±1. It is obvious already this form that the small
q−form of the bubble depends on the combination ω/vF q for any D. The final
result depends on the dimensionality. Performing analytic continuation to real
frequencies iω → ω + i0+ , we obtain
ΠR (ω, q) = −Ns νD 1 −
Taking the imaginary part
ImΠR (ω, q) = −πNs νD ω

dΩ
ω
ΩD ω − vF q cos θ + i0

dΩ
δ (ω − vF q cos θ) .
ΩD

.

(A. 1)

From here
cos θ = ω/vF q,
which means that θ ≈ π/2 for ω ≪ vF q. Thus, the fermionic momentum p is
almost perpendicular to the bosonic one, q, in this limit.

Fundamental aspects of electron correlations and quantum transport

95

Appendix B. Polarization bubble in 1D
Appendix B.1. Small q
Free time-ordered (causal) Green’s function in 1D is equal to
G0 (ε, k) =
±

ε−

±
ξk

1
±,
+ i0+ sgnξk

where
±
ξk = ±vF (k ∓ kF ) ,

and ± signs correspond to right/left moving fermions. We will be measuring the
momenta from the corresponding Fermi points. For +branch :k − kF → k and
for -branch: k + kF → k. Consequently,
G0 (ε, k) =
±

1
.
ε ∓ vF k + i0+ sgnk

I assume the Ns − fold degeneracy (Ns = 2 for electrons with spin, Ns = 1 for
spinless electrons), so that
Π± (ω, q) = −

i

2 Ns

dε

dkG0 (ε + ω, k + q) G0 (ε, k) .
±
±

2 Ns

dε

dk

2 Ns

dε

(2π)

Calculate, e.g., Π+ :
Π+ (ω, q) = −
= −

i

(2π)
i
(2π)

1
1
ε + ω − vF (k + q) + i0+ sgn(k + q) ε − vF k + i0+ sgnk
1
dk
(B. 1)
ω − vF q + i0+ sgn(k + q) − i0+ sgnk

× G0 (ε, k) − G0 (ε + ω, k + q) .
+
+

The integral of the Green’s over the frequency gives a Fermi distribution function
[23]
dε 0
n+ (k) = −i
G (ε, k) .
2π +
For free fermions,
n+ (k) = θ (−k)
Now,
Π0 (ω, q) =
+

Ns
2π

dk

1
[θ (−k) − θ (−k − q)] .
ω − vF q + i0+ sgn(k + q) − i0+ sgnk

(B. 2)

96

Dmitrii L. Maslov

The integral is not equal to zero only if the arguments of the θ -functions are of
the opposite signs. Consider different situations.
1) k > 0; k + q < 0 → 0 < k < −q → q < 0. In this case,
Π0 (ω, q) =
+

q
Ns
, q < 0;
2π ω − vF q − i0+

2) k < 0, k + q > 0 → −q < k < 0 → q > 0
Π0 (ω, q) =
+

q
Ns
, q > 0.
2π ω − vF q + i0+

Combining the results for q > 0 and q < 0 together,
Π0 (ω, q) =
+

q
Ns
.
2π ω − vF q + i0+ sgnq

(B. 3)

Similarly,
Π0 (ω, q) = −
−
The total bubble

q
Ns
.
2π ω − vF q + i0+ sgnω

Π0 (ω, q) = Π0 (ω, q) + Π0 (ω, q) =
+
−

Ns
vF q 2
.
2
π ω 2 − vF q 2 + i0+

(B. 4)

(B. 5)

In what follows, we will also need the retarded and advanced form of the
bubble. These forms can easily be obtained by repeating the calculation above in
Matsubara frequencies and analytically continuing iωm → ω +i0 . Even simpler,
one can use the general relation between time-ordered and retarded propagators
[23] (which works equally well for fermionic and bosonic quantities)
ΠR (ω, q) = Π± (ω, q) , for ω > 0
±
= Π∗ (ω, q) , for ω < 0.
±
Using Eqs. (B. 3) and (B. 4) we obtain
ΠR (ω, q) = ±
±

q
Ns
2π ω − vF q + i0+

(B. 6)

and
ΠR (ω, q) =
=

vF q 2
Ns
2
π ω 2 − vF q 2 + i0+ sgnω
vF q 2
Ns
.
2
π (ω + i0+ )2 − vF q 2

(B. 7)

Fundamental aspects of electron correlations and quantum transport

97

Appendix B.2. q near 2kF
We will also need the 2kF bubble. This time, I choose to do the calculation in
Matsubara frequencies:
Π2kF (iω, q) =
=

Ns
dk dεG+ (iε + iω, k + q) G− (iε, k)
(2π)2
1
1
Ns
.
dk dε
−
(2π)2
ε + ω + i (k + q) ε − ik

Poles in ε1 = ik and ε2 = −i (k + q) − ω have to be on different sides of the real
axis, otherwise the integral is equal to zero. Choose q > 0. Then this condition
is satisfied in two intervals of k : k > 0 and −Λ/2 < k < −q, where Λ is the
ultraviolet cut-off

Π2kF

=
=
=

−

iNs
2π

Λ/2
0

dk −

−q

dk

−Λ/2

1
ω + 2ivF k + ivF q

Ns
iΛvF
ω − ivF q
ln
− ln
4π
ω + ivF q
−iΛvF
2 2
Ns
Λ vF
−
.
ln 2
4π
ω + vF q 2
−

(B. 8)

Because the result depends on q 2 there is no need for a separate calculation for
the case q < 0.

Appendix C. Some details of bosonization procedure
Appendix C.1. Anomalous commutators
ρ (q)

=
p

ρ±

a†
p−q/2 ap+q/2 = ρ+ + ρ− ;
a†
p−q/2 ap+q/2 .

=
p>0(p<0)

The operators of full density commute. The operators of left-right densities have
non-trivial commutators. For example, let us calculate [ρ+ (q) , ρ+ (q ′ )]

98

Dmitrii L. Maslov

C++ (q, q ′ ) =

†
a†
p−q/2 ap+q/2 , ak−q′ /2 ak+q′ /2

[ρ+ (q) , ρ+ (q ′ )] =
p>0,k>0



=

a†
p−q/2

ap+q/2 a† ′ /2
k−q

ak+q′ /2




=δp+q/2,k−q′ /2 −a† ′ /2 ap+q/2
k−q

 −a†
ap+q/2
ak+q′ /2 a†
k−q′ /2
p−q/2
p>0,k>0 

=δk+q′ /2,p−q/2 −a†
a
p−q/2 k+q′ /2






.




The first δ− function means that k = p + q/2 + q ′ /2 > 0 and the second one
that k = p − q/2 − q ′ /2.
C++ (q, q ′ ) =
p>0

†
′
′
′
a†
p−q/2 ap+q/2+q ϑ (p + q/2 + q /2) − ap−q/2−q′ ap+q/2 θ (p − q/2 − q /2)

− [f (q, q ′ ) − f (q ′ , q)] ,
where
f (q, q ′ )

†
′
a†
p−q/2 ak−q′ /2 ap+q/2 ak+q /2

=
p,k>0

a† a† ap+q ak+q′
p k

=
p,k>0

It is easy to show that f (q, q ′ ) = f (q ′ , q) . Indeed,
f (q ′ , q)

a† a† ap−q′ ak+q
p k

=
p,k>0

= re − labelling k ←→ p =

a† a† ak−q′ ap+q
k p
p,k>0

a† a† ap+q ak+q′
p k

= anticommuting =

= f (q, q ′ ) .

p,k>0

Thus
C++ (q, q ′ ) =
p>0

†
′
′
a†
p−q/2 ap+q/2+q′ θ (p + q/2 + q /2)−ap−q/2−q′ ap+q/2 θ (p − q/2 − q /2) .

Introduce a new momentum
Q=

q + q′
.
2

Fundamental aspects of electron correlations and quantum transport

99

In the first sum, shift p + q ′ /2 → p and in the second sum shift p − q ′ /2 → p.
Then
C++ (q, 2Q − q)

=
p>0

a† ap+Q [θ (p + q/2) − θ (p − q/2)]
p−Q

=
p>−q/2

a† ap+Q −
p−Q

a† ap+Q
p−Q
p>q/2

If the main contribution to the sum is given by the states which lie either deep
below or far above the Fermi levels, then the quantum fluctuations in the occupancy of these states are small, and the operators a† ap+Q can be replaced by
p−Q
their expectation values a† ap+Q = δQ,0 np = δQ,0 θ (pF − p). Doing this,
π−Q
we find


pF
pF
pF
pF
 = δQ,0 L
−
C++ (q, 2Q − q) = δQ,0 
dp
dp −
2π
q/2
−q/2
p>−q/2

= δQ,0

p>q/2

qL
.
2π

Therefore,

qL
, spinless.
2π
The same procedure for fermions with spin gives
[ρ+ (q), ρ+ (−q)] =

[ρ+,σ (q), ρ+,σ′ (−q)] = δσσ′

(C. 1)

qL
, with spin.
2π

Similarly,
[ρ−,σ (q), ρ−,σ′ (−q)] = −δσσ′

qL
, with spin.
2π

and
[ρ+,σ (q), ρ−,σ (−q)] = 0.
Combining these results together
[ρα,σ (q) , ρα′ ,σ′ (−q)] = αδα,α′ δσ,σ′

qL
,
2π

where α = ± is the chirality index. For full charge density and current, it means
that
[ρc (q) , ρc (−q)] =
=

ρc (q) + ρc (q) , ρc (−q) + ρc (−q)
+
−
+
−
qV
qV
qV
qV
+
−
−
= 0.
2π
2π
2π
2π

100

Dmitrii L. Maslov

Similarly,
[j c (q) , j c (−q)] = 0,
whereas
[ρc (q) , j c (−q)] =

qV qV qV qV 2
+
+
+
= qL.
2π 2π 2π 2π π

In 4-notations,
2
[j µ (q) , j ν (−q)] = ǫµν qL,
π
where ǫ00 = ǫ11 = 0, ǫ01 = −ǫ10 = 1.
Appendix C.2. Bosonic operators
Let’s check that the representation of density operators via standard bosonic operators does reproduce commutation relation for density. Expand the density
operators over the normal modes
ρ+ (x)

=

1
Aq bq eiqx + b† e−iqx ;
q
L q>0

ρ− (x)

=

1
Aq bq eiqx + b† e−iqx ,
q
L q<0

where, without a loss of generality, Aq can be chosen real and even function of
q. Fourier transforming ρ+ (x)
ρ+ (q)

=

∞

dxe−iqx

−∞

1
L

′

′

Aq′ bq′ eiq x + b†′ e−iq x ,
q
q′ >0

= Aq θ (q) bq + θ (−q) b† .
−q
Choose q > 0 and substitute (C. 2) into the commutation relation
[ρ+ (q) , ρ+ (−q)] = A2 bq , b† = A2 =
q
q
q
Aq

=

qL
.
2π

qL
→
2π

(C. 2)

Fundamental aspects of electron correlations and quantum transport

101

Appendix C.2.1. Commutation relations for bosonic fields ϕ and ϑ
Using
ϕ (x)
ϑ (x)

= −i

1
−∞<q<∞

= i
= −

1

2 |q| L

−∞<q<∞

ϑ′ (x)

2 |q| L

sgnq bq eiqx − b† e−iqx ;
q

eiqx bq − b† e−iqx ;
q

1
q eiqx bq + b† e−iqx ,
q
2 |q| L

−∞<q<∞

we find
1

1

[ϕ (x) , ϑ′ (x)] = i

2 |q| L

q,q′

′

2 |q| L

|q ′ |
′

′

′

× bq eiqx − b† e−iqx , bq′ eiq x − b†′ eiq x
q
q

′

=2δq,−q′

= i

1
L

′

q

eiq(x−x ) = iδ (x − x′ ) .

Appendix C.3. Problem with backscattering
As it was pointed out in the main text, straightforward bosonization of the Hamiltonian for the spinless case encounters a problem if one tries to account for
backscattering. As backscattering (g1 ) is just an exchange process to forward
scattering of fermions of opposite chiralities (g2 ), the Luttinger liquid parameters
with g1 = 0 should be obtained from those with g1 = 0 by a simple replacement:
g2 → g2 − g1 . However, if we do this, we cannot satisfy the Pauli principle which
says that for a contact interaction, when g2 = g4 = g1 , all the interaction effects
should disappear. Indeed, Eqs.6.2) and (6.11) for u and K, correspondingly,
change to
u2

=

K2

=

1+
1+
1+

g4
2π

2

g2 − g1
2π

−

g4 −g2 +g1
2π
g4 +g2 −g1
2π

.

2

;

102

Dmitrii L. Maslov

For contact interaction, when g2 = g4 = g1 , we get
u2

=

K

=

1+

g
2π

2

=1

1.

The charge velocity is different from 1. In addition, the product uK is renormalized from unity–this is also a problem, as it means that the current operator is
renormalized by the interactions. How to fix this problem? Ref. [77] shows how
to arrive at the expressions for u and K which satisfy all necessary constraints
just on the basis on Galilean invariance and dimensional analysis. Ref. [78] arrives at the same result by using a careful point-splitting of the operators. Here, I
present the method of Ref. [78].
Recall that the density operator, represented in terms of bosonic fields, contains not only the lowest harmonic (q → 0), corresponding to long-wavelength
excitations, but also harmonics oscillating at q = 2kF , 4kF , etc. Indeed, taking
into account only the 2kF − oscillations, we have
ρ (x)

†
†
ψ+ (x) e−ikF x + ψ− (x) eikF x

=
=

ψ+ (x) eikF x + ψ− (x) e−ikF x

†
†
†
ψ+ (x) ψ+ (x) + ψ− (x) ψ− (x) + e−2ikF x ψ+ (x) ψ− (x) + H.c.

The first term in this equation has to be treated using the point-splitting procedure, because it involves two fermionic operators at the same point. The result is
an infinite constant, ρ0 , which is just a uniform density, plus the gradient term.
The 2kF -component can be bosonized without a problem, as it involves products
of different fermions. The result is
√
1
1
exp 2 πϕ + 2kF x + H.c.
ρ (x) − ρ0 = √ ∂x ϕ +
π
2πα
Using this expression for the interaction part of H, we have
Hint

1
dx dx′ V (x − x′ ) [ρ (x) − ρ0 ] [ρ (x′ ) − ρ0 ]
2
= H F + HB ,
=

where the forward and backscattering parts of the Hamiltonian are given by
HF

=

1
2π

HB

=

1
2

dx′ V (x − x′ ) ∂x ϕ∂x′ ϕ;

dx
1
2πa

2

dx

dx′ V (x − x′ )

√
√
′
× exp 2i πϕ (x) exp −2i πϕ (x′ ) e2ikF (x−x ) + H.c .

Fundamental aspects of electron correlations and quantum transport

103

In HB , we neglected the terms that oscillate with x, x′ ,and x + x′ , and kept only
those terms that oscillate with x−x′ . As our potential is sufficiently short-ranged,
the oscillations of the first group of terms will average out, whereas the second
group will survive. Introducing new coordinates
R
r

x + x′
;
2
x − x′ ,

≡
≡

and assuming that |R| ≫ |r| , the forward-scattering part of the Hamiltonian
reduces to
HF =

1
2π

dR (∂R ϕ)

2

drV (r) =

V (0)
2π

2

dR (∂R ϕ) .

The product of the two exponentials needs to be evaluated with care. Applying
the Baker-Hausdorff identity
eA eB =: eA+B : e

AB− 1 A2 − 1 B 2
2
2

,

we get
√
√
exp 2i πϕ (x) exp −2i πϕ (x′ )

=

√
exp 2i π (ϕ (x) − ϕ (x′ )) :

× exp[4π ϕ (x − x′ ) ϕ (0) − ϕ2 (0)],

Using the expression for the free bosonic propagator
ϕ (x − x′ ) ϕ (0) − ϕ2 (0)] =

1
a2
,
ln
4π (x − x′ )2

and expanding in r = x − x′ under the normal-ordering sign, we obtain
√
√
1
a2
2
2
exp 2i πϕ (x) exp −2i πϕ (x′ ) = − 4π (∂R ϕ) r2 2 = −2π (∂R ϕ) a2 .
2
r
(While expanding, we neglected the first derivative of ϕ which can be always
eliminated by choosing appropriate boundary condition.). HB reduces to
2

HB

=
=
=

1
1
2
2πa2 dR (∂R ϕ)
drV (r) 2 cos 2kF r
2 2πa
1
2
dR (∂R ϕ)
−
drV (r) cos 2kF r
2π
V (2kF )
dR (∂R ϕ)2 .
−
2π
−

104

Dmitrii L. Maslov

Therefore, the bosonized form of the total Hamiltonian
Hint =

V (0) − V (2kF )
2π

dR (∂R ϕ)

2

manifestly obeys the Pauli principle. The Luttinger-liquid parameters are now
given by
u=

1+

V (0) − V (2kF )
; K=
2π

1
1+

V (0)−V (2kF )
2π

.

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