Quantum interference and spin-charge separation
in a disordered Luttinger liquid
A.G. Yashenkin1,2 , I.V. Gornyi1,3 , A.D. Mirlin1,4,2 , and D.G. Polyakov1
1

arXiv:0808.1093v1 [cond-mat.str-el] 7 Aug 2008

Institut f¨r Nanotechnologie, Forschungszentrum Karlsruhe, 76021 Karlsruhe, Germany
u
2
Theory Division, Petersburg Nuclear Physics Institute, Gatchina, 188300 St. Petersburg, Russia
3
A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia
4
Institut f¨r Theorie der kondensierten Materie, Universit¨t Karlsruhe, 76128 Karlsruhe, Germany
u
a
(Dated: February 23, 2013)
We study the influence of spin on the quantum interference of interacting electrons in a singlechannel disordered quantum wire within the framework of the Luttinger liquid (LL) model. The
nature of the electron interference in a spinful LL is particularly nontrivial because the elementary bosonic excitations that carry charge and spin propagate with different velocities. We extend
the functional bosonization approach to treat the fermionic and bosonic degrees of freedom in a
disordered spinful LL on an equal footing. We analyze the effect of spin-charge separation at finite temperature both on the spectral properties of single-particle fermionic excitations and on
the conductivity of a disordered quantum wire. We demonstrate that the notion of weak localization, related to the interference of multiple-scattered electron waves and their decoherence due to
electron-electron scattering, remains applicable to the spin-charge separated system. The relevant
dephasing length, governed by the interplay of electron-electron interaction and spin-charge separation, is found to be parametrically shorter than in a spinless LL. We calculate both the quantum
(weak localization) and classical (memory effect) corrections to the conductivity of a disordered
spinful LL. The classical correction is shown to dominate in the limit of high temperature.
PACS numbers: 71.10.Pm, 73.21.-b, 73.63.-b, 73.20.Jc

I.

INTRODUCTION

Interacting electrons in one dimension (1D) are
a paradigmatic example of the strongly correlated
fermionic systems. Electron-electron (e-e) interactions
drive the 1D system into a non-Fermi liquid state known
as the Luttinger liquid (LL) (for review see, e.g., Refs. 1,
2,3,4,5,6,7). In recent years, progress in nanofabrication
technologies has made it possible to manufacture a variety of single- and few-channel quantum wires connected
to the electric leads and to perform systematic transport measurements on the very narrow wires. The latter include single–wall carbon nanotubes8,9,10,11,12,13,14 ,
semiconductor-based15,16,17,18 and metallic19 quantum
wires, polymer nanofibers20 , as well as quantum Hall
edge states.21,22 The LL nature of these strongly correlated quantum wires has been supported by a wealth of
experimental findings.8,9,10,11,12,13,14,15,16,19,20,21,22 Another class of strongly-correlated quantum wires that has
recently attracted a lot of interest is ultracold atomic
gases confined to 1D geometry, for review see Ref. 23.
Mesoscopic physics of strongly correlated electrons is
one of the most important and promising directions of
current research on 1D electron systems. Recent transport measurements on carbon nanotubes24 reported both
sample-dependent conductance fluctuations and strong
magnetoconductivity, in qualitative similarity to the
mesoscopic phenomena in higher-dimensional disordered
electron systems. In Ref. 24, the sample size ∼ 1 µm
was of the order of the mean free path limited by impurity scattering. Electron transport through the nanotubes displayed, therefore, features characteristic of

the crossover from ballistic conduction to a disorderdominated regime. On the other hand, in the past few
years techniques to grow nanotubes of size up to the millimeter scale25,26 —much larger than the typical value of
the disorder-induced mean free path—have been developed. First transport measurements25,26 on the ultralong nanotubes provided evidence for disorder-induced
diffusive motion of electrons in a wide range of temperature. Altogether, these advances have paved the way for
systematic experimental study of interference-induced localization phenomena in 1D.
The theory of weak localization (WL) in a disordered
LL of spinless electrons was developed in Refs. 27 and
28 (for recent advances in the ballistic σ-model framework, see also Ref. 29). In the limit of weak interaction between electrons, the phase breaking length lφ that
sets up the infrared cutoff of WL was shown to obey
lφ ∼ α−1 (lT l)1/2 , where α ≪ 1 is the dimensionless interaction constant, lT ∼ v/T the thermal length, v the
Fermi velocity, T the temperature, l the elastic mean free
path. At sufficiently high temperatures, when lφ ≪ l,
the system is in the WL regime.30 The WL correction
∆σWL to the Drude conductivity σD behaves then as
∆σWL /σD ∼ −(lφ /l)2 ln(l/lφ ) ∼ −α−2 (lT /l) ln(α2 l/lT ).
Note that the WL dephasing length of spinless electrons
originates from the interplay of interaction and disorder:
both ingredients are necessary to establish a nonzero dephasing rate. As a result, lφ in the WL regime is much
longer than the total decay length of fermionic excitations
lee ∼ α−2 lT . The latter is the dephasing length relevant
to the damping of Aharonov-Bohm oscillations27,28,31,32
and to the smearing of a zero-bias anomaly in the tunnelling density of states.33

2
In a spinful LL, the most prominent spin-related manifestation of non-Fermi liquid physics is a phenomenon of
spin-charge separation (SCS) (see, e.g., Refs. 1,2,3,4,5,6,
7). The essence of the SCS in the LL model is that the
spin and charge sectors of the theory in a bosonic representation are completely decoupled from each other and
characterized by different interaction coupling constants.
Correspondingly, the elementary bosonic excitations carrying spin and charge propagate with different velocities, independently of each other. Experimentally, the
effect of the SCS on the spectral properties of a LL (as
measured in electron tunnelling experiments) has been
studied in Refs. 15,17. In the last few years, much attention has been given to the so-called “spin-incoherent
regime”34 in 1D systems with strongly different spin and
charge velocities.35,36,37 For recent transport measurements in ballistic quantum wires which show signatures
of the spin-incoherent behavior, see Ref. 18 and references therein.
While the difference in the spectral properties of spin
and charge collective modes is not at all specific to 1D
and is also characteristic of higher-dimensional Fermi
liquids,38 the peculiarity of 1D is that the “factorization” of the bosonic modes modifies the single-particle
fermionic properties in an essential way.39 Our purpose
is to investigate how the SCS affects the quantum interference phenomena in a disordered LL. In particular, we
analyze the dynamical properties of fermionic excitations
at finite temperature and employ the results of this analysis to calculate the WL correction to the conductivity
of a spinful LL. The problem is rather nontrivial conceptually since the spin and charge degrees of freedom
that constitute the electron in a LL acquire different velocities and the very notion of a specific quasiclassical
electron trajectory characterized by a certain velocity—
conventionally invoked in a description of WL—becomes
ambiguous.
We show below that, despite the intricate nature of
a single-electron motion in a spinful LL, the basic notions of WL remain applicable even when the SCS is
incorporated in the calculation. However, the spin degree of freedom has dramatic consequences for the effects
of e-e scattering. Most importantly, the decay rate of
fermionic excitations is strongly enhanced in the presence
of spin: the single-particle length lee ∼ α−1 lT becomes
parametrically shorter than for spinless (spin-polarized)
electrons. Moreover, in contrast to the spinless case, the
WL dephasing length lφ ∼ α−1 lT becomes of the order of lee . As a result, also the WL dephasing turns
out to be much stronger than without the SCS being
included. The temperature at which spinful electrons
get strongly localized (lT /l ∼ α) is therefore much lower
(for small α) than for spinless electrons. Furthermore,
we find that the WL correction to the conductivity is
given by ∆σWL /σD ∼ −(lφ /l)2 ∼ −α−2 (lT /l)2 , showing
a much faster temperature dependence of ∆σWL than in
the spinless case.
We also demonstrate that a classical “memory effect”

(ME) in the electron scattering off disorder contributes to
the T dependence of the conductivity. Moreover, it gives
the leading (larger than ∆σWL ) correction to the Drude
conductivity in the limit of high T . The obtained ME
contribution ∆σME /σD ∼ −lT /l is essentially not related
to e-e interactions and exceeds ∆σWL only when the latter is sufficiently suppressed by the interaction-induced
dephasing. Specifically, |∆σME | ≫ |∆σWL | for lT /l ≪ α2
(i.e., for lφ /l ≪ α). Otherwise, |∆σWL | ≫ |∆σME |.
Technically, the ME manifests itself in the same set of
Feynman diagrams for the conductivity as the WL, so
that we actually treat the two effects—the essentially
classical ME and the essentially quantum WL—on an
equal footing. What distinguishes them from each other
is that the main contributions to ∆σWL and ∆σME come
from scattering on different impurity configurations. The
WL correction stems from scattering on rare, compact
three-impurity complexes in which the characteristic distance between all three impurities is of the order of the
single-particle length lee . The ME correction is associated with impurity configurations in which two of impurities are located very close to each other, with a characteristic distance between them of the order of the thermal
length lT .
The remainder of the paper is organized as follows. In
Sec. II we formulate the model of a disordered LL, discuss
the basic physics of the SCS relevant to the e-e scattering,
and describe the method of functional bosonization used
in our calculation. Section III is devoted to an analysis of
the spectral properties of fermionic excitations in various
representations; in particular, in a “space-energy representation” employed for the calculation of the conductivity. In Secs. IV and V we evaluate the WL and ME corrections to the conductivity, respectively, by means of the
functional bosonization method. In Sec. VI we present a
complementary analysis based on the more conventional
path-integral approach. Our results are summarized in
Sec. VII.

II.

MODEL AND METHOD

We begin by formulating the model of a disordered
spinful LL in Sec. II A. In Sec. II B we present a simple
argument which demonstrates the peculiarity of 1D geometry in that the spin degree of freedom affects the rate
of e-e scattering in the LL in a crucial way. Section II C
is devoted to an overview of the functional bosonization
method.

A.

Disordered Luttinger liquid

Throughout the paper we consider a single-channel infinite quantum wire. Linearizing the dispersion relation
of electrons about two Fermi points at the wavevectors
k = ±kF with the velocity v, the Hamiltonian of a clean

3
LL is written as ( = 1)
HLL =
kµσ

+

1
2

v(µk − kF )Ψ† (k)Ψµσ (k)
µσ
dx nµσ g4 nµσ′ + nµσ g2 n−µ,σ′ .

µσσ′

where the backscattering amplitudes Ub (x) are correlated
∗
as Ub (x)Ub (x′ ) = U (x)U (x′ ) and Ub (x)Ub (x′ ) = 0.
Forward scattering off impurities can be gauged out in
the calculation of the conductivity7,43 and will therefore
be neglected from the very beginning. The total Hamiltonian H that defines our model of a disordered spinful
LL is thus

(2.1)
Here Ψµσ (k) are the electron operators at the wavevector
k, the index µ = ± denotes two branches of chiral excitations (right and left movers), and σ =↑, ↓ stands for
two spin projections. The e-e interaction enters Eq. (2.1)
through the coupling constants g4 and g2 . These describe
forward e-e scattering with small momentum transfer
(much smaller than kF ) between electrons from the same
(g4 ) or different (g2 ) chiral branches. We assume that the
e-e interaction is short-ranged and represent the interaction part of the Hamiltonian in terms of the local in space
†
electron density operators nµσ (x) = ψµσ (x)ψµσ (x). The
parameters of the Hamiltonian with the linearized dispersion should be understood as effective (phenomenological) couplings of the low-energy theory, which include
possible high-energy renormalization effects, similar to
Fermi-liquid theory.
The LL Hamiltonian (2.1) does not contain the term
Hbs =

1
2

†
†
dx ψµσ ψ−µ,σ g1 ψ−µ,σ′ ψµσ′ ,
µσσ′

†
†
∗
dx Ub ψ+σ ψ−σ + Ub ψ−σ ψ+σ , (2.3)
σ

(2.4)

Throughout the paper we consider a quantum wire
with spin and chiral channels not separated spatially in
the transverse direction, so that the constants g2 = g4 ≡
g are spin-independent and equal to each other. The
plasmon velocity u for the spinful case then reads
u = v/Kρ = v(1 + 2g/πv)1/2 ,

(2.5)

with Kρ being the Luttinger constant in the charge sector, whereas the velocity of elementary spin excitations is
equal to v. It is convenient to characterize the strength of
e-e interaction by the dimensionless coupling constant28
2
2
α = (1 − Kρ )/(1 + 3Kρ ), which in the limit of weak
interaction α ≪ 1 is written as
α ≃ (1 − Kρ )/2 ≃ g/2πv .
B.

(2.2)

which describes backward e-e scattering with large momentum transfer resulting in a change of chirality µ. If
one begins with a microscopic model of electrons interacting via a finite-range externally screened Coulomb potential, the constants g2,4 in Eq. (2.1) and g1 in Eq. (2.2)
are related to the Fourier transforms of this potential
at zero and 2kF momenta, respectively. The forward
scattering dominates provided that the radius of external
screening d (e.g., the distance to a metallic gate) is much
−1
larger than kF . We assume that this is the case and
neglect Hbs throughout the paper below.40 Treating the
Coulomb potential in Eq. (2.1) as short-ranged is legitimate for scattering processes with momentum transfer
much smaller than d−1 . This same scale d−1 fixes the
ultraviolet momentum cutoff in our formulation of the
low-energy theory.
The only source of electron backscattering in our model
is thus a static random potential U (x) due to the presence
of impurities. We assume that fluctuations of U (x) are
Gaussian and characterized by the correlation function
U (x)U (x′ ) = δ(x − x′ )v 2 /2l0 (“white noise”). Here l0
is the transport elastic mean free path in the absence
of interaction. The disorder is considered to be weak,
kF l0 ≫ 1. The disorder-induced backscattering term in
the Hamiltonian is given by
Himp =

H = HLL + Himp .

(2.6)

Why spin matters

To qualitatively understand the nature of dephasing of
fermionic excitations in a spinful LL, it is instructive to
recall the perturbative expansion6,27 of the self-energy of
the single-particle Green’s function in the limit of weak
interaction α ≪ 1 and discuss the e-e scattering rate at
the Golden-rule level, first in the absence of disorder. For
the spinless case, such an analysis has been performed
in Refs. 27,28, and 32—see also Refs. 27,28,31,32 and
Refs. 27,32 for a closely related calculation of the temporal decay of the single-particle Green’s function for spinless and spinful electrons, respectively. Since the physics
of dephasing is governed by inelastic e-e scattering, a natural first step is to calculate at lowest (second) order in
−1
α the e-e scattering rate τee given by the imaginary part
of the self-energy.
The Golden-rule expression for the e-e collision rate
reads
1
=
GR
τee (ε)

h
h
h
dωdε′ K(ω) fε−ω fε′ fε′ +ω + fε−ω fε′ fε′ +ω ,

(2.7)

where
H
H
K(ω) = ηs [ K++ (ω) + K+− (ω) ] + K F (ω)

(2.8)

is the kernel of the e-e collision integral and fε is the
h
Fermi distribution function, fε = 1 − fε . In Eq. (2.8),
H
2
H
2
the Hartree terms K++ ∝ g4 and K+− ∝ g2 are related
to scattering of two electrons from the same (++) or difH
ferent (+−) chiral branches, respectively, K F = −K++

4
H
is the exchange counterpart of K++ , ηs is the spin degeneracy, ηs = 1 for the spinless case and ηs = 2 for the
spinful case.
To order O(α2 ), the Golden-rule scattering rate and
the self-energy on the mass shell coincide with each other.
At the Fermi level (ε = 0) we have

1
GR
τee (0)

= −2 ηs (ImΣH + ImΣH ) + ImΣF , (2.9)
++
+−

where the Hartree terms are given by
π
ImΣH = − α2 v
+±
2
×

dω ω coth

ω
ω
− tanh
2T
2T

dq δ(ω − vq)δ(ω ∓ vq)

(2.10)

and the exchange term ΣF = −ΣH . Peculiar to 1D are
++
highly singular contributions to K(ω) related to scattering of electrons moving in the same direction. One
sees that the contribution of ΣH contains a δ-function
++
squared and thus diverges.6,27,28 The divergency of the
perturbative expression for the probability of scattering
of two electrons of the same chirality simply means that
the energy and momentum conservation laws for this kind
of scattering give a single equation ω − vq = 0.
For spinless (spin-polarized) electrons, the divergency
in ImΣH is canceled by the same divergency in the ex++
change term. The remaining term ImΣH yields27,28,32
+−
1
= α2 πT .
GR
τee (0)

(2.11)

Note that for scattering of electrons from different chiral branches on each other, the energy and momentum
conservation laws lead to two equalities: ω − vq = 0 and
ω +vq = 0, which combine to give ω, q = 0 for allowed energy and momentum transfers. This “quasi-elastic”27,28
character of e-e scattering is a peculiarity of 1D: in higher
dimensionalities, the characteristic energy transfer that
−1
determines τee in a clean system is of order T .
For spinful electrons, the Fock contribution cancels
only the part of the Hartree term ImΣH that comes
++
from interaction between electrons with the same spin.
The divergent second-order Hartree term that arises from
interaction between electrons with opposite spins remains
uncompensated. This indicates that the main contribu−1
tion to τee is now related to scattering of electrons from
the same chiral branch. Thus, already the perturbative
expansion demonstrates6,27,28 a qualitative difference between the cases of spinless and spinful electrons.
In fact, for spinful electrons, the perturbative expan−1
sion of τee in powers of α is diverging in the clean limit
at each order. We will analyze the finite-T damping of
the single-particle Green’s function for α ≪ 1 in Sec. III.
GR
Here, we stick to the calculation of 1/τee within a “generalized Golden-rule” scheme. The term “generalized”
means that we go beyond second order in α by introducing the dynamically screened e-e interaction V (ω, q)—
which is exactly44 given by the random phase approximation (RPA) [see Eq. (2.25) below]. The second δ-function

in the integrand of Eq. (2.10) comes precisely from the
imaginary part of the retarded propagator ImV (q, ω) if
one takes the propagator at second order in α,
ImV (q, ω) ≃ −(2πα)2 ωv[ δ(ω − vq) + δ(ω + vq) ] .
(2.12)
Using the full RPA propagator ImV (q, ω) ∝ [ δ(ω − uq) +
δ(ω + uq) ], the leading at α ≪ 1 Golden-rule expression
for the e-e scattering rate of spinful electrons is written
as
1
≃ 2πα2 vT
GR
τee (0)

dω

dq δ(ω − vq)δ(ω − uq) , (2.13)

which only differs from −2ImΣH in Eq. (2.10) in that
++
one of the δ-functions has a shifted velocity v → u. Note
GR
that, similarly to the spinless case, 1/τee (0) in Eq. (2.13)
is determined by ω, q = 0. The relative shift between the
arguments of the δ-functions makes the expression for
GR
1/τee (0) finite:
1
v
≃ 2πα2 T
≃ |α|πT ,
GR
τee (0)
|u − v|

(2.14)

where we used Eq. (2.5) for u − v ≃ 2αv for small α.
Remarkably, the e-e scattering rate for spinful electrons
turns out to be of first order in α, in contrast to the
spinless case (2.11), where it is of order α2 .
As we will see below in a more consistent treatment
which does not rely on the generalized Golden-rule apGR
proach, the scattering rate 1/τee (0) gives a characteristic decay rate for single-particle excitations and also the
characteristic dephasing rate for WL. What the above
consideration teaches us is that in 1D the spin degree
of freedom strongly enhances the e-e scattering rate for
weakly interacting electrons. The parametric difference
between the spinless and spinful cases is in stark contrast to higher dimensionalities, where taking spin into
account typically yields for relaxation rates in a clean
system only numerical factors of order unity.
C.

Functional bosonization

The method we use here to study the quantum interference in a disordered spinful LL is functional bosonization.
It was introduced for the clean LL model in Refs. 45,46
and further developed in Refs. 47,48,49,50. In the earlier work27,28 by three of us, the functional bosonization
framework was extended to deal with disordered problems and applied to study the transport properties of
a disordered spinless LL. In this subsection we present
a brief outline of the formalism (for more details see
Sec. VII in Ref. 28), highlighting the differences between
the spinless and spinful cases.
In contrast to “full bosonization”, conventionally used
for a theoretical description of the LL, the functional
bosonization technique preserves both fermionic (electrons) and bosonic (collective excitations – plasmons,

5
spinons) degrees of freedom.
This feature of the
method is of great advantage when one has to deal
with interacting problems which are most naturally described in terms of fermionic excitations, e.g., quantum interference (Refs. 27,28 and the present work) or
nonequlibrium33,51,52 phenomena in a LL. In particular, the functional bosonization allows for a straightforward treatment of e-e interaction while residing in the
fermionic basis, which is especially cost-efficient in the
disordered case.
The key steps in setting up the formalism for a LL at
thermal equilibrium are as follows:
• a conventional Hubbard-Stratonovich decoupling of
the four-fermion interaction term in the Matsubara action is performed by means of introducing a
bosonic field ϕ(x, τ );
• interaction of fermions with the field ϕ is gauged
out by means of a local transformation (µ = ±)
ψµσ (x, τ ) → ψµσ (x, τ ) exp [ i θµ (x, τ )],

(2.15)

where the phase θµ (x, τ ) obeys
(∂τ − i µ v ∂x ) θµ (x, τ ) = ϕ(x, τ ).

(2.16)

This transformation completely eliminates the coupling between the fermionic and bosonic fields from
the action (this property is peculiar to 1D);
• upon this transformation, the bosonic part of the
action remains Gaussian. It is this point at which
the peculiarity of the LL model—the exactness of
the RPA—comes into play. The correlation function of the field ϕ is given by
ϕ(x, τ )ϕ(0, 0) = V (x, τ ),

(2.17)

where V (x, τ ) is the dynamically screened interaction [see Eq. (2.25) below];
• an arbitrary time-ordered fermionic average is expressed through a product of free electron Green’s
functions and Gaussian averages of the phase factors exp[iθµ (x, τ )] (taken at different space-time
points). The bosonic averages are represented in
terms of the correlation functions
Bµν (x, τ ) = [θµ (0, 0) − θµ (x, τ )] θν (0, 0)

(2.18)

which are related to the Fourier component
V (q, iΩn ) of the interaction propagator (2.17) as
B+± (x, τ ) = T
n

dq iqx−iΩn τ
e
−1
2π

V (q, iΩn )
,
(vq − iΩn )(±vq − iΩn )
B−− (x, τ ) = B++ (−x, τ ) ,
B−+ (x, τ ) = B+− (x, τ ) .

Here Ωn = 2πnT is the bosonic Matsubara frequency;
• while calculating observables (closed fermionic
loops), e-e interaction is completely accounted
for by attaching the fluctuating gauge factors to
backscattering vertices. If the number of fermionic
loops is larger than one, each of them has to contain
at least one pair of backscattering vertices in order not to be disconnected (Wick’s theorem for the
functional bosonization diagrammatic technique).

e

-iθ+(1)

e

+iθ+(2)

e-B++(x2-x1,τ2-τ1)

+
1

+
2

1

2

FIG. 1: The Green’s function of a right mover propagating
between space-time points 1 = (x1 , τ1 ) and 2 = (x2 , τ2 ) before (a) and after (b) averaging over fluctuations of the gauge
factors. Solid line: the bare Green’s function. The wavy
lines at the end points represent the factors exp[−iθ+ (1)] and
exp[iθ+ (2)]. The wavy line connecting points 1 and 2 denotes
averaging over fluctuations of θ+ (1) and θ+ (2).

As a simple example, consider the single-particle
Green’s function for, say, a right mover G+ (x, τ ). Upon
gauge transformation (2.15), one gets the free Green’s
function g+ (x, τ ) dressed by two phase factors as shown
in Fig. 1(a). Pairing of the two bosonic fields yields
[Fig. 1(b)]
G+ (x, τ ) = g+ (x, τ ) exp [−B++ (x, τ )] .

(2.20)

More complex quantities are calculated in a similar
way. Each impurity backscattering vertex at spacetime point N generates a phase factor of the type
exp{±i[θ+(N ) − θ− (N )]}, as illustrated in Fig. 2. Upon
averaging, the phase factors are paired in all possible
ways (Fig. 3). In closed fermionic loops, the correlators
Bµν (x, τ ) [Eq. (2.18)] only appear in the combination
M (x, τ ) = B++ (x, τ ) + B−− (x, τ ) − 2B+− (x, τ ) .
(2.21)
As a result, each pair of backscattering vertices at points
(xN , τN ) and (xN ′ , τN ′ ) contributes either the factor
Q(x, τ ) = exp[M (x, τ )] ,

(2.22)

where x = xN − xN ′ , τ = τN − τN ′ , or Q−1 (x, τ ), depending on whether chirality of incident electrons at the
vertices is the same (Q) or different (Q−1 ).
The RPA dynamically screened interaction V (q, iΩn )
obeys

×

(2.19)

V −1 (q, iΩn ) = g −1 + Π(q, iΩn ) ,

(2.23)

6

e-iθ+(1)

ei[θ+(2)-θ-(2)]

where

eiθ-(3)

(πT /Λ)2
,
sinh[ πT (x/u + iτ ) ] sinh[ πT (x/u − iτ ) ]
v sinh[πT (x/v + iτ )]
(2.27)
η(x, τ ) =
u sinh[πT (x/u + iτ )]
ς(x, τ ) =

+

--

1

2

3

FIG. 2: Backscattering of a right mover off an impurity (denoted by a cross) at point 2. The impurity vertex is dressed
by a local gauge factor exp{i[θ+ (2) − θ− (2)]} which contains
two fluctuating fields of different chirality.

1

B--(3,2)
2

+

--

B+-(2,1)

αb =

3

B+-(3,2)

FIG. 3: Backscattering off an impurity as shown in Fig. 2
after averaging over fluctuating bosonic fields. Each of the
wavy lines represents a factor of the type exp(±Bµν ).

u2 − v 2
.
2uv

(2.28)

2 2

2
v q
,
πv v 2 q 2 + Ω2
n

(2.24)

v 2 q 2 + Ω2
n
u 2 q 2 + Ω2
n

(2.25)

which gives

with u from Eq. (2.5).
As will be seen below, it suffices, when calculating the
dephasing rate for weak localization, to deal with the
ballistic interaction propagator (2.25) which does not
include backscattering of electrons off disorder. This
should be contrasted with the spinless case, where the
dephasing of localization effects is absent altogether unless the disorder-induced damping of V (q, iΩn ) is taken
into account27,28 (“dirty RPA”).
Substituting Eq. (2.25) in Eqs. (2.19) yields
1
αb
B++ (x, τ ) = − ln η(x, τ ) −
ln ς(x, τ ) ,
2
4
αr
(2.26)
B+− (x, τ ) = − ln ς(x, τ ) ,
4

(2.29)

since αb is quadratic in α, whereas αr is linear. Using
this hierarchy will greatly simplify the calculation below.

III. SINGLE-PARTICLE SPECTRAL
PROPERTIES AND SPIN-CHARGE
SEPARATION

where Π(q, iΩn ) is the polarization operator. In a spinful
LL, the latter is written as

V (q, iΩn ) = g

αr =

αb ≪ αr ≪ 1 ,

B+-(2,2)

Π(q, iΩn ) =

(u − v)2
,
2uv

Inspecting Eqs. (2.26) and (2.27), we see that there is
an extra factor of 1/2 in front of both B++ and B+− as
compared to the spinless28 case. It is this factor that is
responsible for the SCS.
The exponents αb and αe = αr +αb determine a powerlaw suppression of the tunneling density of states (zerobias anomaly) for tunneling in the bulk and in the end of
a LL, respectively (see, e.g., Refs. 1,2,3,4,5,6,7). In the
rest of the paper, we will treat the interaction strength
α as a small parameter. The hierarchy of the constants
(2.28) is then as follows

B+-(3,1)

B++(2,1)

and the constants αb and αr are given by

In this section, we use the functional bosonization formalism to study the single-particle spectral properties
of electrons at finite temperature. We begin with the
space-time representation in Sec. III A. Then we transform to a “mixed” space-energy (Sec. III B) and the
momentum-energy (Sec. III C) representations, using approximations appropriate in the weak-interaction limit
α ≪ 1. The analysis of the various representations of the
single-particle Green’s function will serve as a starting
point for the calculation of the WL and ME terms in the
conductivity in Secs. IV and V.

A.

Green’s function in the (x, τ ) representation:
Weak-interaction approximation

In the absence of interaction, the single-particle
Green’s function of right (+) and left (−) movers g± (x, τ )
is given by
g± (x, τ ) = ∓

iT
1
.
2v sinh[πT (x/v ± iτ )]

(3.1)

Plugging Eqs. (2.26), (2.27), and (3.1) into Eq. (2.20), we
have for the Green’s function of right movers in a spinful

7
LL:
G+ (x, τ ) = −

i
√
2π uv

πT
πT
sinh[πT (x/v + iτ )] sinh[πT (x/u + iτ )]

1/2

×

πT /Λ
πT /Λ
sinh[πT (x/u + iτ )] sinh[πT (x/u − iτ )]

αb /4

×

,
(3.2)

in agreement with the result obtained by conventional
bosonization (see, e.g., Refs. 2,3,4,5,6,7) and by purely
fermionic methods (see, e.g., Ref. 1). The Green’s function of left movers G− (x, τ ) = G+ (−x, τ ). The analytical structure of G+ (x, τ ) in the complex plane of τ
(0 < Re τ < 1/T ) is shown in Fig. 4 for positive x. There
are three branch points at τ = ix/u, ix/v, and −ix/u.
One way to choose branch cuts is shown in the top left
panel of Fig. 4: those starting at τ = ix/u and ix/v are
sent upwards, whereas that starting at τ = −ix/u is sent
downwards. If α ≪ 1, the first two cuts are much different from the third one. In the limit of small α, the cut
that connects the points τ = ix/u and ix/v corresponds
to an almost square-root singularity, so that the main
change the Green’s function experiences when crossing
this cut is a change of sign (“strong cut”). On the other
hand, the cut that goes from −ix/u to −i∞ is “weak” in
the sense that the discontinuity of G+ (x, τ ) across this
cut is proportional to αb ∼ α2 ≪ 1. Similarly, crossing the axis of imaginary τ between τ = ix/v and i∞ is
associated with a weak discontinuity.
The main approximation we make in this paper consists in sending αb to zero everywhere in the calculation
while keeping the effects of leading (linear) order in the
interaction strength. That is, below we retain the difference between u and v, Eq. (2.5),
u ≃ v(1 + 2α) ,

α≪1,

(3.3)

as the only effect of e-e interaction.53 The Green’s function G+ (x, τ ) then reads
G+ (x, τ ) ≃ −
×

i
√
2π uv

πT
πT
sinh[πT (x/v + iτ )] sinh[πT (x/u + iτ )]

1/2

.
(3.4)

The velocities u and v in Eq. (3.4) coincide with the velocities of the elementary collective excitations (plasmons
and spinons). The appearance of the two velocities in the
single-particle correlator signifies SCS. Within the approximation (3.4), the two velocities enter the fermionic
Green’s function in a symmetric way. The analytical
structure of G+ (x, τ ) in Eq. (3.4) is simplified to a single
square-root cut between the points τ = ix/u and ix/v,
as illustrated in the center top panel of Fig. 4. We will

FIG. 4: Left top panel: the analytical structure of the rightmover Green’s function G+ (x, τ ) in the complex plane of the
Matsubara time τ . The bold solid and the dashed lines represent “strong” and “weak” branch cuts, respectively [see the
text below Eq. (3.2)]. Central top panel: within the approximation (3.4), only the square-root branch cut between the
points τ = ix/u and ix/v survives. Last three panels: the
contour transformation used to calculate the Green’s function in the space-energy representation in Sec. III B.

use the approximation (3.4), which captures the essential
physics of SCS, throughout the paper below.
It is instructive to compare Eq. (3.4) with the Green’s
function of spinless electrons Gsl (x, τ ) (see, e.g., Refs. 1,
+
2,3,4,5,6,7,28). Since the correlator B++ (x, τ ) in the
spinless case is twice as large, Gsl (x, τ ) is given by
+
Gsl (x, τ ) = −
+
×

πT
i
2πu sinh[πT (x/u + iτ )]

πT /Λ
πT /Λ
sinh[πT (x/u + iτ )] sinh[πT (x/u − iτ )]

αb /2

.
(3.5)

One sees that in the absence of spin, interaction leads to
a replacement of the velocity v → u in the bare Green’s
function and generates two brunch cuts: one with an exponent close to 1 and the other with the small exponent
αb /2. It follows that for spinless electrons the approximation αb → 0, analogous to Eq. (3.4), would eliminate
all dephasing effects—since the latter only originate from
the factors in the second line of Eq. (3.5). By contrast,
dephasing in the spinful case arises already at order O(α),
as will be shown below. Consequently, the approximation (3.4) allows us to obtain—in a controllable way—
analytical results valid in the limit α ≪ 1.
Let us now identify two important spatial scales. For
this purpose, we perform the Wick rotation τ → i(t + i0)
in Eq. (3.4). For large |x/u−t|, |x/v−t| ≫ 1/T , Eq. (3.4)
yields
G+ (x, it) ∝ exp [−πT (|x/u − t| + |x/v − t|) /2] . (3.6)

8
Within the interval x/u < t < x/v, the Green’s function
given by Eq. (3.6) decays as
G+ (x, it) ∝ exp (−x/2lee ) ,

(3.7)

independently of t, whereas outside this interval the
Green’s function is suppressed much more strongly; in
particular, at t = 0:
G+ (x, 0) ∝ exp (−x/2lT ) ,

(3.8)

merely due to the thermal smearing. Here we have introduced
u−
(3.9)
lee =
2πT
and
lT =

u+
,
2πT

(3.10)

which are the length scale of spatial decay of fermionic excitations due to e-e interaction and the “thermal smearing length”, respectively. The velocities u± are given by
1
1
=
u±
2

1
1
±
v u

.

(3.11)

The length lee has also been termed the Aharonov-Bohm
dephasing length.27,28,31,32 Note that this length agrees
with the Golden-rule estimate (2.14), up to a numerical
factor.
For α ≪ 1 we have
lT ≃ v/2πT,

lee ≃ lT /α,

(3.12)

i.e., lee for weak interaction is much longer than lT . In
Secs. IV and V, when considering the system in the presence of disorder, there will appear one more characteristic length scale: the electron mean free path due to
backscattering off impurities l. We will assume that T is
sufficiently large, so that lee ≪ l. As will be seen below,
this condition means that the disordered system is in the
WL regime. For lower temperatures, strong localization
sets in.30 Altogether, the hierarchy of length scales in our
problem is
lT ≪ lee ≪ l .

and x positive, we transform the contour of integration
as shown in Fig. 4. Closing the contour upwards, the
integral along the real axis of τ is represented as a sum of
two integrals along the imaginary axis at τ = +0 and τ =
1/T − 0. In view of the periodicity of G+ (x, τ ) in τ , the
sum gives the integral along the contour around the cut.
Closing similarly the contour of integration downward if
both εn and x are negative, we get

(3.13)

1/T
0

dτ exp(iεn τ ) G+ (x, τ ) = Gr (x, iεn ) − Ga (x, iεn ) .
+
+

(3.14)
Here εn = 2π(n + 1 )T is the fermionic Matsubara fre2
quency,
Gr (x, iεn ) = θ(εn )θ(x)G (x, εn ) ,
+
Ga (x, iεn ) = θ(−εn )θ(−x)G (x, εn ) ,
+

(3.15)

and the function G (x, εn ) depends on the absolute values
of the coordinate and energy:
2|x|/u−
T
√
dt
exp(−|εn x|/u)
i uv
0
exp(−|εn |t)
×
.
[ sinh(πT t) sinh(2πT |x|/u− − πT t) ]1/2
(3.16)

G (x, εn ) =

For left movers, Gr,a (x, iεn ) = Gr,a (−x, iεn ). Integration
−
+
in Eq. (3.16) yields (in the rest of the subsection, let both
εn and x be positive):
exp (−εn x/u + x/2lee )
√
i uv
× 2 F1 [ 1/2 + ξn , 1/2, 1; χ(x) ] , (3.17)

G (x, εn ) =

where 2 F1 (a, b, c; z) is the hypergeometric function,
χ(x) = 1 − exp(2x/lee ),

ξn = εn /2πT .

(3.18)

We now analyze the asymptotic behavior of G (x, εn )
as a function of two dimensionless parameters x/lee and
εn /2πT . For x/lee ≫ 1 and εn /2πT ≥ 1, Eq. (3.17) gives
G (x ≫ lee , εn ) ≃

exp (−εn x/u − x/2lee ) Γ(ξn )
√ √
,
1
i π uv
Γ( 2 + ξn )
(3.19)

B.

Green’s function in the (x, ε) representation

We now turn to the single-particle Green’s function
in the space-energy representation, which is obtained by
Fourier-transforming G+ (x, τ ) with respect to τ . Within
the small-α approximation (3.4), the only singularity of
G+ (x, τ ) is a branch cut between τ = |x|/u and |x|/v in
the upper or lower half-plane of τ depending on the sign
of x. Since G+ (x, τ ) in this approximation is analytical in
one of the half-planes of τ , its Fourier transform vanishes
for εn < 0 if x > 0 or for εn > 0 if x < 0. For both εn

where Γ(z) is the gamma-function. After the analytical
continuation to real energies iεn → ε + i0, Eq. (3.19)
reveals oscillations of Gr (x, ε) as a function of εx/u
+
and an exponential decay as a function of x/lee . Using
Eq. (3.19) for the analytical continuation is only accurate
for (|ε|/T )(x/lee ) ≫ 1. In the opposite limit, one has to
analytically continue already in Eq. (3.17), which yields
the “static limit” for the Green’s function with
G (x, 0) ≃

2 exp(x/2lee )
√
K χ(x) ,
iπ uv

(3.20)

9
where K(z) is the complete elliptic integral. For x/lee ≫
1, Eq. (3.20) reduces to
G (x ≫ lee , 0) ≃

x
2
√
exp(−x/2lee ) .
iπ uv lee

(3.21)

Continued to real energies, Eqs. (3.19) and (3.21) match
onto each other at (|ε|/T )(x/lee ) ∼ 1. Finally, the highenergy short-distance asymptotic behavior of the Green’s
function is given by
G (x ≪ lee , εn ≫ T ) ≃

exp(−εn x/u+ )
√
I0
i uv

εn x
u−

,
(3.22)

where I0 (z) is the Bessel function of the imaginary argument.
The imaginary part of Gr (x, ε) [obtained as an ana+
lytical continuation of Gr (x, iεn ) onto the real axis of
+
energy from the upper half-plane] as a function of x for
small and large ε/T is shown in Figs. 5 and 6, respectively. While in the former case there are only simple oscillations which are suppressed exponentially on the scale
of lee [cf. Eq. (3.19)], the behavior of Gr (x, ε) in the lat+
ter case is richer. Specifically, Fig. 6 exhibits beatings
and an intermediate power-law decay, in agreement with
Eq. (3.22). The real part of Gr (x, ε) behaves similarly.
+

FIG. 6: Imaginary part of the Green’s function Gr (x, ε) as
+
a function of x for large energies ε ≫ T . One sees oscillations with a period 2πu+ /ε, beatings with a period 2πu− /ε,
and a power-law decay x−1/2 (taking place up to x ∼ lee ).
The asymptotic behavior for large x ≫ lee at which Gr (x, ε)
+
is suppressed exponentially is not shown in the figure. The
parameters of the plot are: ε/T = 20 and u/v = 1.1.

real axis from below, iεn → ε − i0, yields the advanced
Green’s function GA (q, ε).
+
The retarded and advanced Green’s functions of right
movers can be written in the form
2 lee
GR,A (q, ε) = √ P(±κu )P(±κv ) ,
+
uv

(3.23)

where the signs + and − correspond to the retarded (R)
and advanced (A) Green’s functions, respectively,
κu = (ε/u − q)lee ,

κv = (ε/v − q)lee ,

(3.24)

and
P(z) =

FIG. 5: Imaginary part of the Green’s function Gr (x, ε) [ob+
tained as an analytical continuation of Eq. (3.16) onto the
real axis of ε from above] as a function of x for small energies
ε ≪ T shows oscillations cos(xε/u) exp(−x/2lee ) with a period 2πu/ε, exponentially suppressed on the scale of lee . The
parameters of the plot are: ε/T = 0.25 and u/v = 1.1.

C.

Green’s function in the (q, ε) representation:
Spectral weight

Here, we complete the analysis of the single-particle
Green’s function in a spinful LL by inspecting its spectral properties in the momentum-energy representation.
Fourier-transforming Eq. (3.14) with respect to x, and
analytically continuing the result onto the real axis of
ε from the upper half-plane, iεn → ε + i0, we get the
retarded Green’s function GR (q, ε) in the (q, ε) represen+
tation. Similarly, the analytical continuation onto the

Γ [(1 − 2 iz)/4]
.
Γ [(3 − 2 iz)/4]

(3.25)

As a function of complex variable z, P(z) has a series of
simple poles (originating from the gamma-function in the
numerator) at z = −i(4m + 1)/2, where m is a positive
integer. The pole that is closest to the real axis [corresponding to complex ε = uq−iu/2lee and ε = vq−iv/2lee
in Eq. (3.23)] determines the spatial/temporal decay of
the Green’s functions considered in Secs. III A and III B.
An alternative representation, which straightforwardly
splits GR,A (q, ε) into the real and imaginary parts, is
+
lee
GR,A (q, ε) = √ LR,A (κu , κv )K(κu )K(κv ),
+
uv
(3.26)
where
LR,A (x, y) = sinh [ π(x + y)/2 ]
∓ i cosh [ π(x − y)/2 ]

(3.27)

and the real function K(z) is given by
K(z) =

1
Γ [(1 − 2 iz)/4] Γ [(1 + 2 iz)/4] .
2π

(3.28)

10
The upper and lower signs in Eqs. (3.26)–(3.28) correspond to the retarded and advanced functions, respectively.
Note that the influence of interaction on the behavior
of GR,A (q, ε) is twofold. Firstly, it factorizes the single+
particle fermionic Green’s function into two parts characterized by different velocities—v for the spin factor and
u for the charge factor. This is the essence of the SCS.
Secondly, at finite T , interaction leads to a broadening
of the singularities in the spectral weight, i.e., to a shift
of the singularities of GR,A (q, ε) into the complex plane.
+
In the (x, ε) representation, this shift manifests itself in
the exponential damping of the Green’s function on the
spatial scale of lee , as discussed in Sec. III B.
Figures 7 and 8 illustrate how the imaginary and real
parts of GR (q, ε) as a function of ε evolve with varying
+
temperature. At T = 0 one gets:54
1
√
GR (q, ε) = √
+
ε − uq ε − vq

FIG. 7: Spectral weight ImGR (q, ε) (arbitrary units) for right
+
movers as a function of energy for different temperatures and
u/v = 1.1. At low T , the SCS manifests itself in the doublepeak structure, with two peaks located at ε = vq and ε = uq.
When T increases (T /vq = 0, 0.05, 0.1, 0.5 and 1) the singularities are rounded off and eventually a single peak emerges.

(3.29)

(for u → v, ε is understood as ε + i0). At low T , there
is a double-peak structure which represents the SCS,
with square-root singularities at ε = vq and ε = uq,
weakly smeared by temperature. With increasing T ,
the broadening becomes more pronounced and eventually two peaks in the spectral weight merge into a single
peak of width ∼ αT .
At this point, it is worth recalling that we have neglected effects of interaction which are related to the
exponent αb ∼ O(α2 ) in Eq. (3.2). Retaining αb , i.e.,
including the last factor in Eq. (3.2) would only lead
to the following two effects, both of which are of minor importance in our consideration at weak interaction.
Firstly, there will be an additional small asymmetry between two peaks in Figs. 7 and 8. Secondly, an additional,
weak singularity (characterized by the exponent αb ) will
arise at ε = −uq (cf. Refs. 55 and 54, where the singleparticle Green’s function in a spinful LL was investigated
at T = 0 beyond the weak-interaction limit).
To conclude, in Sec. III we have analyzed the behavior of single-particle excitations in a spinful LL at finite
T . We have demonstrated that, because of the SCS, it is
dramatically modified as compared to the spinless case.
In particular, the decay length lee turned out to be parametrically shorter than for spinless electrons. However,
a priori it is not immediately clear to what extent the
modification of the single-particle properties will affect
the transport (i.e., two-particle for fermions) properties
of a spinful LL. Indeed, as shown in Refs. 27,28 for the
spinless case, the WL dephasing length lφ is parametrically longer than lee . Calculation of the conductivity of
a disordered quantum wire in the presence of spin is a
subject of the next section.

FIG. 8:
Real part of the right-mover Green’s function
GR (q, ε) as a function of energy. Parameters are the same
+
as in Fig. 7. The singularities are smoothened and the peakdip structure broadened with increasing temperature.

IV.

WEAK LOCALIZATION

So far we have analyzed the single-particle properties
of a spinful LL at finite T in the absence of disorder.
Now we introduce disorder and turn to the calculation
of a two-particle quantity, namely, the conductivity. At
high T , the leading term in the conductivity is given by
the Drude formula,
σD =

2e2
l,
π

(4.1)

with a renormalized56,57,58,59 by interaction—therefore
temperature dependent—mean free path l ∝ T αr .
We consider now a correction ∆σ to the Drude conductivity, associated with the quantum interference of
electron waves multiple-scattered off disorder. In 1D,
the leading contribution to ∆σ comes from a Cooperontype scattering process which, in contrast to higher dimensionalities, involves a minimal possible number of

11

(a)

Cooperon”).27,28 The peculiarity of 1D in this respect is
that a single-channel quantum wire is in the WL regime—
not strongly localized—only if the dephasing length lφ
that cuts off the WL correction is shorter than the mean
free path l. That is, the WL correction is accumulated
on ballistic scales—hence the shortest possible Cooperon
ladder with three impurity legs.

(b)

(c)
A.

FIG. 9: Diagrams giving the leading contribution to the interference correction to the conductivity. The dashed lines
represent the impurity scatterings and the solid lines denote
the electron Green’s functions (with the disorder effects incorporated at the self-energy level) and the dashed lines represent the impurity-induced backscattering. The diagrams are
understood as “dressed” by interaction as shown in Fig. 10.

scatterings on impurities, namely three (“three-impurity

∆σ(iΩm ) = 2 × 2 × 2 × (ev)2 ×

v2
2l0

3

×

General expression for Cooperon

The leading term in ∆σ is given by the diagrams27,28
in Fig. 9. These are understood as dressed by interactioninduced fluctuating gauge factors exp[±iθµ (x, τ )] attached pairwise to the backscattering vertices, as described in Sec. II C and illustrated in Fig. 10 for the case
of diagram (a). The sum of contributions of diagrams
(b) and (c) is equal to that of diagram (a). At this level,
there is no difference in the structure of the diagrams between the spinful and spinless cases—the only difference
stems from the particular form of the correlators of the
phases θµ (x, τ ).
Averaging over the fields θµ (x, τ ), we get for the interefernce correction at Matsubara frequency Ωm :

1
Ωm

1/T
1/T
1/T
1/T
1/T
T 1/T
dτ1
d¯1
τ
dτ2
d¯2
τ
dτ3
d¯3 dx1 dx2 dx3
τ
L 0
0
0
0
0
0
× g+ (x1 − x3 , τ1 − τ3 )Q−1 (x1 − x3 , τ1 − τ3 ) g− (x2 − x1 , τ2 − τ1 )Q−1 (x2 − x1 , τ2 − τ1 )
¯
¯

×

×

g+ (x3 − x2 , τ3 − τ2 )Q−1 (x3 − x2 , τ3 − τ2 )

g− (x1 − x3 , τ1 − τ3 )Q−1 (x1 − x3 , τ1 − τ3 )
¯
¯

× g+ (x2 − x1 , τ2 − τ1 )Q−1 (x2 − x1 , τ2 − τ1 ) g− (x3 − x2 , τ3 − τ2 )Q−1 (x3 − x2 , τ3 − τ2 )
¯
¯
¯
¯
¯
¯
¯
¯
× Q(x1 − x3 , τ1 − τ3 ) Q(x1 − x3 , τ1 − τ3 ) Q(x1 − x2 , τ1 − τ2 )
¯
¯
¯
× Q(x2 − x1 , τ2 − τ1 ) Q(x3 − x2 , τ3 − τ2 ) Q(x2 − x3 , τ2 − τ3 )
¯
¯
¯
−1
−1
−1
× Q (0, τ1 − τ1 ) Q (0, τ2 − τ2 ) Q (0, τ3 − τ3 )
¯
¯
¯
f
i
× W+ (x1 − x3 , τ1 , τ3 , Ωm )W− (x1 − x3 , τ1 , τ3 , Ωm ) ,
¯
¯

(4.2)

where L is the system size, the free Green’s functions g± (x, τ ) are given by Eq. (3.1), the interaction-induced factors
i,f
Q(x, τ ) are defined by Eq. (2.22), and the factors W± (x, τ, τ ′ , Ωm ) come from integration of the two Green’s functions
attached to the current vertices over the external coordinates and times:
i
W+ (x, τ, τ ′ , Ωm ) =

×

sgnΩm
|Ωm | + v/l

e−iΩm τ − e−iΩm τ
′

′

+

e−iΩm τ θ(Ωm x) − e−iΩm τ θ(−Ωm x)

i,f
i,f
W− (x, τ, τ ′ , Ωm ) = −W+ (−x, τ, τ ′ , Ωm ),
f
i
W+ (x, τ, τ ′ , Ωm ) = W+ (x, τ, τ ′ , −Ωm ).

(4.4)

v
1 − e−|Ωm x|/v
|Ωm |l
,

(4.3)

In Eq. (4.3), we have included vertex corrections for the
current vertices, which arise from the anisotropy of impurity scattering [recall that only backscattering off impurities (2.3) is retained in the model]. The vertex cor-

12
in the limit of small α is then written as

-

+
Q(x, τ ) ≃

-

+

sinh[πT (x/u + iτ )] sinh[πT (x/u − iτ )]
.
sinh[πT (x/v + iτ )] sinh[πT (x/v − iτ )]

(4.5)

The second term in Eq. (4.3), proportional to 1/l, can
be omitted for lφ /l ≪ 1, so that one more approximation
we make is to take the factors (4.3) in Eq. (4.2) at x = 0:

-

+

-

+

i
W+ (0, τ, τ ′ , Ωm ) =

FIG. 10: The same diagram as in Fig. 9a with the interactioninduced factors exp(±iθµ ) shown explicitly. The solid lines
with arrows stand for bare Green’s functions, the crosses
for the impurity vertices, and the wavy lines for the factors
exp(±iθµ ). The space-time coordinates of the backscattering
¯
vertices are denoted by N = (xN , τN ) and N = (xN , τN ).
¯
Averaging over the fields θµ couples all wavy lines with each
other, cf. Fig. 3.

′
sgnΩm
e−iΩm τ − e−iΩm τ
|Ωm | + v/l

.
(4.6)

It is convenient to introduce new variables
xa = x1 − x3 ,
xb = x3 − x2 ,
xc = x1 − x2 ,

(4.7)

and
rections result in a replacement of the total scattering
rate v/2l by the transport scattering rate v/l. Note also
that Eq. (4.2) is written in terms of the contribution of
the Cooperon with chiralities of the current vertices as
shown in Fig. 10. The numerical coefficient in Eq. (4.2)
takes into account that ∆σWL is a factor of 2 × 2 × 2 = 8
larger than the contribution of the diagram in Fig 10 [one
of the factors of 2 comes from the spin, another from a
summation over chiralities of the current vertices and the
third one from diagrams (b) and (c) in Fig. 9].
The approximation (3.4) means that the terms in the
exponent of Q(x, τ ) [Eqs. (2.21),(2.22), and(2.26)] that
come from B++ (x, τ ) and B−− (x, τ ) and are proportional
to αb are neglected. As for the term that comes from
B+− (x, τ ) and is proportional to αr , it leads to a renormalization of the impurity strength (see Ref. 28 for details) but does not contribute to the dephasing rate to
first order in α, similarly to the spinless case. Therefore,
in the calculation below we put both αb and αr in Q(x, τ )
equal to zero, while l0 is replaced by l which is understood
as the renormalized mean free path. The factor Q(x, τ )

∆σ
≃ lim
Ω→0
σD

−

2πT
v4
Ωm (|Ωm | + v/l)2

τa
τc
τb
τa
¯
τc
¯
τb
¯

=
=
=
=
=
=

τ1 − τ3 ,
¯
τ2 − τ1 ,
τ3 − τ2 ,
τ1 − τ3 ,
¯
τ2 − τ1 ,
¯
¯
τ3 − τ2 .
¯
¯

(4.8)

These satisfy the constraints xc = xa + xb and τa + τb +
τc + τa + τb + τc = 0. We thus represent Eq. (4.2) as an
¯
¯
¯
integral over the variables (4.7),(4.8) and insert, instead
of T /L, in the integrand the factor
T
n

τ
τ τ
eiεn (τa +τb +τc +¯a +¯b +¯c ) δ(xc − xa − xb )

(4.9)

which contains summation over fermionic Matsubara frequencies. By extracting the vertex functions (4.3) at
x = 0, Eq. (4.2) in the limit α ≪ 1 is rewritten in the
new variables as

1/T

d τa d τb d τc d τa d τb d τc
¯ ¯ ¯
n

d xa d xb d xc

0

× G+ (xa , τa ) G− (xa , τa ) G+ (xc , τc ) G− (xc , τc ) G+ (xb , τb ) G− (xb , τb )
¯
¯
¯
× C+ (xa , τa ) C− (xa , τa ) C+ (xc , τc ) C− (xc , τc ) C+ (xb , τb ) C− (xb , τb )
¯
¯
¯
× Q(xa , τb + τc ) Q(xa , τb + τc ) Q(xc , τa + τb ) Q(xc , τa + τb ) Q(xb , τa + τc ) Q(xb , τa + τc )
¯
¯
¯
¯
¯
¯
× exp[ i(Ωm + εn ) ( τa + τb + τc ) − iεn ( τa + τb + τc ) ] δ (xa + xb − xc )
¯
¯
¯

iΩm →Ω+i0

,

(4.10)

13
where G± (x, τ ) is given by Eq. (3.4), Q(x, τ ) by Eq. (4.5), and we introduce
C± (x, τ ) =

sinh[πT (x/v ∓ iτ )]
,
sinh[πT (x/u ∓ iτ )]

(4.11)

such that g± Q−1 ≃ G± C± . When deriving Eq. (4.10) we have used the approximation (4.6)
f
i
W+ (0, τ1 , τ3 , Ωm )W− (0, τ1 , τ3 , Ωm ) →
¯
¯

1
(|Ωm | + v/l)2

τ
τ
− eiΩm (τ1 −¯1 ) − eiΩm (¯3 −τ3 )

τ τ
+ eiΩm (τ1 −τ3 ) + eiΩm (¯3 −¯1 ) → −

The “diagonal” terms exp[iΩm (¯1 −τ1 )] and exp[iΩm (τ3 −
τ
τ3 )] yield identical contributions [which is accounted for
¯
by the factor of 2 in Eq. (4.10)]. The cross-terms are
neglected, since, after the integration over times (see
Secs. IV B and V below), they produce the products of
Green’s functions in the (x, ε) representation of the type
r(a)

r(a)

G+ (x, εn )G− (x, εk ) = 0

(4.13)

[see Eq. (3.15)]. This corresponds to retaining only those
Cooperon diagrams that contain an equal number of the
retarded and advanced Green’s functions, even with e-e
interaction included.
We are thus left with the functions G± (x, τ ), Q(x, τ ),
and C± (x, τ ) that are all given by various combinasinh[πT (x/u ± iτ )] and
tions of square-root factors
sinh[πT (x/v ± iτ )]. Note that G± (x, τ ) and C± (x, τ )
enter Eq. (4.10) only in the combination G± C± with the
same arguments. Equation (4.10) will be analyzed in the
next two subsections.
B.

Regular impurity configurations: Weak
localization

In Eq. (4.10), we transform the integration contours
for each of the time variables similarly to Fig. 4. As
a result, we obtain integrals along square-root branch
cuts in the vertical direction, each of which connects two
points whose coordinates can be written as τ + ix/u and
τ + ix/v with different τ and x. For example, let us
assume, here and throughout the paper below, that in
Eq. (4.10) all
xa , xb , xc > 0

(4.14)

(the region of integration xa < 0 and xb , xc > 0 gives the
same contribution to ∆σWL ). Then, starting with the
integration over τa and closing the contour upwards, we
represent the integral along the real axis of τa as a sum of
three integrals around vertical cuts: between ixa /u and
ixa /v, between −¯b + ixc /u and −¯b + ixc /v, and beτ
τ
tween −¯c + ixb /u and −¯c + ixb /v. The first cut comes
τ
τ

τ
2 eiΩm (¯3 −τ3 )
.
(|Ωm | + v/l)2

(4.12)

from the Green’s function G+ (xa , τa ), whereas the last
two from the factors Q(xb , τa + τc ) and Q(xc , τa + τb ), re¯
spectively. Since G+ (x, τ ) as a function of x for a given
τ falls of on the scale of lT , while Q(x, τ ) on the scale of
lee , one sees that in the limit of small α the main contribution to ∆σWL comes from the branch cuts that are
associated with the Green’s functions. The cuts related
to the factors Q(x, τ ) can be neglected.
The selection of singularities at α ≪ 1 is closely analogous to that in the spinless case, see Appendix F of
Ref. 28. For spinless electrons, the main contribution to
∆σWL stems from singularities (“nearly poles” for weak
interaction) of the single-particle Green’s functions at the
classical trajectory of an electron moving with the velocity u. Other close-to-pole singularities (those in the factors Q and C), which are related to “nonclassical trajectories”, yield subleading corrections small in the parameter lT /lφ ≪ 1. The spinful problem is very much similar
in this respect. The main difference is that the dominant
singularities are now pairs of close square-root branching
points rather than the poles. The transformation of the
poles into the branch cuts, induced by the SCS, can be
viewed as a “smearing” of the classical trajectories: all
velocities between v and u become accessible.
Let us first consider the contribution to Eq. (4.10) from
typical impurity configuration for which the characteristic scale of
xa ∼ xb ∼ xc

(4.15)

is of the order of lφ ≫ lT . We can then expand all sinh’s
in the C and Q factors as
sinh(πT y) ≃

1 πT |y|
e
sgn y ,
2

|y| ≫ 1 .

(4.16)

It is immediately seen that using Eq. (4.16) reduces the
product of six C factors to a simple exponential:
C ≃ exp(2xc /lee ) .
Similarly, the product of six Q factors

(4.17)

14

2xb + xa
¯ ¯
Q xa , i tc + tb +
u
2xa + xb
Q xb , i ta + tc +
u

2xb + xa
u
xb − xa
¯
× Q xc , i tb − ta +
u

Q = Q xa , i tc + tb +

xa − xb
u
2xa + xb
¯
¯
Q xb , i ta + tc +
u
¯
Q xc , i ta − tb +

(4.18)

is represented as
Q ≃ exp(πT q/2) ,

(4.19)

where
q =
+
+
+
+
+

|2xb /u + tc + tb | − |2xb /u + tc + tb − 2xa /u− | + |2xc /u + tc + tb | − |2xc /u + tc + tb + 2xa /u− |
¯ ¯
¯ ¯
¯ ¯
¯ ¯
|2xb /u + tc + tb | − |2xb /u + tc + tb − 2xa /u− | + |2xc /u + tc + tb | − |2xc /u + tc + tb + 2xa /u− |
¯
¯
¯
¯
|2xb /u − ta + tb | − |2xb /u − ta + tb + 2xc /u− | + |2xa /u + ta − tb | − |2xa /u + ta − tb + 2xc /u− |
¯a − tb | − |2xa /u + ta − tb + 2xc /u− | + |2xb /u − ta + tb | − |2xb /u − ta + tb + 2xc /u− |
¯
¯
¯
|2xa /u + t
|2xa /u + ta + tc | − |2xa /u + ta + tc − 2xb /u− | + |2xc /u + ta + tc | − |2xc /u + ta + tc + 2xb /u− |
¯
¯
¯
¯
¯
¯
¯
¯
|2xa /u + ta + tc | − |2xa /u + ta + tc − 2xb /u− | + |2xc /u + ta + tc | − |2xc /u + ta + tc + 2xb /u− | . (4.20)

In Eqs. (4.18) and (4.20) we have shifted the time variables according to
iτj = −tj − xj /u,
¯
i¯j = tj + xj /u,
τ

(4.21)

¯
with j = a, b, c. The shifted variables tj and tj in Eq. (4.21) are real on the branch cuts corresponding to the Green’s
functions in Eq. (4.10) and change along these cuts from 0 to 2xj /u− ∼ αxj /u, where u− is given by Eq. (3.11).
Inspecting Eq. (4.20), we observe that for xa > 2xc u/u− (or similarly for xb ) all the moduli can in fact be omitted
on the Green’s function branch cuts, which yields
Q ≃ exp(−4πT xc /u− ) = exp(−2xc /lee )

(4.22)

(the opposite case xa,b < 2αxc is addressed in Sec. V). As a result, the factors C and Q compensate each other:
QC ≃ 1 .

(4.23)

The integrand of Eq. (4.10) thus reduces to a product of the single-particle Green’s functions in the (x, ε) representation
[Eqs. (3.15), (3.17)]:
∆σWL
= lim
Ω→0
σD

−

2πT
v4
Ωm (|Ωm | + v/l)2

∞
n

dxa dxb dxc δ (xa + xb − xc )

0

× Gr (xa , iεn + iΩm )Gr (xc , iεn + iΩm )Gr (xb , iεn + iΩm )Ga (xa , iεn )Ga (xc , iεn )Ga (xb , iεn )
+
+
+
−
−
−

iΩm →Ω+i0

,

(4.24)
where we have taken into account that only terms with εn < 0, εn + Ωm > 0 survive, in view of Eq. (4.13), which
follows from Eq. (3.15). Performing the analytical continuation to real frequencies Ωm → Ω + i0, we get in the dc
limit Ω → 0:
lee 2
∆σWL
,
(4.25)
= −AWL
σD
l
where the numerical factor AWL is defined by
∞

AWL = π
−∞

dz
cosh2 πz

∞

∞

dx
0

0

dy R(x, z) R(y, z) R(x + y + xy, z),

(4.26)

with
R(x, z) = 2 F1 ( 1/2 + iz, 1/2, 1; −x ) 2 F1 ( 1/2 − iz, 1/2, 1; −x ) .

(4.27)

We have estimated AWL ≃ 0.13 by taking the integral (4.26) numerically.
The small factor (lee /l)2 in Eq. (4.25) is due to the

exponential decay exp(−2xc /lee ) of the product of six

15
Green’s functions in the integrand of Eq. (4.24) on the
scale of lee . One sees that the dephasing factor that suppresses the interference term in the conductivity behaves
as exp(−LC /lφ ), where LC = xa + xb + xc = 2xc is the
total length of the Cooperon loop and the WL dephasing
length lφ reads
lφ = lee .

(4.28)

Schematically, Eq. (4.24) can be estimated as
∆σWL
∼ −
σD

∞
0
∞

×

0

dxa
exp(−2xa /lee )
l
dxb
lee
exp(−2xb /lee ) ∼ −
l
l

2

,

so that for typical realizations of disorder with xa ∼ xb ∼
xc each of the distances is of the order of lee , see Fig. 11a.
For comparison, in the spinless case,27,28 the relevant distances obey xa xb ∼ llee , which yields

∼ −

l

0

dxa
l

l

0

dxb
exp(−xa xb /llee )
l

lee
l
∼−
ln
l
lee

lφ
l

2

ln

l
.
lφ
(4.30)

It follows that lφ for spinful electrons is much shorter
than the dephasing length for spinless electrons (recall
that for the latter, lφ diverges27,28 in the limit of vanishing disorder). In fact, in the spinful case lφ is equal to
the single-particle (electron) decay length lee , in contrast
to the spinless case, where lφ ≫ lee .
V. ANOMALOUS IMPURITY
CONFIGURATIONS: MEMORY EFFECTS

The WL contribution to the conductivity, calculated
in the preceding section, is associated with scattering on
compact three-impurity configurations which are “regular” in the sense that the characteristic distances between
all three impurities are the same. Below, we will see
that “anomalous” (strongly asymmetric) configurations
in which two of the impurities are anomalously close to
each other, i.e., xa ≪ xb or xb ≪ xa (see Fig. 11b),
give rise to a conductivity correction which is larger than
∆σWL if T is sufficiently high. As mentioned already in
Sec. I, the relevance of the asymmetric configurations is
related to the classical ME60 in electron kinetics, in contrast to the quantum interference of scattered waves that
yields ∆σWL .
A.

l ee

l ee

(b)
lT

(4.29)

sl
∆σWL
∼ −
σD

(a)

Qualitative consideration: Identifying scales
and parameters

To demonstrate the peculiarity of the asymmetric impurity configurations, it is instructive to consider first

l

FIG. 11: Three-impurity configurations that give the main
contribution to the (a) quantum (WL) and (b) classical (ME)
corrections to the conductivity. The characteristic distances
between the impurities are shown.

the limit of two scattering events occurring at the same
point, by setting
x1 = x3

(5.1)

in Eq. (4.2). As discussed at the beginning of Sec. IVB,
for typical impurity configurations (4.15) the main contribution to the correction (4.2) comes from “smeared”
classical trajectories, meaning that all trajectories with
velocities between v and u contribute to ∆σWL . The
case (5.1) is, however, special in that only one velocity
remains and that is the velocity of noninteracting electrons v. Indeed, at x1 = x3 the four Q factors that depend on x1 − x3 drop out of the integration over times—
since Q(0, τ ) [Eq. (4.5)] does not depend on τ . After
this, the integrals in Eq. (4.2) over τ1 and τ1 are dom¯
inated by the poles of the noninteracting Green’s functions g+ (0, τ1 − τ3 ) and g− (0, τ1 − τ3 ), which yields
¯
¯
τ1 = τ3 ,
¯

τ1 = τ3 ,
¯

(5.2)

and then all the remaining factors Q compensate each
other. We thus end up with a product of six bare Green’s
functions g± multiplied by the factors W± , which constitutes the noninteracting limit of the problem.61
In fact, the compensation of the dephasing factors
is evident even before the averaging over the fluctuating fields θ± (x, τ ) (Fig. 10). Indeed, the factors
exp{±i[θ+(x, τ ) − θ− (x, τ )]}, dressing two impurity vertices, cancel each other when taken at the same spacetime point [Eqs. (5.1) and (5.2)]. As a result, the third
impurity located at x2 becomes decoupled with respect
to e-e interaction from the two-impurity complex at
x1 = x3 .
In the cyclic variables (4.7), Eq. (5.1) means xa = 0.
Since interaction completely drops out of the problem at
xa = 0, the integration over the remaining spatial coordinate xb = xc in Eq. (4.10) is not cut off by dephasing, in
contrast to the regular impurity configurations, for which
it is restricted to xc
lee ≪ l. In this situation, we
have to take into account a disorder-induced damping
of the single-particle Green’s functions g± (x, τ ), which
results in an additional factor exp(−|x|/4l) attached to

16

(a)

1

3

xa ~

+

+
+

+

xb

~

3

1
2 3

lT
lT
3 2

(b)
+

+

+

+
+

lT
xa ~

1

3

3

1

xb

~

lT

3

2

3

2

1

(c)

xa ~

lT
3

1

+

+
+

xb

3

2

~

lT

+

+

3

2

3

mote third impurity in the three-impurity Cooperon diagram happens to be in the crossover region between the
ballistic and diffusive motion, we should extend the single
scattering on the third impurity to an infinite sequence
of scatterings on other impurities,62 i.e., to a diffuson
ladder, as shown in Fig. 12. One sees that the diagram
takes the form characteristic of a quasiclassical ME:60 an
electron is scattered at x1 ≃ x3 , then moves around diffusively, and returns to x1 ≃ x3 where it is scattered once
again. Clearly, this is a non-Markovian process which is
beyond the conventional Boltzmann description. However, the non-Boltzmann type of kinetics associated with
the return processes is classical in origin: as discussed
above, when the points x1 and x3 are sufficiently close to
each other, dephasing becomes irrelevant. We will first
analyze the simplest three-impurity diagram at finite but
small xa and include the diffusive returns (which will only
renormalize the numerical prefactor) later in the end of
this section.
It is instructive to begin with a semi-quantitative analysis which will give correctly the parametric dependence
of the result but not the numerical coefficient. To this
end, we replace all hyperbolic sines in both the Green’s
functions G± and the Q and C± factors in Eq. (4.10) by
their exponential asymptotics, Eq. (4.16). This approximation is parametrically correct, since all integrals in
Eq. (4.16) are determined by the regions of integration
in which the arguments of the hyperbolic sines are of the
order of or larger than 1. Within the “exponential” approximation, the product C of six factors C± is given by
Eq. (4.17), while the product G of six Green’s functions
in Eq. (4.16) becomes
G ∝ exp(−2xc /lee ) ,

(5.3)

FIG. 12: Each diagram for ∆σ in Fig. 9 has two ME contributions, coming from xa ∼ lT (¯ → 1, ¯ → 3) and xb ∼ lT
3
1
(2 → 3, ¯ → ¯ The remote-impurity line should be replaced
2
3).
by the full diffuson ladder.

so that the exponential factors in G and C cancel each
other, similarly to the regular impurity configurations
in Sec. IV B. What is different, however, is that when
calculating the factor Q [Eqs. (4.18)–(4.20)] one can no
longer omit the moduli in Eq. (4.20) if

each g± (x, τ ). On the Cooperon loop, these combine
to give the overall factor exp(−xc /l) in the integrand
of Eq. (4.2), so that xb = xc is then limited by the mean
free path, see Fig. 11b.
Since at xa = 0 the characteristic distance to the re-

xa < 2αxc

(5.4)

[or xb < 2αxc —we will proceed with the estimate for the
case of Eq. (5.4)]. For the strongly asymmetric configurations (5.4), Eq. (4.20) is rewritten as

¯
¯
q ≃ −8xc /u− + 2xa /u − tb − 2xa /u − tb + 2xc /u− + 2xa /u − tb − 2xa /u − tb + 2xc /u−
¯
¯
+ 2xa /u + tc − 2xa /u + tc − 2xc /u− + 2xa /u + tc − 2xa /u + tc − 2xc /u− ,

(5.5)

¯
where we have neglected terms of second order in α. In particular, the integration over ta and ta goes along very
short branch cuts of length ∝ αxa ∼ α2 xc . To our accuracy, the cuts reduce to poles. For this reason, we have set
¯
ta = ta = 0 in Eq. (5.5).
The integrals over remaining time variables in Eq. (4.10) decouple from each other and can be easily calculated
¯
¯
(note that the integrals over tb and tc are identical—in the dc limit—to those over tb and tc , respectively). Let us

17
denote ∆σasym the contribution to the conductivity that comes from the strongly asymmetric impurity configurations
with |xa,b | < 2αxc . Carrying out the analytical continuation for both fermionic and bosonic frequencies and taking
the dc limit, we get, within the exponential approximation:
∞

xc
2xc
dε dxa dxc
exp −
exp −
l
lee
T cosh2 (ε/2T )
0
4
xa xc ε
× θ(xa − αxc )θ(2αxc − xa ) f1
,
,
2lT 2lee πT
2
xa xc ε
xa xc
ε
+ θ(αxc − xa ) f2
f3
,
,
,
,
2lT 2lee πT
2lT 2lee πT

1
∆σasym
∼ − 2
σD
l

2

,

(5.6)

where the functions f1,2,3 are defined as follows
f1 (x, y, z) =

exp(izy) − 1
,
iz

exp[(iz + 1)(y − x)] − 1
exp(izx) − 1
− i exp(x − y)
,
iz
(iz + 1)
exp[(iz + 1)y] − exp[(iz + 1)x]
exp(izx) − 1
+ i exp(−x)
.
f3 (x, y, z) =
iz
(iz + 1)

f2 (x, y, z) = exp[iz(y − x)]

(5.7)

In Eq. (5.6), the θ-functions split the domain of integration over xa into two, αxc < xa < 2αxc and
0 < xa < αxc .

(5.8)

In the first interval of xa , the integration over xc is limited by the factor exp(−2xc /lee ) in the first line of Eq. (5.6),
which yields a contribution to ∆σasym /σD of the order of −lee lT /l2 . This is much smaller than the contribution of
the regular impurity configurations [Eq. (4.22)] and can be neglected. However, the situation is qualitatively different
for the interval (5.8). Indeed, for 1, x ≪ y, the functions |f2 (x, y, z)|2 and |f3 (x, y, z)|2 take the form
1
exp(izx) − 1
+
z
1 + iz
exp[2(y − x)]
≃
.
(z 2 + 1)

|f2 (x, y, z)|2 ≃
|f3 (x, y, z)|2

2

,

(5.9)
(5.10)

As a result, the factor exp(−2xc /lee ) from the first line of Eq. (5.6) is canceled by the factor exp(2y) from Eq. (5.10).
The remaining integral over xc is no longer restricted to xc lee but rather extends to xc ∼ l, in agreement with our
qualitative consideration of the ME effect for xa = 0 at the beginning of this section. The ME contribution ∆σME
to the conductivity thus arises from the impurity configurations obeying Eq. (5.8) and is given by the second term in
Eq. (5.6). It can be estimated as
∆σME
∼−
σD

∞
0

xc
dxc
exp −
l
l

αxc
0

2xa
dxa
exp −
l
lT

∼−

lT
l

(5.11)

and becomes much larger than the WL contribution [Eq. (4.22)] if T is sufficiently high, namely if
T ≫ v/α2 l .

(5.12)

The characteristic regions of xa , xb that control the quantum (WL) and classical (ME) corrections to the conductivity
are shown in Fig. 13a. For comparison, an analogous plot for the spinless problem is presented in Fig. 13b. One
sees that in the spinful case the “quantum” and “classical” domains are parametrically separated from each other.
2
As a result, their areas lee ∝ T −2 and llT ∝ T −1 , respectively, can “compete” with each other. On the contrary, in
the spinless case, the ME strips of area ∼ llee are directly adjacent to the WL domain, whose area ∼ llee ln(l/lee )
sl
is (logarithmically) larger. It follows that for spinless electrons the ME correction ∆σME gives only a subleading
sl
contribution as compared to the WL correction ∆σWL [Eq. (4.30)]:
sl
sl
∆σME
lee
lee
∆σWL
∼
∼
≪
ln
σD
l
σD
l

[recall that lφ ∼ (llee )1/2 and lee ∼ α−2 lT for spinless electrons].

l
lee

(5.13)

18

(a)

(b)

xb

l
lee

l

111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000

lφ
111111111111
000000000000
111111111111
000000000000
111111111111
000000000000
111111111111
000000000000

lT

1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
a
1111
0000
1111
0000
b
1111
0000
1111
0000
1111
0000
1111
0000
ee
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111111111111
0000000000000
11111111111
00000000000
1111111111111
0000000000000
11111111111
00000000000
1111111111111
0000000000000
11111111111
00000000000
1111111111111
0000000000000
11111111111
00000000000
1111111111111
0000000000000
11111111111
00000000000

xx

~

WL

WL
lT

xb

lee=l φ

xa

l

lee

ll

lee lφ= llee

xa

l

FIG. 13: Domains of the distances xa , xb that govern the WL (gray-shaded area) and ME (dashed area) contributions to the
conductivity for the spinful (a) and spinless (b) models.

B.

Rigorous calculation of ∆σME

In the preceding subsection, we have adopted the “exponential approximation” by replacing all the hyperbolic sines
in the correlation functions by their asymptotics (4.16). This allowed us to identify the relevant scales and obtain the
¯
¯
parametric dependence of the result. Here, we calculating the integrals over the temporal variables tb , tb , tc , and tc
more accurately and find the numerical prefactor in Eq. (5.11). The estimate in Sec. V A teaches us that the dominant
¯
contribution to ∆σME comes from xa ∼ lT and xb ≃ xc ∼ l, while the integrals over tb,c and tb,c are determined by
the upper limit of integration, i.e., by the time scale αxc /v ∼ αl/v. In the WL regime, this time scale is much larger
than 1/T .
Similarly to the case of regular impurity configurations, we observe that the product of all the C factors can always
be approximated by Eq. (4.17), whereas the hyperbolic sines should be retained in the Green’s functions G± . As
for the factors Q, in contrast to the regular configurations, not all of them can be replaced by their asymptotics.
Specifically, those hyperbolic sines in the Q factors that correspond to the moduli in Eq. (5.5) (i.e., all terms in q
except for the first one) should retain their form and not be replaced by the exponentials. Importantly, within this
“partially exponential” approximation, all integrals over times are still decoupled. Thus, we replace the exponential
factor depending on tc in Sec. V A according to
θ xa /lT + πT tc − xc /lee − iθ − xa /lT − πT tc + xc /lee
× exp
→
→

1
(−xc /lee + |xa /lT + πT tc | − |xa /lT + πT tc − xc /lee |) − |εn + Ωm | tc
2
2 exp(−|εn + Ωm | tc )

sinh1/2 (xa /lT + πT tc )

sinh1/2 (πT tc ) sinh1/2 (xc /lee − πT tc ) sinh1/2 (xa /lT + πT tc − xc /lee )
2 exp(xa /2lT ) exp(−|εn + Ωm | tc )
sinh1/2 (xc /lee − πT tc ) sinh1/2 (xa /lT + πT tc − xc /lee )

(5.14)

¯
and make a similar replacement for tc (with εn + Ωm → εn ). The factors depending on tb are modified as follows:
i exp
→

1
(−xc /lee + |xa /lT − πT tb | − |xa /lT − πT tb + xc /lee |) − |εn + Ωm | tb
2
2 exp (−|εn + Ωm | tb )

sinh1/2 (xa /lT − πT tb )

sinh1/2 (πT tb ) sinh1/2 (xc /lee − πT tb ) sinh1/2 (xa /lT − πT tb + xc /lee )
2 exp (−xa /2lT ) exp (−|εn + Ωm | tb )
→
,
1/2
sinh (xc /lee − πT tb ) sinh1/2 (xa /lT − πT tb + xc /lee )

(5.15)

19
¯
the integral over tb again differs only in that εn + Ωm changes to εn . Using Eqs. (5.14) and (5.15), we get
lT
∆σME
= −AME
,
σD
l

(5.16)

where AME is given by the dimensionless integral (check the first arguments of F)
∞

AME = π

dz
2
−∞ cosh πz

∞
0

dx M(z, x)

2

M(z, −x)

2

.
(5.17)

The function M(z, x) in Eq. (5.17) is defined by
M(z, x) =

1
2π

∞

dy
0

exp(2izy)
1/2

sinh

(y) sinh1/2 (y + x)

.

(5.18)

We have estimated AME numerically as AME ≃ 0.2.
Finally, let us discuss the overall combinatorial factor
in ∆σME (which is already included in AME ). Firstly,
similarly to the WL correction, the contribution of diagram (a) in Fig. 12 should be multiplied by a factor of
2 × 2 × 2 = 8 due to (i) two possible chiralities of the
current vertices; (ii) diagrams (b) and (c) in Fig. 12; and
(iii) two possible anomalous configurations for each diagram: xa ∼ lT ≪ xb and xb ∼ lT ≪ xa . Secondly,
as mentioned above, not only the return after one single
backscattering but rather the entire diffuson ladder contributes to ∆σME [Fig. 12]. The insertion of the diffuson
into the three-impurity diagram effectively generates an
additional velocity-vertex correction, which yields a factor of 1/2 (the ratio of the transport and total scattering
times for the backscattering impurities).

VI.

PATH-INTEGRAL METHOD

So far, we have been treating the conductivity correction within the formalism of Matsubara functional
bosonization. This powerful method treats on an equal
footing28 the real inelastic scattering processes responsible for dephasing and the virtual transitions responsible
for the renormalization effects. An alternative approach,
formulated for the spinless case in Refs. 27 and 28, consists of two steps. Firstly, disorder is renormalized by virtual processes with characteristic energy transfer larger
than T . What is obtained after the renormalization is
an effective “low-energy” theory which is free of ultraviolet singularities characteristic of a LL. The low-energy
theory is treated by means of a path-integral approach,
analogous to the one developed in Ref. 63 for higherdimensional systems. This method is particularly convenient for an analysis of inelastic scattering (dephasing) in problems with a nontrivial infrared behavior. In
Ref. 28, the WL correction to the conductivity was calculated both within the path-integral and the functionalbosonization schemes, with identical results. The path-

integral calculation also allows one to “visualize” the origin of the dephasing processes in terms of quasiclassical
trajectories.
In this section, we present a path-integral analysis of
the conductivity correction in the spinful problem. It
turns out that the situation here is more intricate than
in the spinless case, for two reasons: (i) the characteristic energy transfer in the path-integral calculation is
of the order of T (while it was much smaller than T for
spinless electrons); (ii) because of the SCS, the velocity of
the quasiclassical trajectories is not uniquely defined: the
whole interval of velocities between v and u contributes
to ∆σWL , see Sec. III. As a result, we will only be able
to reproduce the parametrical dependence of the conductivity correction but not the numerical prefactor. Nevertheless, this analysis is useful, since it yields a physically
transparent picture of the quantum interference and dephasing in the spin-charge separated system.
Since the dominant contribution to the dephasing rate
is given by the g4 –processes of scattering between electrons from the same chiral branch, we will set g2 = 0
which is consistent with our central approximation αb →
0. Furthermore, it is convenient to introduce different
⊥
coupling constants,1,2 g4 and g4 , for interaction between
electrons with equal and opposite spins, respectively (at
⊥
the end, we set g4 = g4 = g ≃ 2πvα). In view of the
Pauli principle, the g4 –interaction does not lead to any
real scattering but only renormalizes the velocity of electrons:
v → v ∗ = v(1 + g4 /2πv) .

(6.1)

This allows us to put g4 = 0, simultaneously replacing v
by v ∗ . All the nontrivial interaction-induced physics is
⊥
due to g4 –processes.
Solving the corresponding RPA equations for the interaction propagators, we get for parallel (V ) and anti-

20
where u = v(1 + 2α) and v ∗ = v(1 + α). We note that
V ⊥ does not enter the path integral, since electron spin
is conserved.
The general expression for the WL dephasing action acquired on the exactly time-reversed trajectories
xb (t) = xf (tC − t) for the three-impurity Cooperon
reads27,28

(a)
x(t)
backward

xa
o

t
forward

−xb

tC

tC

Sij (tC , {x(t)}) = −T
×

dt2

dt1
0

0

dω
2π

dq
2π

ImVµν (q, ω)
exp {iq[ xi (t1 ) − xj (t2 )] − iω[t1 − t2 ]}
ω
tC

tC

= −T

dt1
0

0

dt2 Fµν [xi (t1 ) − xj (t2 ), t1 − t2 ] ,
(6.4)

(b)
x(t)
backward

xa

t

o
forward

where each of the indices i, j takes one of the values “f”
(for the forward path of the Cooperon) or “b” (for the
backward path), µ = sgnxi and ν = sgnxj , and Fµν (x, t)
˙
˙
is the Fourier transform of ω −1 ImVµν (q, ω). The main
contribution comes from the diagonal terms with i = j
and t1 = t2 , for which µ = ν, see Fig. 14a. The imaginary part of the corresponding interaction propagator is
written as
ω
uv
× [ u δ(vq ∓ ω) + v δ(uq ∓ ω)] , (6.5)

Im V±± (q, ω) = (πv ∗ α)2

−xb

which gives
FIG. 14: World lines corresponding to the time-reversed trajectories (solid and dashed lines) with velocity v ∗ in the threeimpurity Cooperon. The dash-dotted line describes the propagation of a plasmon with velocity u. (a) Regular configuration (Sec. IV) contributing to ∆σWL ; (b) asymmetric configuration (Sec. V) contributing to ∆σME . The interaction
line gives a contribution to the dephasing action S which is
proportional to α(Nf − Nb )2 . Here Nf,b is the number of the
small-angle intersections (black dots) of a plasmon line with
the forward (f ) and backward (b) paths. The intersections at
a large angle (the plasmon and electrons moving in opposite
directions, unfilled circles) contribute to the dephasing only
at second order in α and hence are neglected. One sees that
Nf = Nb for typical configurations (a), whereas for asymmetric configurations (b) Nf = Nb for most of the plasmon lines.
For a corresponding plot in the spinless case, see Fig. 7 of
Ref. 28.

(v ∗ q − ω)2
= 2πv α
,
(uq − ω)(vq − ω)
v ∗ q(v ∗ q − ω)
V++ (q, ω) = −2πv ∗ α2
(uq − ω)(vq − ω)
∗

(6.2)

and
⊥( )

(v ∗ α)2
[ u δ(x ∓ vt) + v δ(x ∓ ut) ] .
2uv
(6.6)

A graphic illustration of the e-e scattering processes
contributing to the dephasing action S = 2(Sff − Sfb )
is presented in Fig. 14. There, we show by the circles
and black dots the space-time coordinates for which the
arguments of the δ-functions in Eq. (6.6) are zero. The
main contribution to S comes from small-angle intersections (black dots). For electron trajectories characterized
by velocity v ∗ , these give a large factor of the order of
α−1 , either v ∗ /|u − v ∗ | or v ∗ /|v − v ∗ | [cf. Eq. (2.14)]. As
a result, the action for typical impurity configurations
(Fig. 14a) reads
S ∼ α2 T tC /|α| ∼ |α|T tC .

parallel (V ⊥ ) spins :
⊥
V++ (q, ω)

F±± (x, t) = −π

⊥( )

V−− (q, ω) = V++ (−q, ω) ,

(6.3)

(6.7)

For comparison, in the spinless case27,28 , intersections
between the interaction and particle propagators running in the same direction give zero dephasing because of
the cancellation between the Hartree and exchange terms
(Sec. II B). On the other hand, at large angles the ballistic (straight line) interaction propagator always intersects
a pair of (forward and backward) electron trajectories,
which gives zero dephasing as well. As a result, only the
intersections at large angles that arise from scattering of

21
the interaction propagator off disorder contribute to the
dephasing rate.27,28
More rigorously, substituting Eq. (6.6) in Eq. (6.4), we
find the dephasing action for the trajectory characterized
by the velocity v ∗ ,

0 < xa < αxc /2 ,
 xa ,
8πT 
αxc /2,
αxc /2 < xa < (1 − α/2) xc ,
S≃
v 

xc − xa ,
(1 − α/2) xc < xa < xc ,
(6.8)
which is illustrated in Fig. 15. Any other velocity between v and u yields a qualitatively similar action, but
with different numerical factors. This is another indication of the previously discussed fact that all velocities
from this interval contribute to the conductivity correction.

S
WL

ME

1

ME

xa /xc
0

α/2

1−α/2

1

FIG. 15: Dephasing action S (in units of 2xc /lee ) corresponding to the velocity v ∗ , Eq. (6.8), as a function of xa for fixed
xc .

One sees from Eq. (6.8) (middle line) and Fig. 15 (flat
region) that for typical impurity configurations the dephasing action becomes of the order of unity at tC =
2xc /v ∗ ∼ 1/αT . This gives a parametric estimate for
the dephasing length lφ ∼ α−1 lT , and for the WL correction ∆σWL /σD ∼ −(lee /l)2 , in agreement with the results of Sec. IV. Equation (6.8) (first and third lines) and
Fig. 15 also demonstrate the suppression of the dephasing action in the anomalous (strongly asymmetric) impurity configurations (Fig. 14b) responsible for the ME,
∆σME /σD ∼ −lT /l, as discussed in Sec. V.
VII.

SUMMARY

To conclude, we have analyzed, within the functional
bosonization formalism, the quantum interference of interacting electrons in a disordered spinful LL. Our results
are summarized in Table I and in Fig. 16.
The single-particle properties of fermionic excitations
in this model have been studied in Sec. III in several
representations. Two most important and interrelated

features of the single-particle spectral characteristics are:
(i) the SCS, as a result of which the whole range of velocities between the charge velocity u and the spin velocity
v contributes to the spectral function, and (ii) the singleparticle decay on the spatial scale of lee , Eq. (3.9).
In Sec.IV we have calculated the leading quantum interference correction to the conductivity, Eq. (4.25). The
corresponding dephasing length lφ is given by Eq. (4.28).
Two qualitative differences as compared to the spinless
case should be emphasized in this context. Firstly, for
spinful electrons, the decay length of single-particle excitations is inversely proportional to the first order in the
interaction strength α, lee ∼ α−1 lT , whereas lee ∼ α−2 lT
in the absence of spin. Secondly, the WL dephasing
length lφ is equal to the single-particle dephasing length
lee in the spinful model,√whereas lφ depends on the
strength of disorder, lφ ∼ lee l, for spinless electrons.
In Sec. V we have analyzed the contribution to the
Cooperon diagrams of “nontypical” configurations of disorder with two impurities being anomalously close to
each other. We have shown that this contribution,
Eq. (5.16), describes the quasiclassical ME. It is parametrically smaller than the WL term at sufficiently low
T and gives the leading correction to the conductivity in
the limit of high T .

σ(Τ)

Drude

ME

strong
localization

WL
Τ

v/l

v/α l

v/α l
2

FIG. 16: Schematic plot (log-log scale) of the T dependence
of the conductivity σ(T ) of spinful electrons.

Our findings, demonstrating the strong dependence of
the quantum interference effects on spin in a LL, imply
that the Zeeman splitting by magnetic field should lead
to strong effects in the conductivity of a single-channel
quantum wire. Results obtained in this direction will be
reported elsewhere.42
We thank D. Aristov, D. Bagrets, and A. Morpurgo for
useful discussions and gratefully acknowledge support by
EUROHORCS/ESF (AGY and IVG), by the DFG Center for Functional Nanostructures, and by RFBR [Grant
No. 06-02-16702] (AGY).

22

e-e scattering
length lee

spinless

spinful

v
α2 T
v
αT

dephasing
length lφ

1
α

„

vl
T

«1/2

WL correction
∆σWL /σD

−

“ l ”2

v
αT

φ

l

−

ln

l
v ln(α2 T l/v)
∼−
lφ
α2 T l

“ l ”2
φ

l

∼−

“ v ”2
αT l

ME correction
∆σME /σD

−

lee
v
∼− 2
l
α Tl

−

lT
v
∼−
l
Tl

TABLE I: Characteristic spatial scales induced by interaction and the conductivity corrections for spinless and spinful disordered
LLs.

1
2
3

4

5

6

7

8

9

10

11

12

13

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Neglecting Hbs is not always justified despite g1 being
much smaller than g2,4 . First, in the presence of spin, g1 processes “added” to the clean LL model are a “marginally
relevant” or “marginally irrelevant” perturbation in the
renormalization-group sense, depending on the sign of the
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single-channel quantum wire,41,42 which we will consider
elsewhere.42 However, for the effects studied in the present
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43

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