Conductance through a potential barrier embedded in a Luttinger liquid:
nonuniversal scaling at strong coupling.
D.N. Aristov1, ∗ and P. W¨lfle1, 2
o

arXiv:0902.4170v2 [cond-mat.str-el] 14 Jul 2009

1

Institut f¨r Theorie der Kondensierten Materie and Center for Functional Nanostructures,
u
Universit¨t Karlsruhe, 76128 Karlsruhe, Germany
a
2
Institut f¨r Nanotechnologie, Forschungszentrum Karlsruhe, 76021 Karlsruhe, Germany
u
(Dated: May 26, 2018)

We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary
interaction g2 , and subject to a potential barrier of arbitrary strength, as a function of temperature.
We map the Hamiltonian in the basis of scattering states into an effective low energy Hamiltonian in
current algebra form. First the renormalization group (RG) equation for weak interaction is derived
in the current operator language both using the operator product expansion and the equation of
motion method. To access the strong coupling regime, two methods of deducing the RG equation
from perturbation theory, based on the scaling hypothesis and on the Callan-Symanzik formulation,
are discussed. The important role of scale independent terms is emphasized. The latter depend on
the regulaization scheme used (length versus temperature cutoff). Analyzing the perturbation theory
in the fermionic representation, the diagrams contributing to the renormalization group β-function
are identified. A universal part of the β-function is given by a ladder series and summed to all orders
in g2 . First non-universal corrections beyond the ladder series are discussed and are shown to differ
from the exact solutions obtained within conformal field theory which use a different regularization
scheme. The RG equation for the temperature dependent conductance is solved analytically. Our
result agrees with known limiting cases.

I.

INTRODUCTION

The quantum theory of charge transport in onedimensional (1D) systems of interacting fermions is a fundamental building block of our understanding of electron
transport in nanostructures and of a future nanoelectronics. One-dimensional interacting fermion systems have
been studied since the 1950s, beginning with the pioneering works of Tomonaga and Luttinger. The early
work exploited the fact that the elementary excitations in
these systems are charge and spin excitations of bosonic
character, and led to the formulation of the method of
bosonization.1,2 This method offers a natural description
of the fractionalization of the usual fermionic quasiparticles existing in higher dimensions, into spinon and holon
quasiparticles in 1D. While the bosonization method is
highly successful in accounting for the properties of systems in equilibrium, it is somewhat problematic when
applied to transport situations. For example, an ideal
quantum wire attached to two reservoirs is expected to
2
have a two-point conductance of RQ = e per channel.
h
In contradiction to this general law it was found in early
work on transport in clean Luttinger liquids3,4,5 that the
conductance (here and in the following conductance will
be measured in units of the conductance quantum RQ ) is
given by the Luttinger parameter K, and thus depends
on the interaction strength. It was later pointed out that
by carefully choosing the order of limits of frequency,
ω → 0, and wire length, L → ∞,6,7 or else by taking
into account the screening of the external field by the interacting electron system8 one recovers the correct value
of unity for the two-point conductance. Whether the former or the latter explanation is the correct one remains
disputed to this day.

The more relevant case of a Luttinger liquid with potential barrier has been considered first by Kane and
Fisher4,5 and by Furusaki and Nagaosa,9 again using the
bosonization method. These authors found that interaction has a dramatic effect on the conductance: for repulsive interaction the conductance is found to tend to zero
as a fractional power of the temperature T , in the limit
T → 0 . This was shown in the two limits of a weakly
scattering barrier and a strong (tunneling) barrier. In
addition, results in the complete temperature range are
available at special values of the interaction parameter,
1
K = 1 and K = 3 .4,5,10,11,12 All these works suffer from
2
the drawback that in the clean limit the conductance
is found to tend to K rather than 1 . It has been argued that these results may be applied to the four-point
conductance. However, the four-point conductance is expected to tend to infinity in the limit of vanishing barrier
strength, a behavior not shown.
For the reasons noted above we think it worthwhile
to develop an alternative formulation of the transport
theory of Luttinger liquids, formulated entirely in the
fermionic language. The fermionic representation offers
the possibility to connect the fermionic degrees of freedom in the (non-interacting) leads smoothly with the
degrees of freedom in the interacting system. In other
words, it allows to use the scattering states of the system in the non-interacting limit as a basis of description.
On the simplest level, in lowest order in the interaction,
this program has been carried out in a seminal paper by
Yue, Matveev and Glazman.13 The physics of this problem lies in the scale-dependent build-up of a polarization
potential around the bare barrier, induced by the Friedel
oscillations of the density. For repulsive interaction, the
polarization potential is found to extend farther and far-

2
ther out as the temperature is lowered, until at T = 0
it is infinitely extended, leading to a vanishing transmission probability across the barrier. The process of gradual growth of the effective potential barrier may be described in the language of renormalization group theory
applied to the transmission probability as a function of
the temperature. A generalization of the approach of Yue
et al. to the case of two barriers has been given in Ref.
[14]. These predictions, based on the continuum version
of the theory, were thoroughly checked and confirmed by
the fermionic functional renormalization group method,
starting from the Hubbard models on a lattice.15,16
In this paper we report on a significant extension of
the work of Yue et al. to the case of systems of spinless fermions in 1d interacting via arbitrarily strong forward scattering (parameter g2 ) and subject to a short
range barrier potential (width a). The principal tool we
will be using is perturbation theory in the interaction,
summed to infinite order. To achieve this goal, and to
gain insight into the structure of perturbation theory,
we map the problem first onto an effective model of interacting currents of chiral fermions. The reformulation
allows to describe the effect of the barrier potential as
a local magnetic field at the position of the barrier, acting on the pseudospin vector of the chiral current. The
logarithmic corrections to this field, stemming from the
fermionic interactions can be addressed in a simplified
poor-man scaling approach; however this approach becomes ambiguous in higher orders. Therefore at a later
stage we will apply the Callan-Szymanzik type of analysis, by first introducing the counter terms to magnetic
field part of the Hamiltonian to compensate the effect of
the interaction, and secondly requiring the “bare” value
of the field to be independent of the high-energy cutoff. Equivalently, we may assume the scaling property
of the conductance to hold, entailing the existence of a
renormalization group equation for the conductance as a
function of the scaling variable Λ (at finite temperature
Λ = ln(T0 /T ) ). By comparison with the structure of perturbation theory for the conductance, which is given by a
power series in the interaction and in the scaling parameter Λ , one may extract the RG β-function. In this way
we derive a universal renormalization group equation for
the conductance as a function of the scaling parameter.
The most important consequence of the current algebra formulation is, however, to allow for an efficient organization of perturbation theory. After presenting the
formulation of perturbation theory as well as a few low
order results we analyze the structure of the perturbation expansion with respect to logarithmically divergent
terms containing powers of Λ = ln(L/a) or ln(T0 /T ).
We identify a class of universal diagrams contributing the
principal terms linear in Λ and derive an integral equation resumming these contributions (“ladder approximation”). These terms are shown to dominate in both limits, small and large conductance. We identify the prefactor of the terms in the perturbation series for the conductance G linear in Λ with the renormalization group

β-function of the flow equation of G(Λ). There is, however, one important new feature: there appear scale independent terms (from third order on), which lead to a
correction of the conductance even at Λ = 0, the ultraviolet cutoff. These terms must be taken into account,
in order to preserve the scaling property of the conductance, and lead to a starting value of the conductance
G(Λ = 0), different from the conductance in the absence
of interaction.
The RG equation may be solved analytically in the ladder approximation, to give the conductance as a function
of temperature for any interaction strength and barrier
potential. The result completely agrees with earlier work,
where a comparison is appropriate We emphasize that
our result for the conductance reduces to unity in the
absence of a barrier. Furthermore, the effect of screening
of the external field8 is included by omitting all polarization diagrams. We checked the renormalizability of
the theory explicitly by calculating all terms up to third
order in Λ and g2 .
At intermediate temperatures (semi-transparent barrier) additional diagrams contribute small corrections to
the β-function. We calculate these terms in lowest (third)
order in g2 . A proper treatment of these contributions
requires taking into account scale independent terms in a
way sketched above. We demonstrate explicitly that the
resulting β-function is uniquely determined for a given
physical quantity. However, β turns out to be a slightly
different function for the two cases of interest here, the
temperature dependent and the length dependent conductance. The result of the solution of the RG-equation
for the conductance, using the approximate β-function
thus obtained, is found to agree quantitatively with all
known results on the scaling behavior of the conductance
in the limits G → 0 and G → 1 . In the intermediate
regime G ∼ 1/2 we find small discrepancies with the re1
sults of conformal field theory in the case K = 2 , which
we can trace back to a different cutoff scheme used there
and the effect of higher order terms neglected in our work.
As an excellent interpolation formula between two exact
limits our result goes beyond all previous works: it is
valid for any interaction strength K and any potential
scattering strength.

A.

Formulation of the problem

The Hamiltonian includes the kinetic energy of freely
right- and left-moving (R,L) spinless fermions with linearized dispersion, H0 , the interaction energy between
the left- and right-moving densities, H1 , and the impurity part decribing scattering off the barrier, placed at

3
Hamiltonian Himp in (1) is then written as

the origin, Himp :
H = H0 + H1 + Himp
∞

H0 = vF

dx
−∞
L

H1 = g 2
−L

(1)

†
ψR (i∂x )ψR

−

†
ψL (i∂x )ψL

†
†
dx (ψR ψR )(ψL ψL )

In order to regularize the theory, we assume that the
region of interaction |x| > a, where g2 = 0, and the region
over which the impurity potential is nonzero, |x| < a ∼
kF (kF is the Fermi wave vector) do not overlap spatially.
The length scale a will serve as an ultraviolet cutoff in
the scale dependent logarithmic corrections considered
below.
The last term, Himp , in (1) is not explicitly specified
here (but see below). We will rather take the one-particle
scattering states to be known. This means that we characterize the barrier by transmission and reflection ampli˜
tudes t = t = cos θ, r = −˜∗ = i sin θeiφ , with negligible
r
energy dependence in the energy range of interest (here
˜
t, t, r, r are the usual components of the impurity scat˜
tering matrix). Notice also that we assume the system
to be non-interacting in the leads, |x| > L, which allows
for an asymptotic single-particle scattering states representation.
We define the creation operator of scattering states as
†
˜
ψk (x) = (eikx + rk e−ikx )c† + tk e−ikx c† ,
1k
2k

=

tk eikx c†
1k

+ (˜k e
r

ikx

+e

Himp = vF

−ikx

)c† ,
2k

x<0
x > 0 (2)

with c† (c† ) the creation operators of asymptotically
1k
2k
right-going (left-going) fermions of momentum k. Such
a representation requires the kinetic energy part of the
Hamiltonian to √ of the form −∇2 /(2m), and the mobe
mentum, k = 2mE > 0, before the linearization of
the dispersion around the two Fermi points. In appendix
A we clarify the correspondence between the scattering
states representation (2) and our linearized Hamiltonian
(1) written in the basis of chiral fermions.
For simplicity we do not consider the so-called g4 part
of the fermionic interaction in this work, i.e. the term
L
†
†
1
2
2
2 g4 −L dx [(ψR ψR ) + (ψL ψL ) ]. For L → ∞ the g4 interaction can be absorbed into the redefinition of the
Fermi velocity, vF = vF + g4 /2π. For finite L one can
˜
show that g4 does not lead to a renormalization of d.c.
conductance, which is the quantity of interest below. The
effect of g4 on the a.c. conductance can be analyzed,
following the guidelines in7,17 .
It appears that in the case of weak potential scattering, U (x) ≪ EF , and for electrons with energies close to
the Fermi energy, EF , the situation is characterized two
limits of the Fourier transform U (q) of the potential: the
forward scattering amplitude, U (q ≃ 0) = u1 , and the
backward scattering amplitude, U (q ≃ 2kF ) = u2 , with
real-valued u1 and complex-valued u2 . The fermionic

†
†
dx u1 (x)(ψR ψR + ψL ψL )

(3)

†
+(u2 (x)ψL ψR + h.c.) ,

where the dimensionless functions u1,2 (x) are short-range
impurity potentials, which means that u1,2 (x) = 0 for
|x| > a and the above amplitudes u1,2 = dx u1,2 (x).
In Appendix A we show that the connection between
the microscopic Hamiltonian and the S-matrix can be
clarified in some simple cases, but requires a detailed
knowledge of u1,2 (x), when one goes beyond the lowest
Born approximation.
Returning to (2), we define Fourier transforms,
∞ dk
∞ dk
+
+
ψ1 (x) = 0 2π eikx c+ and ψ2 (x) = 0 2π e−ikx c+ .
˜1,k
˜2,k
Then we have for the electron creation operator at position x
ψ + (x) =

+
+
˜ +
Θ(−x) ψ1 (x) + rψ1 (−x) + tψ2 (x)

+
+
+ Θ(x) tψ1 (x) + rψ2 (−x) + ψ2 (x)
˜ +

, (4)

with the step function Θ(x) = 1 at x > 0.
II.

INTERACTING CASE, PREVIOUS WORKS
A.

Boundary sine-Gordon model

In bosonization technique, the chiral fermions are rep†
resented as exponentials of chiral boson fields, ψR ∼
†
iϕL
−iϕR
, ψL ∼ e . with the fields ϕR(L) = ϕ ∓ θ.
e
Here the primary field ϕ and its canonically conjugate
momentum Π = π −1 ∂x θ obey the commutation relation [ϕ(x), Π(y)] = iδ(x − y). We have for the density
†
†
ψR ψR + ψL ψL = ∂x ϕ/π and the Hamiltonian (1), (3)
can be represented as
H =

vF
˜
2π

dx K (∂x θ)2 + K −1 (∂x ϕ)2

+2u1 (x)∂ϕ(x) + u2 (x) cos 2ϕ(x)

(5)

One has vF = vF and K = 1 for free fermions, whereas
˜
the short-range interaction H1 between the fermions, Eq.
(1), leads to the renormalized Fermi velocity vF = vF (1−
˜
g 2 )1/2 and the Luttinger parameter K = [(1 − g)/(1 +
g)]1/2 . Here and below we use
g = g2 /(2πvF )

(6)

One usually argues, that the term u1 can be absorbed
into a redefinition of ϕ(x) → ϕ(x) + u1 sgn(x) (cf. Appendix A). After this redefinition one concentrates on
the u2 term which constitutes the boundary sine-Gordon
(BSG) problem. A solution to this problem, in the lowest
order of u2 was presented in the early work.4,5 Focussing
on the small-energy sector of the problem, one uses the

4
renormalization group approach, removing the higher energy states of the problem while simultaneously rescaling
the parameters of the action. It turns out that under this
procedure the amplitude of the BSG term is renormalized
as
u2 → u2 e(1−K)Λ

(7)

where the RG scaling variable Λ = ln EF /ǫ > 0 . As a
result, the conductance at small u2 , given by G = |t|2 ≃
1 − u2 , undergoes a similar renormalization
2
G ≃ 1 − |u2 |2 e2(1−K)Λ

−1

−1)Λ

Fermionic approach, RG for S-matrix

Yue, Matveev and Glazman13 have developed a theory,
starting from the formalism of scattering states and taking into account the fermionic interaction in lowest order
of perturbation. They arrived at the RG equation for the
transmission amplitude t in the form
dt
= −gt(1 − |t|2 )
dΛ

(10)

(8)

with the lower cutoff in Λ defined by temperature or voltage bias, Λ = ln EF / max[T, V ]. We shall confine ourselves to the the linear response regime, V ≪ T in the
following.
For repulsive interaction K < 1 the conductance decreases with falling temperature. When the renormalized
impurity potential becomes strong, |u2 |2 e2(1−K)Λ ∼ 1,
the weak-impurity assumption is violated and one should
use a different line of reasoning. One may think of two
semi-infinite parts of the wire, connected by a weak tunneling link t. An appropriate RG treatment shows that
the repulsive interaction leads to a further decrease of the
conductance, which acquires the form
G ≃ |t|2 e2(K

B.

(9)

These two regimes, Eqs. (8), (9) show different scaling exponents, which are approximately equal for small
interaction
1 − K ≃ K −1 − 1 ≃ g.
A smooth crossover between the two asymptotes (8) and
(9) was first explicitly provided for a special case of interaction strength, K = 1/2, which is known as LutherEmery refermionization point. The full curve for the
conductance was obtained4,5,10 in the form of a oneparameter scaling function.
In subsequent works11,12 the BSG model was analyzed
at arbitrary values of Luttinger parameter, K. The set
of coupled integral equations was shown to be finite for
1/K = 1, 2, 3 . . . and the solution for K = 1/3 was derived, in particular. Corresponding scaling functions for
the conductance were obtained for finite T and voltage
bias on the contact V either by numerical solution of the
corresponding integral equation or as a series in powers of
the scaling parameter, see also18 . For practical purposes
this form of representation is not very convenient.
In principle, the statement of a one-parameter scaling function for the conductance and the knowledge
of its actual form provides a solution for the temperature renormalization of the barrier apparently at arbitrary strength. However, it was noted in11,12 , that
”the strong-barrier problem which is at the end of our
renormalization-group trajectory follows formally from
dimensional continuation of the integrals for the weak
barrier problem. It is not in any case a generic strongbarrier problem.”

The solution to this equation is found as
t =

t0
(|t0 |2 + (1 − |t0 |2 )e2gΛ )1/2

(11)

which, in leading order in g, agrees with Eqs. (8), (9).
The advantage of this approach is its applicability for
arbitrary strength of impurity potential. One may thus
avoid the step involving the transition from the initial
Hamiltonian (5) to the observed conductance, which as
we saw above can depend on the short-distance (ultraviolet) details of relevant quantities. This view is further
corroborated by the work by Callan et al.19 who discussed the solution of the non-interacting (K = 1) BSG
theory, and arrived at a current algebra formulation similar to our formulation below. These authors show that
the connection of the initial model (5) to the observable
quantities depends on the ultraviolet regularization. We
will return to this point in our discussion below.

III.

SCATTERING STATES AND CURRENT
ALGEBRA

In the above we defined a scattering states representation (4) in chiral representation. In order to describe the
effects of interaction, we need to define fermionic densities in the same basis. One may form four bilinear density
combinations out of two chiral fermions. It is convenient
to group these combinations into chiral densities or “currents”, according to

J(x) =
≡

†
†
ψ1 (x)ψ1 (x) ψ1 (x)ψ2 (−x)
†
†
ψ2 (−x)ψ1 (x) ψ2 (−x)ψ2 (−x)

(12)

J0 + J3 J1 − iJ2
J1 + iJ2 J0 − J3

(13)

We call J0 the pseudocharge current and the vector J =
(J1 , J2 , J3 ) the pseudospin current. These operators obey
U (1) and SU (2) Kac-Moody algebras, respectively1,20 , as
we discuss below.
The particle density operators for incoming (i) and
outgoing (o) particles in terms of the Jµ ’s are given by

5
(here and in the following x > 0 )
†
ρiR (−x) = ψ1 (−x)ψ1 (−x) = J0 (−x) + J3 (−x)

ρiL (x) =
ρoR (x) =
=
ρoL (−x) =
=

(14)

†
ψ2 (x)ψ2 (x) = J0 (−x) − J3 (−x)
(15)
†
†
∗
∗
(tψ1 (x) + r ψ2 (−x))(t ψ1 (x) + r ψ2 (−x))
˜
˜
†
(S.J(x).S )11 = J0 (x) + J3 (x)
(16)
†
†
∗
∗
˜
˜
(rψ1 (x) + tψ2 (−x))(r ψ1 (x) + t ψ2 (−x))
†
(S.J(x).S )22 = J0 (x) − J3 (x)
(17)

Here J3 = (RJ)3 is the third component of the pseudospin current vector J rotated by the orthogonal matrix
Rµν given by
Rµν =

1
T r[σµ .S.σν .S † ].
2

(18)

with Pauli matrices σµ . It is seen from (18) that the
global phase of S drops out completely. Omitting this
phase, we parametrize
S=

t r
˜
˜ =
r t

cos θ i sin θe−iφ
i sin θeiφ
cos θ

(19)

In this notation, the components of interest below read
R33 = |t|2 − |r|2 = cos 2θ, R32 = Im{tr∗ + tr∗ } =
− sin 2θ cos φ, and R31 = Re{tr∗ − tr∗ } = sin 2θ sin φ.
We see that the description of the barrier is given by
the eigenmodes of the current Ji (x), with negative and
positive x and i = 0, . . . , 3. In this picture, the incoming current is given by the diagonal components of the
J matrix, and the outgoing currents correspond to the
diagonal components of the rotated matrix S.J .S † . Evidently, the pseudocharge component J0 is not affected
by this rotation.
A.

Formal identities and Green’s functions

The chiral densities introduced above satisfy the KacMoody commutation rules
i
∂x δ(x − y)
(20)
4π
i
[Jj (x), Jk (y)] =
δjk ∂x δ(x − y) + iεjkl Jl (y)δ(x − y)
4π
[J0 (x), J0 (y)] =

with j, k, l = 1, . . . 3 and totally antisymmetric εjkl . The
short-distance operator product expansions (OPEs) are

[2π(x − y)]−1 . A bilinear form in the currents is a product of four fermion operators; the complete convolution
of these gives a product of two two-point Green’s function (modulo normally ordered operators), and the partial convolution produces a Green’s function multiplied
by bilinear forms in the fermion operators, i.e. current
operators. The position argument of the latter current
operators (last term in (21)) is subject to ambiguity, but
usually this concerns only less relevant terms. In our
analysis below we deal with an action non-local in the
currents, and to handle such an ambiguity becomes an
increasingly difficult matter. This is why we resort to the
perturbative treatment in terms of chiral fermions.
B.

Observable current and conductance

The physical charge density is ρ(x) = ρiR (x) + ρoL (x)
at x < 0 and ρ(x) = ρoR (x) + ρiL (x) at x > 0. A simple
calculation shows that
ρ(x) = ρc (x) + ρs (x),
ρc (x) = J0 (−x) + J0 (x),

(22)

ρs (x) = sign(x) −J3 (−|x|) + J3 (|x|) .
The physical current operator is also decomposed into
two parts, in the pseudocharge and pseudospin sectors.
We use the continuity equation −∂x j(x) = ∂t ρ(x) =
−i[ρ(x), H] to obtain
j(x) = jc (x) + js (x),
jc (x) = vF (−J0 (−x) + J0 (x)),

(23)

js (x) = vF (J3 (−|x|) + J3 (|x|)),
It follows then that the pseudocharge current corresponds to the part of physical density which is even with
respect to the reflection at the scatterer, ρc (−x) = ρc (x).
The pseudospin current corresponds to the odd part of
the density, ρs (−x) = −ρs (x). This symmetry property simplifies the discussion of a voltage bias, applied
to the barrier. One sees immediately, that applying the
same electrochemical potential to both leads of the wire
corresponds to its coupling to the pseudocharge current.
It results in an overall increase of the fermionic density,
symmetric with respect to the barrier, and not to a net
physical current. By contrast, an electric potential of the
1
form V (x) = 2 V sign(x), couples only to the pseudospin
part, leading to a term in the Hamiltonian
∞

1
−1
,
J0 (x)J0 (y) =
8π 2 (x − y − i0)2
−1
δjk
1 εjkl Jl (y)
Jj (x)Jk (y) =
+
(21)
.
8π 2 (x − y − i0)2
2π x − y − i0
In simple terms, relevant to our discussion below, one can
consider (20) as a consequence of (21). In their turn, the
OPEs (21) simply represent fermionic Green’s functions,

HV = V
0

dx (J3 (x) − J3 (−x))

The physical current induced by a bias voltage is therefore given in linear response theory as
j(x, ω) = G(x, ω)V (ω)
∞

G(x, t) = −iΘ(t) [js (x, t),

dy ρs (y, 0)] (24)
0

6

1

1
g2

1

2

g2

2

(a)

2
g2

1

1
2

J

J

1
(b)

1a the g2 -interaction processes are shown in the usual
scattering configuration. The representation in terms of
the chiral currents Jµ (all particles moving to the right)
is shown in Fig. 1b and leads to a seemingly nonlocal
interaction. Notice that in the case of perfect reflection,
˜
t = 0, we have J3 = −J3 and the observable densities
(see below) form an Abelian U (1) sub-algebra of SU (2).
This case of “open boundary bosonization”, considered
by Fabrizio and Gogolin, allows a complete and rather
simple analysis21 . In the next subsection we elucidate
what happens at the origin, x = 0. .

2

D.

2

g2

FIG. 1: (Color online) (a)The interaction of the left- and
right-going densities in the basis of scattered waves. (b) The
interaction of the left- and right-going densities in the chiral
basis, corresponding to non-local interaction, Eq. (26).

A remark is in order here. Strictly speaking, the prefactor
vF in the definition (23) applies for non-interacting leads.
In the interacting region we have to replace vF → vF −
g2 /2π in (23). We may avoid this complication, analyzed
elsewhere7,17 , by assuming in (24) that both js (x, t) and
ρs (y, 0) are placed outside the interacting region, x, y >
L. The change of the limits of integration over y in (24)
is not important in the limit of d.c. conductance |ω| ≪
vF /L, considered below.
C.

Using the parametrization (19), one finds that Rij =
exp(εijk Bk ) i.e., it is a finite rotation characterized by
the dual vector B = 2θ(cos φ, sin φ, 0). We cannot symmetrize the Hamiltonian in the pseudospin space, reducing it to the so-called Sugawara form,1 because the vector
B breaks the SU (2) symmetry of the problem. We may
symmetrize the Hamiltonian in the plane perpendicular
to B, but that provides little advantage .
The above formulation of the problem, incorporating
the knowledge about the scatterer in the asymptotic scattering states, is equivalent to the the alternative formulation of the pseudospin current being rotated by a fictitious magnetic field B located at the origin. The pseudocharge sector remains unchanged and the pseudospin
sector of the problem can be equivalently described by
the following Hamiltonian
H′ =

Hamiltonian

The barrier as magnetic field

∞
−∞

It is known that fermions with linearized dispersion in
one spatial dimension can be described by a Hamiltonian, quadratic in chiral fermionic densities. This observation, usually attributed to Tomonaga, is the basis of
the bosonization approach.1 Similarly to (5), we have
∞

H0 = πvF
0

2
dx 2πvF O3 (x) − g2 O3 (−x)O3 (x)

+vF Bj Oj (x)δ(x)) ,

(27)

where the operators Oj (x) ≡ U † Jj (x)U , with j = 1, 2, 3
obey Kac-Moody relations (20). The two Hamiltonians,
(25), (26) and (27), are connected by a canonical transformation H ′ = U † HU , where
∞

dx(ρ2 (−x) + ρ2 (x) + ρ2 (x) + ρ2 (−x))
iR
iL
oR
oL

which can be represented as
0

H0 = 2πvF

2
2
dx (J0 (x) + J3 (x))

−∞
∞

+2πvF
0

(25)

2
2
dx (J0 (x) + J3 (x))

The interaction part is
L

H1 = g 2

dx (ρiR (−x)ρoL (−x) + ρoR (x)ρiL (x))
a
L

= 2g2
a

dx (J0 (−x)J0 (x) − J3 (−x)J3 (x)) (26)

In Fig. 1 we show in a pictorial way our parametrization
of the fermionic densities and the interaction, H1 . In Fig.

U = exp i

dxBj Jj (x).
0

We derive the relation between Oj (x) and Jj (x) in Appendix B.
Equation (27) resembles the Hamiltonian for the
Kondo problem in the current algebra representation.20
However there are important differences. The first one
is the non-local interaction g2 between the incoming and
outgoing waves in our case. The second difference is the
classical nature of the field B, as opposed to the quantum
nature of the Kondo spin S.
The matrix Green’s function for chiral fermions is diagonal in pseudospin space when defined in terms of the
asymptotic eigenstates. However in terms of the usual
right- and left-movers its matrix structure is evidently
given by Eq. (19), which shows the action of the mag1
netic field S = exp isB, with s = 2 σ. We will use this
fact in Sec. VI below.

7
IV.

Using the definition (12), and free fermionic Green’s functions, one obtains e.g. the precise relation

RG EQUATION FOR S-MATRIX

Let us show how the renormalization group equation
for the many-body S−matrix is simply reproduced in
the current operator formalism. Following Affleck and
Ludwig,20 we write the g2 term of the S−matrix in the
interaction representation as
L

Tt exp 2ig2

dt

dxJ3 (−x)(RJ)3 (x)
a

Expanding the latter object in first power of g2 , we find
at given t
L

−2g2

dxJ3 (−x)R3j Jj (x)
a

Applying OPE rules (21) to the last expression one finds
an operator term
J3 (−x)Jj (x) ∼

ε3jk Jk (0)
2π −2x

After eliminating the distances close to the scatterer, i.e.
integrating over x from a to a certain a′ , which serves
now as new lower cutoff, we obtain the term which can
be re-exponentiated to the action giving a contribution
of the form δH = vF δBk Jk (0), with
δBk = ε3jk R3j

g2
2πvF

a′
a

dx
.
x

We notice further that ε3jk R3j
≡
Bk
=
sin 2θ(cos φ, sin φ, 0), i.e., the additional magnetic
field at the origin is parallel to the above B, and we
may restrict the discussion to the change in modulus,
a′
B → B + δB. Denoting dΛ = a dx/x, we obtain
d(2θ)/dΛ = g sin 2θ, or
d sin2 θ
= 2g sin2 θ cos2 θ
dΛ

(28)

which is equivalent to Eq. (10).
After this verification of the previous result in the first
order in g, we are tempted to obtain the next-order corrections to the RG equation (28). It turns out however
that working entirely in the current operator formalism
gives no advantage in such analysis, as we explain in the
next subsection.

A.

Higher orders in current operator formalism

Let us return to the last term in (21)
Jj (x)Jk (y) ∼

εjkl Jl (y)
.
2π(x − y)

J2 (x)J3 (y) =

†
†
ψ1 (x)ψ2 (−y) + ψ1 (y)ψ2 (−x) + h.c.
, (29)
8π(x − y)

modulo the normal ordered part. Evidently, the numerator in (29) is reduced to 4J1 (y) only in the limit x = y.
In other words, the relation (21) is the ”short-distance”
expansion and must be used with caution. The meaning
of the relation (21) becomes clearer when we compare
the results obtained within the purely fermionic (and
thus unambiguous) approach with the results provided
by the use of (21). When available, this comparison
yields identical results in the limit of large (external)
times and distances. Hence, the relations (21) provide a
good mnemonic rule for a quick estimate of a given contribution to a desired quantity. However, if we want to
obtain concrete prefactors and verify the renormalizability of the theory, it is better to use the standard fermionic
approach, treating the currents Jl as pseudospin vertices.
Proceeding this way below, we use a symmetry of the
problem, namely the unitarity of the S-matrix related to
the conservation of charge, which leads to a pure rotation
of the vector current operator after the scattering event.
A different attempt to explore the current structure
of the theory would be the use of the equation of motion method, which we describe in the Appendices C and
D. If successful, this strategy would account for the influence of interaction in all orders. The main difficulty
arising in this approach is the nonlocal character of interaction (26) and the appearance of an infinite series of
coupled equations. We show in Appendix D, that the
attempt to truncate this series by considering the ”most
relevant” contributions leads, perhaps not surprisingly,
to the restoration of the previous result (11). In order to
go beyond this level of consideration, we use the regular
perturbation theory in chiral fermions in the remainder
of the paper.

V.

CORRECTIONS TO CONDUCTANCE:
PERTURBATION THEORY
A.

Diagrammatic technique

Let us first formulate the rules for the calculation of
corrections to the conductance in perturbation theory in
the interaction g.
The contributions to G(Ωm ) in n-th order of g2 may
be calculated with the help of Feynman diagrams in the
position-energy representation (Ωm is the external Matsubara frequency). We draw n vertical wavy lines in parallel, each carrying the factor −2g2, the upper endpoint
1 3
of the i-th line at −xi with pseudospin matrix 2 ταβ , the
µ
lower one at xi with matrix 1 R3µ ταβ attached ; α, (β) are
2
pseudospin indices of incoming (outgoing) fermion lines.
1
The external vertices are at −y with matrix 2 τ3 and at

8
1
x with matrix 2 (Rτ )3 . The vertex points are connected
by Green’s functions
−1
G(x ; ωn ) = −ivF sign(ωn )Θ(ωn x)e−ωn x/vF ,

(30)

where the ωn are Matsubara fequencies ωn = (2n+ 1)πT .
All internal x -variables are integrated on the positive
semi axis. The trace over the product of all isospin matrices in each fermion loop is taken and a factor of 1/n! is
applied to each n-th order diagram. The limit Ωm → +0
is taken at the end.
After this brief overview we provide further details
about our calculation.
(i) From Eqs. (23), (24) it follows that only two out
of four correlation functions in G(x, ω) are non zero.
Namely, the chiral character of the currents (21) leads
to J3 (x, t)J3 (y, 0) ω = 0 at x < y. Then assuming
y > x = L in (24) we get
G(x, t) ∼

dy [J3 (−x, t) + J3 (x, t), J3 (−y, 0)]

The first term here gives a contribution G(1) (ω → 0) =
1/2, it is not affected by interaction. The second term is
G(2) (ω → 0) = R33 /2 = cos(2θ)/2 for free electrons, so
that G = G(1) + G(2) = cos2 θ = |t|2 as expected.
(ii) It is this second correlation function, ∼
J3 (x, t)J3 (−y, 0) , which undergoes renormalization.
Let us denote this quantity by Y :
Y ≡ 2G − 1 = R33 = cos 2θ

(31)

The decrease of the conductance can be viewed as the
increase of the rotation angle θ. Below we discuss the
corrections to Y .
(iii) In order g n , we draw a diagram which contains
2n + 2 current vertices. Two currents are external, and
2n come from n pairs of currents in each interaction term.
We fix the coordinates of the interaction terms and calculate the diagrams. In the limit of external frequency
Ω → 0 each diagram is ∝ Ωm with m ≥ 1, the integration
over external y in (24) reduces this power m → m − 1
so that only the linear-in-Ω contribution survives in this
limit ; see also paragraph (viii) below. After this procedure we have finite contributions, dependent on the
positions of interaction points, xi , and should integrate
over these positions.
(iv) These last integrals over xi (i = 1, . . . n), if taken
over the entire semiaxis (0, ∞), diverge logarithmically
and it is precisely at this step when we have to introduce
the cutoff, as we discuss at length below.
(v) Each current vertex has matrix structure and is
σ3 for a current at negative coordinate (before scattering) and cos 2θσ3 + sin 2θσ2 for positive coordinate (after
ˆ
scattering). We denote these matrices by Vj . Corrections are generated by connecting all current vertices by
Green’s functions. Technically, we employ Mathematica
and generate all possible permutations {j1 , j2 , . . . , j2n+2 }
of vertices and make one fermionic loop, connecting these

vertices in given order. The advantage of current representation is that the information about the scattering is
encoded in Pauli matrices and the kinetics is given by
chiral Green’s functions (30). The matrix structure produces an overall prefactor before the diagram of the form
1
ˆ
ˆ ˆ
T r[Vj1 .Vj2 . · · · .Vj2n+2 ]
2
(vi) Other corrections are generated when 2n + 2 current vertices are connected by more than one fermionic
loop. We denote these structurally different sets of diagrams by listing the number of current vertices in internal
loops. In the zeroth order of g we have the trivial set {2},
in the first order of g we have only a set {4}. In second
order of g we have two sets, {6} and {4, 2}. In the third
order of g we have four different sets : {8}, {6, 2}, {4, 4}
and {4, 2, 2}. Some of these diagrams are shown in the
Figures 2 and 3.
(vii) To exclude double counting of different graphs in
this procedure, we should fix one element in each loop.
We always fix one external vertex as a first element in the
first loop. In addition, we assume that the coordinates of
internal vertices are ordered, i.e. L > x1 > x2 > x3 > 0,
etc. Then. e.g. the number of graphs in the {8} set becomes 7! = 5040. In the {6, 2} set, one has 7!/2 graphs,
with factor 2 stemming from the necessity to fix one element in the second loop. Similarly, the set {4, 4} is obtained by taking 7! permutations, then partitioning each
permutation into two parts of length 4, summing all possible diagrammatic contributions and dividing the result
by factor 4 to account for the necessity to fix one element
in the second loop.
(viii) We do not explicitly require that both external
vertices belong to the same loop, e.g. in the diagrams of
the {4, 2} type. However, before the final integration over
the internal coordinates, we seek contributions, which are
proportional to the first power of external frequency Ωn ,
this linear dependence coming from two external vertices
belonging to the same loop. If these vertices belong to
two different loops, then such diagram is proportional
at least to Ω2 . After analytic continuation iΩn → Ω
n
and (non-logarithmic) integration over interaction region
(−L, L) we have an extra factor, ΩL, which is small in
the limit Ω → 0. In other words, we neglect the screening of external field, which is given by RPA series where
fermion polarization loops are (1-reducibly) connected by
interaction lines. It is allowed when measurement times,
∼ ω −1 , are large compared to times of travel through the
interacting region, L/vF . Thus we find the value unity
for the conductance in the absence of the barrier.6,7,17

B.

From perturbative corrections to the RG
equation

In this subsection we discuss the general form of corrections to the conductance and outline basic ideas of

9
the renormalization group approach, which we will use
below.
Starting with the conductance 0 < G < 1 in the noninteracting limit, the corrections δG induced by the interaction take the form of a power series in g. More
precisely, we discuss the equivalent quantity Y = 2G − 1
, equal to cos 2θ in the limit g → 0 . We denote the
quantity Y , renormalized by interaction effects, by Yr
and recall that it may be represented as a power series
in g . The coefficients of this series contain terms varying as powers of Λ = ln(L/L0 ) (at zero temperature)
or Λ = ln(T0 /T ) at finite temperature T in the limit of
large L . Here L0 and T0 are ultraviolet cutoff parameters. Accordingly, Yr may be represented as a double
series,
Yr = Y0 + b11 gΛ
+b22 g 2 Λ2 + b21 g 2 Λ
+b33 g 3 Λ3 + b32 g 3 Λ2 + b31 g 3 Λ + . . .

(32)

(33)

Naively speaking, the correction should be small δY =
Yr − Y ≪ Y , in order to be considered as a correction. More precisely, the series (32) should be converging. When all |bij | ∼ 1 and Λ ≫ 1, this convergence is
achieved for gΛ ≪ 1 which in turn imposes stricter conditions on the smallness of g. However, assuming that
Yr (g, Λ) is an analytic function of Λ, we expect that an
analytical continuation from the region of small Λ to large
Λ should be possible23 . Then we can relax the requirement of the smallness of g. Rearranging the above series
we write
Yr (Λ) = Y0 + β1 Λ +

β2 2 β3 3
Λ + Λ + ...
2
3!

(34)

where the coefficients βi are functions of Y .
If the theory is renormalizable, the following scaling
relation holds:
Yr (g, Y, T ) = F (g, T /Θ)
where Θ = Θ(g, Y ) is the correlation temperature of the
problem (an analogous relation holds for the scaling with
L at zero temperature). The scaling property implies the
existence of a renormalization group equation
∂Yr (g, Y, Λ)
= β(g, Yr (Λ))
∂Λ

dYr (g, Λ)
dΛ

= β1 (g, Y ) = β(g, Y0 )

(35)

where β(g, Yr ) is the so-called β-function. To obtain β
from perturbation theory we take the first derivative of

(36)

Λ=0

In the last equation we have replaced Y by Y (Y0 ) , obtained by inverting the series (33) for Y0 in powers of
Y , which transforms the function β1 (g, Y ) into β(g, Y0 ) .
Now we see that (36) is nothing else but the RG equation
(35) at Λ = 0 where Yr = Y0 . Consequently, the function
β(g, Y0 ) is the true β-function.
To check whether the scaling and therefore the RG
equation holds, we may integrate (35) to recover the perturbation series, by employing the inverse function
Yr

Λ=
Y

with bij functions of Y . The terms of the form (gΛ)n
in (32) are conventionally called “leading” contributions,
and the terms of the form g n Λm are called “subleading”
for n > m. The first term Y0 is the sum of all scale
independent contributions
Y0 ≡ Yr (Λ = 0) = Y + b20 g 2 + b30 g 3 + . . .

(34) with respect to Λ and put Λ = 0

dY
β(g, Y )

(37)

d
d
Alternatively, noting that dΛ = β(Y ) dY we may obtain
the Taylor expansion in powers of Λ directly in the form

1
Yr (Λ) = Y + Λβ(Y ) + Λ2 β(Y )β ′ (Y )
2
1 3
+ Λ β(Y )[β(Y )β ′ (Y )]′ + . . .
6

(38)

with a prime denoting the differentiation with respect to
Y.
In what follows we, firstly, verify the structure (38) of
the theory up to third order in g, using computer algebra.
Thus we show the applicability of the RG approach. Secondly, on the basis of this analysis we propose a method
of resummation of the most important contributions to
the β-function in all orders of g.
But before presenting these results we return to the
significance of the scale independent terms in the perturbation theory (the difference between Y0 and Y ), as this
problem has been sometimes overlooked in the literature.
In the first order of g we can absorb such finite terms into
the definition of Λ. However, after this fixing of the definition of Λ, higher order scale independent terms cannot
be removed by redefinition. For example, we should expect a term b21 g 2 (Λ + c) with c ∼ 1 in the second line of
(32). It is clear that in the third order we can find a corresponding multiplicative contribution ∼ b11 b21 g 3 Λ(Λ + c),
which contains a linear-log term and thus affects the definition of the β−function in the form of (36). Evidently,
such terms arise only at the level of g 3 , or “in the third
loop”, and do not affect the leading-log contributions. As
this problem is of principal importance, we present in the
following a second line of derivation of the β-function.
Within the Callan-Symanzik (CS) formulation of the
RG method24,25 one may start with the perturbation series of Yr in terms of the bare conductance Y , (32). The
latter relation is inverted and Y is expressed in terms of
the “running” parameter Yr
Y = Yr + ¯11 gΛ
b
+¯22 g 2 Λ2 + ¯21 g 2 Λ
b
b
¯33 g 3 Λ3 + ¯32 g 3 Λ2 + ¯31 g 3 Λ + . . .
+b
b
b

(39)

¡¢£

10

with ¯ij functions of Yr . The expansion (39) is the essence
b
of the CS formulation, appropriate for our problem. Here
Y is understood to be a function of the two variables
Yr (Λ) and Λ . Since the quantity Y is independent of Λ
we have the relation
0=

∂Y
∂Y ∂Yr
dY
=
+
dΛ
∂Λ
∂Yr ∂Λ

(40)

∂Yr
∂Y /∂Λ
=−
∂Λ
∂Y /∂Yr

(41)

or
β=

In practice, the calculation of the β-function from (41)
requires the calculation of the coefficients ¯ij from perturb
bation theory, which is analogous to the transformation
of β1 (Y ) to β(Y0 ) discussed above. If everything is done
correctly, then the expression (41) does not depend on Λ
explicitly.
The CS procedure described above is different from
the Gell-Mann–Low (GL) formulation of RG. In the latter one considers the value of conductance used in the
calculation to be defined at the scale Λ ; which means Yr
in our notation. To eliminate the large terms involving
Λ one introduces counterterms into the Hamiltonian (cf.
below), in order to guarantee a finite value of the bare
conductance, with a resulting equation similar to (39).
Usually the insertion of the counterterms is done in the
“minimal subtraction” scheme, meaning that scale independent contributions are omitted. As we show in Appendix G, this GL formulation leads to ambiguities in
the definition of the β-function at the level of the third
loop, in contrast to the CS formulation or the scaling
formulation presented in Eq. (36).

C.

−x

−z

(a)

−x

In the perturbation theory calculation the logarithmic
factors appear at the last step, when integrating over the
points of interaction. Prior to this step the calculations
are straightforward. We proceed first with the zero temperature case. In this case the sums over discrete Matsubara frequencies transform into integrals over imaginary energies, which are rather easily done by computer
symbolic computation.
Each diagram contains a prefactor Ωe−Ω(x+y) with the
two external vertices at x, y > 0. Integrating over y and
putting Ω → 0 makes this prefactor unity. Equivalently,
we may keep only the first, linear-in-Ω, term in each diagram and let Ω = 0 in its remaining part. In the first order of g the correction is given by the diagrams in Fig. 2.
The vertex correction part in Fig. 2c cancels and the two
first diagrams give an expression which contains 1/x1 .
This should be integrated over the point of interaction
x1 in the limits (a, L) which produces a factor
(42)

−z

−z

(b)

z

(c)

z

y

z

y

y

FIG. 2: Three Feynman diagrams, depicting the first order in
g2 contribution to the conductance. The other three diagrams
are obtained from these by reversing the direction of fermionic
propagation.

Similarly, in the second order of g we have two contributions, (x1 x2 )−1 and (x1 + x2 )−2 , which lead to Λ2 and
0
Λ0 , respectively. We provide further details in Appendix
E and list here only the summary of our calculation.
Grouping corrections in powers of g, as in (32) we have
for the renormalized quantity Yr :
Yr = Y − c{4} gΛ(1 − Y 2 )

+ g 2 Y (1 − Y 2 )[−c{6} (Λ2 − a4 ) + c{4,2} (Λ − a1 )/2]

+ g 3 (1 − Y 2 ) − c{8} (Λ3 − 3Λa4 )(Y 2 − 1/3)
+ c{8} (1 − Y 2 )Λa2 + c{6,2} Y 2 Λ(Λ − a1 )

− c{4,4} (1 − Y 2 )(Λ2 /4 − a3 Λ)
− c{4,2,2} (1 + Y 2 )Λ/4

(43)

where a = (2 ln 2, π 2 /12, (ln 2 − 1)/2, 0). All c{·} = 1
and these coefficients are shown only for reference to the
underlying group of diagrams, according to the above
conventions.
Using the above CS scheme we come after some algebra
to the RG equation dYr /dΛ = β(Yr ) with
β(Y ) = (1 − Y 2 ) −g + g 2

Summary for the zero temperature case

Λ0 = ln L/a

−x

c3

Y2+1
Y
− g3
2
4

+ c3 g 3 (1 − Y 2 )2 + O(g 4 )
π2
1 − ln 2
= a2 + a3 − a4 =
−
12
2

(44)

Here the value of c3 is that of the T = 0 case (cf. below).
This value differs from our previously reported result27 ,
which accounted only for the set {8} of third order diagrams. Here we found an aditional contribution from the
{4, 4} set.
It is important to keep the scale independent contributions in the terms {6} and {4, 2} in (43) in order to have
the agreement with the result by Kane and Fisher in the
limits of weak scattering and weak tunneling Y 2 → 1.
One can check that the terms with higher powers of Λ
in (43) are reproduced by Eqs. (38) and (44). In fact, this
applicability of the RG equation is manifested already
in the absence of Λ in the right-hand side of (44). This
means that, when discussing diagrammatic contributions
in nth order in g we should concentrate only on the leastleading logarithm, i.e. the terms ∼ g n Λ. These linear-inΛ contributions arise in two qualitatively different cases.

¡¢£
¤¥¦
§¨
+

+

+

+



+


+

+

+



symm.


FIG. 3: Skeleton Feynman graphs, leading to lowest order
logarithmic contribution, g 3 Λ, beyond ladder series.

In the first case, the entire Feynman diagram is proportional to a single logarithm. This happens with the
three first terms in (44) stemming from diagrams with
the maximum number of fermionic loops. In the above
notation, these are the sets {4}, {4, 2}, {4, 2, 2} for corrections in order g, g 2 , g 3 , respectively. We discuss these
corrections in the next section. The finite terms, O(1),
do not contribute to the β-function as can be checked by
the absence of coefficient a1 in (44).
In addition to these sets, there are contributions in the
third order, whose leading divergence is also linear logarithmic. They are depicted by a first skeleton graph in
Fig. 3, with the maximally crossed wavy lines of interaction. These graphs provide for one half of the value of a2
in Eqs. (43), (44).
In the second case, the linear-log contributions arise
as accompanying weaker divergence in the stronger divergent graphs. In simple cases, it may be a combination Λ3 + ΛO(1) ; in more complicated situations, it
may be a stronger singularity at internal points, e.g.
∼ dz1 /(z1 − z2 )−1 which is eventually removed upon
symmetrization, whereas the subleading log-divergence
survives and contributes to (43), cf.26 . This second
type of linear-log contributions happens in the remaining
graphs in Fig. 3 and provides for the rest of the value of
the last term c3 in (44).

D.

Finite temperatures

Let us discuss the effects of temperature, first in the
lowest order in g. At T = 0 the large logarithm of the
theory (42) came from the integration dx1 /x1 over the
range of interaction [a, L]. When the integration over internal frequencies is replaced by a summation over Matsubara frequencies, we find that the integrand contains
the fermionic Green’s function G(2x, t = 0). of the form
G(2x) =

T
1
↔
.
2vF sinh(2πT x/vF )
4πx

(45)

with an appropriate change in the definition
L

Λ0 → 4π

a

dx G(2x) = ln

tanh L/ξ
≡Λ
tanh a/ξ

(46)

11
with the temperature correlation length, ξ = vF /πT . Explicitly we have
Λ = ln coth(a/ξ) ≃ ln

T0
ξ
≃ ln ,
a
T

(47)

where the last relation takes place at larger temperatures,
when L ≫ ξ. Here we introduced the ultra-violet energy
−1
cutoff T0 = vF /πa; one may think of a ∼ kF so that
T0 ∼ EF . The results presented below refer to this case
of relatively high temperatures, T0 ≫ T ≫ vF /L.
We calculated the terms of perturbation theory by
means of computer algebra, similarly to the T = 0 case
above. The summary of the result is given by the same
expression (43), but the subleading coefficients are now
given by a = (3 ln 2, π 2 /12, ln 2 − 1/2, π 2 /24). In its turn,
it leads to a different numerical value of the coefficient
(44), we obtain for T ≫ vF /L :
c3 =

π2
1
+ ln 2 − .
24
2

(48)

Let us comment on this result.
We observe that the diagrams with the maximum number of fermionic loops lead to expressions linear in Λ with
the same prefactors as in the T = 0 case. They generate
the same part of the β function, given by the first line in
(44), and correspond to one-loop RG approximation. In
the next section we show that these terms form a series
which can be resummed in all orders of g.
The maximally crossed diagrams of the third order
which had only linear-log divergence at T = 0 lead to
the same a2 /2 coefficient in (43) at T ≫ vF /L.
Let us now consider the modifications in the second
order. The complete expression for the diagrammatic set
{6} is given by the expression (F2) which shows that a4
in Eq. (43) is a smooth finite function of temperature.
At finite T each particular diagram in order g 3 can lead
to a rather complex expression, and the resulting sum of
all diagrams becomes too complicated for a brute-force
approach. The analysis can be best done in the following way. We add all the terms of a given structural set
of diagrams, thus removing the internal singularities at
z1 = z2 , etc. As discussed above the leading logarithmic
behavior of the set (higher power of Λ) is determined by
the RG equation. We may hence subtract a simpler expression, with the same leading behavior, from the complicated general expression.
For instance, for the {8} set we pick all diagrams proportional to cos 8θ, and notice that
the zero-T expression has a leading contribution
(z1 z2 z3 )−1 with a certain coefficient. Then we expect that the finite-T expression should have a leading
term 8ξ −3 (sinh(2z1 /ξ) sinh(2z2 /ξ) sinh(2z3 /ξ))−1 with
the same coefficient. We subtract this term from our
expression and analyze the degree of singularity of the
remaining expression. It turns out that in the third order after the subtraction of such leading Λ3 terms we are
left with linear-log expressions. The coefficients before

12
these expressions are also smooth functions of T , as we
explain in the Appendix F.
Concluding this section, we verified the validity of
the RG-equation by checking that the higher-than-linear
powers of Λ are generated by the linear-log terms. This
holds both at T = 0 and at T ≫ vF /L; the resulting
β-function is somewhat different in these cases and we
discuss this difference below. There is a part of the βfunction, which is identical in both cases, and stems particularly from a sequence of diagrams with the maximum
number of polarization loops. We analyze this sequence
in the next Section and show that it is possible to resum
it in all orders of g.
It is important to note that the last term in (44) is vanishing in the limit when the barrier becomes fully transparent or fully reflecting, i.e. at Y → ±1. In order to see
that, let us consider the case (Y +1)/2 = G → 0, then the
first three terms in (44) read β ≃ 2G −g − g 2 /2 − g 3 /2
and lead to power-law scaling of G in accordance to resutls by Kane and Fisher (see below). At the same time
the last term in (44) is ∝ g 3 G2 and does not lead to a
noticeable effect in the limit considered.
It is also worth noting here, that the calculation both
at T = 0 and at T ≫ vF /L confirms the relation between
the subleading coefficients in sets {8} and {6, 2} on one
hand and {6} and {4, 2} on the other hand. This is seen
in Eq. (43) for the a4 and a1 coefficients. Without this
relation the results by Kane and Fisher for weak and
strong barrier would not be recovered.

VI.

COUNTERTERMS IN THE HAMILTONIAN

In the previous section we saw that the corrections to
the conductance are obtained as self-energy insertion into
the fermionic loop, whereas the vertex corrections vanish
in the dc limit Ω ≪ vF /L. This means, in particular,
that the effect of renormalization in this limit can be
found at the level of the one-particle Green’s function, or
by addition of counterterms to the effective low-energy
Hamiltonian.
We calculate the corrections to the matrix Green’s
function for the chiral fermions in the pseudospin sector
in perturbation theory. The rules of the diagrammatic
technique remain the same. This time we do not consider a closed fermionic loop, which described the conductance above. Instead we study the matrix Green’s
function G(x, −y, Ωn ) connecting distant points on different sides of the barrier in the chiral representation. The
initial Green’s function is diagonal in pseudospin space,
and the corrections to it make it non-diagonal. We show
that this non-diagonal form corresponds to additional rotation of pseudospin near the origin, in accordance with
consideration of Sec. IV.
Employing the CS scheme we start with the bare barrier, characterized by the parameter θ and find the renormalized value θr (θ, Λ) as described below. Then we determine the inverse relation θ(θr , Λ) and require the in-

dependence of initial θ at the scale of consideration Λ.
This provides us with the RG equation for θr
The prefactor in G(x, −y, Ωn ) = G exp −Ωn (x + y) is
found as
G = 1 + c{4}

gΛ
iσ1 sin 2θ
2

(49)

g2
(sin2 2θΛ2 − iσ1 sin 4θ(Λ2 − b1 ))
8
g2
−c{4,2} iσ1 sin 4θ(Λ − b3 )
8
g3
3(Λ3 − b1 Λ) sin 2θ sin 4θ
−c{8}
48
Λ3
−iσ1
sin 2θ(9 cos 4θ − 1)
2
−c{6}

+i12σ1Λ sin 2θ(b2 sin2 2θ + b1 )
g3
Λ(Λ − b3 ) sin 4θ(sin 2θ − 2iσ1 cos 2θ)
16
g3
+c{4,4} iσ1 sin3 2θΛ(Λ − b4 )
8
g3
+c{4,2,2} iσ1 Λ(5 sin 2θ + sin 6θ)
32
+c{6,2}

where again all c{·} = 1. The coefficients bj , j = 1, ...4
depend on the case in consideration. For the zero temperature we find b = (0, π 2 /6, 2 ln 2, 2 − 2 ln 2) and for the
finite temperature b = (π 2 /12, 0, 3 ln 2, 2 − 4 ln 2).
In order to establish the connection to the Hamiltonian
we observe that the above form (49) can be represented
as G = exp iσ1 (θr − θ) with
g2
gΛ
sin 2θ +
sin 4θ(Λ2 − Λ − b1 + b3 )
2
8
g3
4
+ sin 2θ Λ3 cos 4θ + Λ2 (1 + 3 cos 4θ)
16
3
+Λ((3 − 4b1 − 2b2 + 2b3 − b4 )

θr = θ +

+(1 + 2b2 + 2b3 + b4 ) cos 4θ) + O(g 3 Λ0 ) (50)
This means that the effect of the interaction is indeed reduced to a change in the magnetic field 2θ, in accordance
with Sec. III D.
We have for the value at Λ = 0
θ0 = θ +

g2
(b3 − b1 ) sin 4θ + . . . ,
8

Expressing θ by θ0 in (50) and inverting the series we
obtain
gΛ
g2
sin 2θr +
sin 4θr (Λ2 + Λ)
2
8
4
g3
− sin 2θr Λ3 cos 4θr + Λ2 (1 + 3 cos 4θr )
16
3
+Λ((3 − 2b1 − 2b2 − b4 )

θ0 = θr −

+(1 + 2b1 + 2b2 + b4 ) cos 4θr ) + . . .

(51)

13
The difference θ0 − θr , expressed in terms of θr , is interpreted as the counterterms to the Hamiltonian, needed
to compensate the divergent diagrammatic contributions.
In our approach we do not re-calculate diagrams with inclusion of the appropriately chosen counterterms, which
would be problematic, given the non-Abelian character
of the theory and the absence of Wick’s theorem. These
counterterms merely reflect the structure of the corrections to the magnetic field, Sec. III D above.
Demanding the independence of θ0 on Λ, we obtain
(here θ(Λ) = θr )
2

∂θ
g2
g3
= g sin 2θ −
sin 4θ +
sin 2θ(1 + cos2 2θ)
∂Λ
4
4
−c3 g 3 sin3 2θ + O(g 4 )
(52)

loops. These terms are ∝ (1 − Y 2 ) and therefore are
important −x any−x −x of the conductance. It is −x
at
value
possible
−x
−x
−x −x
1

1

1

2

1

2

−z

1

2

−z

¡ ¢ £ ¤ ¥
=

x2

+

x1

+

x1

x2

+

x1

z

x2

x1

z

+ ···

x2

¯
FIG. 4: Ladder series, L(x1 , x2 ; ωn ), describing the combined
effect of interaction and barrier, and leading to a linear-inlogarithm ln(T0 /T ) contribution to conductance.

to resum all the contributions of this type, as we discuss
below.

where
1
(2b1 + 2b2 + b4 )
4
ln 2 1
π2
+
− ≃ 0.67, T = 0
=
12
2
2
1
π2
+ ln 2 − ≃ 0.60, T ≫ vF /L
=
24
2

c3 =

which agrees with the above Eqs. (44), (48). This coincidence confirms that the vertex corrections to the conductance are unimportant in the considered dc limit.
Any method of rendering the logarithmically divergent
integrals finite is called a scheme of regularization. As we
explained above the contributions linear in Λ get additional contributions in third order of g from the scale
independent terms appearing in the second order of g.
We removed this ambiguity by using the scaling method
or the appropriately devised CS scheme. A part of the
answer is independent of the regularization scheme and
technically is obtained by summing the one-loop contributions to β-function. One loop means here one independent integration over the internal energy, leading to
a linear logarithm. Indeed the summation of the ladder
series in Sec. VII amounts to a finite renormalization of
the effective interaction constant, g → g. In addition to
that, the quantities calculated above in the order g 3 included linear-log contributions stemming from diagrams
with three independent energy integrations (Matsubara
frequency summations). Particularly, the set of diagrams
{4, 4} contains diagrams with vertex corrections to the
polarization loops in the ladder series Fig. 4, which contribute to the regularization-dependent coefficient b4 in
Eq. (49) and c3 in (52).
VII.

INFINITE RESUMMATION OF TERMS IN
THE β−FUNCTION

We observe that the first three terms in (44), (52) are
given by diagrams with maximum number of fermionic

A.

Ladder series and Wiener-Hopf equation

The above discussed linear-log contributions to the
conductance with the maximum number of fermionic
loops, can be viewed as a “dressing” of the wavy interaction line g in Fig. 2a, Fig. 2b by fermionic polarization
loops. This is shown schematically in Fig. 4, with the
result of the summation denoted by g and depicted by a
double wavy line.
We show below that the result of this summation is
scale independent (without logarithmic terms), so that
using the double wavy line in first-order contributions
Fig. 2a, Fig. 2b will produce a single logarithmic factor,
while containing all powers of g.
In the usual situation, the RPA summation of Fig. 4
is trivial after Fourier transform. In our case of chiral
fermion representation, we have left-right asymmetry of
the vertices whose value in pseudo-spin space depends
on the coordinate. Hence we always integrate over the
semi-axis only, which leads, as shown below, to an integral equation of the Wiener-Hopf type, instead of a
simple Fourier convolution. Still, the ladder equation for
the dressed interaction can be written and solved rather
simply.
Let us denote the interaction between the points −x
¯
and y by L(x, y; ωn ) where ωn is the Matsubara frequency
along the line. Let us define the vector X(x, y; ωn ) in the
following way

X(x, y; ωn ) =

¯
L(x, y; ωn )
¯
L(x, −y; ωn )

(53)

Then the ladder summation, Fig. 4, is expressed for
ωn > 0 by the integral equation :

14

X(x, y; ωn ) = −4πgδ(x − y)

1
− gωn
0

Y e−ωn (x+z)
Θ(x − z)e−ωn (x−z)
.X(z, y; ωn)
−ωn (z−x)
Θ(z − x)e
0

dz

(54)

where we used the fact that the polarization loop in the (ωn , x) representation is J3 (x, τ )J3 (y, 0) ωn =
(4π)−1 ωn Θ(ωn x)e−ωn x . Here and below the integration over coordinates is over the interval (a, L), it is convergent
and may be extended to (0, ∞).
¯
Later we will need the integrated quantity L(x; ωn ) = dye−ωn y L(x, y; ωn ), which obeys the integral equation
g
L(x; ω) = −ge−ωx 4π + ω(Y + )
2
B.

1
dze−ωz L(z; ω) + g 2 ω
2

dz e−ω|x−z| L(z; ω) .

(55)

Solution and its properties

The above equation for L(x; ω) is an integral equation of Wiener-Hopf type, and its solution can be found as follows.
We differentiate both sides of Eq. (55) twice with respect to x and find that d2 L(x; ω)/dx2 = ω 2 (1 − g 2 )L(x; ω).
Demanding that L(x → ∞; ω) = 0 we seek a solution in the form L(x; ω) = C exp(−ωx 1 − g 2 ). Substituting the
latter form into (55), we determine the constant C and find
L(x; ω) = −4πg

1+
1+

1 − g2

1−

g2

+ gY

e−ωx

√

1−g2

.

(56)

Now we are in a position to determine the linear-log correction to the conductance stemming from the diagrams Fig.
2 where the wavy line of interaction is replaced by the double wavy line.
The equation reads :

G(L) =

1−Y2 2
T
4

¯
dx1 dx2 dy L(x1 , x2 ; ω)G(x1 − x; ǫ)G(−x1 − x2 ; ǫ − ω)G(x2 − y; ǫ)G(y + x; ǫ + Ω) , (57)

ǫ,ω

where the superscript L in G(L) denotes ladder summation. Taking here the limit Ω → 0 we find
G(L) =

−2g(1 − Y 2 )

1+

1 − g 2 + gY

Λ0 ≡ −g(1 − Y 2 )Λ0 .

(58)

Here we defined the renormalized interaction constant g.
We observe that
g = 1 − K,
= K −1 − 1,

Y =1
Y = −1

above, we find

1 ¯
ωn g 2 −ωn d|x−y|
L(x, y; ωn ) = −δ(x − y) −
e
(60)
4πg
2d
ωn g 2 −ωn d(x+y) Y (1 + d) + g
+
e
2d
Y (1 − d) + g

(59)

with the Luttinger parameter K = (1 − g)/(1 + g).
In fact, the expression (57) already assumes the limit
Ω → 0, because the prefactor 1 − Y 2 is obtained by
adding two contributions, one shown in Fig. 2b (∝ Ω(1 −
2Y 2 )) and another in Fig. 2a with inverted direction of
fermionic lines (∝ Ω).
The importance of the limit Ω → 0 in the derivation
¯
of (58) is further illustrated by the solution L(x, y; ωn ) of
the initial Eq. (54) rather than Eq. (55). After a long and
straightforward calculation, similar to the one described

with d = 1 − g 2 . The first term in (60) is the bare
interaction, the second one is the result of RPA summation, which is independent of the barrier transparency Y
and finite even in the bulk, at x ≃ y → ∞. The last term
explicitly contains the transparency Y and decays away
from the barrier. Integrating over y we get the above
Eq. (56), and Eq. (60) shows that the “dressing” of the
interaction constant, g → g, in (58) is not reduced to a
simple multiplicative factor, but depends on the coordinates. Finally we note that G(L) = 0 at Λ = 0, i.e. there
are no scale independent contributions in the ladder approximation, and Y0 = Y in that case.

15
VIII.

Integrating Eq. (65) we obtain the following result :

RG EQUATION AND ITS SOLUTION

From the solution of the Wiener-Hopf equation for the
ladder series (58) we then find the β-function in ladder
approximation

βL (Y ) = −

1+

2g(1 − Y 2 )

1 − g 2 + gY

(61)

Inverting this equation and using the definition of K we
obtain the symmetric form
1
1
1
1
dΛ
1
1
=
=−
−
−1 1 + Y
βL (Y )
dY
2 1−K 1−Y
1−K
(62)
invariant under the simultaneous exchange of K ↔ K −1
and Y ↔ −Y . This differential equation may be solved
analytically27 . We recall that it is asymptotically exact
in the limit |Y | → 1 .
Let us now turn to the first correction beyond this
series in Eq. (44). Applying the correction as calculated
we get the improved RG equation
dY
2g(1 − Y 2 )
+ c3 g 3 (1 − Y 2 )2
=−
dΛ
1 + 1 − g 2 + gY

(63)

The correction term is qualitatively different in that it
vanishes quadratically in the limit |Y | → 1 and hence
is irrelevant in this limit (considered in the early work
by Kane and Fisher). The duality property still holds,
i.e. the invariance of (63) under the simultaneous change
g → −g and Y → −Y (or θ → π/2 − θ in (52)). Based
on these observations, we expect that the fourth-order
terms beyond the ladder series in the RG equation will
be proportional to g 4 Y (1 − Y 2 )2 . One should keep in
mind that the correction terms here have been included
only to order g 3 . We now discuss a further improvement.
We may perform yet one more partial resummation by
dressing interaction lines as g → g, where
g=2

1+K
+Y
1−K

−1

(64)

The c3 term in (63) results from non-ladder diagrams.
Therefore without double-counting we can dress the interaction constant there and write c3 g 3 (1 − Y 2 )2 . Then
(62) receives a correction term of the form
dΛ
1
1
1
1
1
= −
−
dY
2 1−K 1−Y
1 − K −1 1 + Y
2(1 − K)
− c3
+ ...
1 + K + Y (1 − K)

(65)

The term proportional to c3 here is of order of g at small
g and we omitted higher order terms.

(T /T0 )2(1−K) = Φ(G)/Φ(G0 ) ,
(66)
K
G
(K + G(1 − K))4c3 (1−K) ,
Φ(G) =
1−G
we assume the initial condition G = G0 = (1 + Y0 )/2 =
−1
2
cos2 (θ) + c3 g0 cos(2θ) sin2 (2θ) at T = T0 , and g0 =
1
2

1+K
1−K

+ cos(2θ) ; here c3 is given by (48) for the case

T ≫ vF /L. Evidently, the renormalization stops at T
vF /L.
IX.

SUMMARY AND DISCUSSION

In this work we have presented a fermionic formulation of the transport theory of interacting electrons in
a onedimensional system with potential barrier in the
linear response regime. We considered the Luttinger liquid model with interaction g2 (forward scattering involving right and left movers), and employed a current algebra represention, leading to a substantial simplification of perturbation theoretical calculations. Using the
renormalization group idea in various forms we showed
that the conductance as a function of temperature (or, at
zero temperature, as a function of system length) obeys
scaling, and therefore may be obtained by integrating a
renormalization group equation. The β-function of the
latter equation has been found within a ladder approximation, which becomes exact in the limit of small or
large conductance, at any interaction strength. Corrections to the ladder approximation up to and including
third order in the interaction, and relevant at intermediate values of the conductance, have been calculated.
These correction terms require the solution of a fundamental problem of renormalization group theory: What
is the significance of and how does one treat scale independent contributions appearing in perturbation theory?
We show that these terms have to be taken into account,
as they guarantee the scaling of the conductance and can
lead to an important redefinition of the β-function. In
that respect our work is of general interest in the context of the renormalization group method, beyond the
particular transport problem considered here.
Eq. (66) is one of the central results of our paper. It
is in agreement with the previous findings4,5,13 in the
limiting cases G → 1, G → 0 (i.e. |Y | → 1 ), K →
1, except for the fact that in these previous works the
conductance tends to K, rather than 1 , in the limit of
vanishing barrier strength at finite temperature. In this
limit ( |Y | → 1 ) our result is exact for any value of K . In
the intermediate regime G ≃ 1 there appear corrections
2
to our result. We have calculated these corrections in
the lowest order, which is third order in the interaction
strength. The parameter c3 determines the strength of
these corrections. We have shown that c3 is nonuniversal
in the sense that it depends on the physical nature of

16

(a)
0.1

2

CFT solution
ladder summation, c3=0
c3=1/4
c3=0.604...

0.0001
0.01

1

0.1

10

100

T /T*

(67)

1

(b)
2

At this point it is appropriate to compare our findings
to the exact results obtained within the conformal field
theory (CFT) approach to the bosonized version of the
problem, Sec. II A. In these earlier works the β-function
of the RG equation was not discussed. However results
equivalent to those by Fendley et al.11,12 are presented in
a recent study by Lukyanov and Werner28 , where particularly the β-function was obtained in a form equivalent
to our (66). According to Lukyanov (private communication), the coefficient c3 = 1/4, which deviates from
our value c3 ≃ 0.6 and we use both values for comparison below. Unfortunately, one rarely, if ever, discusses
the scheme of regularization used in obtaining the CFT
exact solutions. We have to conclude that the regularization used by28 is different from ours. The choice of
cutoff scheme in our work is not based on mathematical convenience, but rather it is following naturally from
perturbation theory. Therefore, as a theoretical result
to be applied to explain experimental data, our (66) has
advantages over the CFT results.
We saw above that passing from the backscattering
amplitude in the Hamiltonian to the conductance (i.e.
S-matrix) may require ultraviolet regularization, i.e. certain prescriptions of dealing with short-distance singular
quantities. Our observation is supported by the study
of the free (K = 1) BSG theory,19 where the authors
arrive at a current algebra formulation similar to our
treatment above. They notice that the description of
the strong coupling limit |u2 | → ∞ corresponds to taking the quantity |θ| → π, (see Eq. (2.23) there), which is
related to our |B| = 2θ and hence to the conductance.
They also notice that the connection between |θ| and
|u2 | = θf (θ2 ) = θ + θ3 O(1) begins to depend on the
type of regularization chosen in the third order of θ. On
the other hand, given that the CFT exact solution in
the interacting case K = 1 is expressed in terms of the
scaling variable |u2 |(T0 /T )1−K , we see that the implied
equation |u2 |(T0 /T )1−K = θf (θ2 ) corresponds in form
exactly to our (66). It would be thus desirable to understand the particular regularization used in the CFT
solution in more detail.

0.01

0.001

G, e / h

C = K 4c3 (1−1/K) ,

1

G, e / h

the ultraviolet cutoff, e.g. the temperature cutoff or the
length cutoff. We are convinced that the scaling functions
of the temperature dependent conductance, G(T ), in the
limit of infinite sample length L and the length dependent
conductance, G(L), at T = 0 are slightly different.
Let us discuss the role of the factor c3 in (66) which
is new as compared to previous studies4,5,13 . It is clear
that c3 in (52) defines the renormalization of the conductance in the intermediate region, G ∼ 1/2, and thus
determines the coefficient relating high-temperature and
low-temperature asymptotes. We may ask the following
question: what is the coefficient C in the asymptotic ex−1
pression G ≃ Cτ 2(K −1) (τ = T /T0 ) if we fix the overall
temperature scale at hight T as G ≃ 1 − τ 2(K−1) ? From
Eq. (66) we have

0.5
CFT solution
ladder summation, c3=0
c3=1/4
c3=0.604...
0
0.1

1

10

T /T*
FIG. 5: (Color online) (a) Conductance as a function of temperature for K = 1/2. The results of ladder summation, (66)
with c3 = 0, are shown by dashed-dotted line ; the result of
Eq. (66), with c3 given by (52) is shown by solid line. The
result available from CFT (multiplied by factor 2) is shown
by pluses.4,5,10 , together with our fit by (66) with c3 = 1/4
shown as dashed line. (b) The same plot in log-linear scale.

Let us now illustrate the details of the behavior of G(T )
in the example of the well-known case K = 1/2. Both
high-temperature and low-temperature asymptotes of G
are interesting from the theoretical viewpoint, but one
can expect the experimental variation of the conductance
to be most visible at lower T , when G 1/2. It is therefore useful to consider the variation of the conductance
for fixed K and different values of c3 . We take K = 1/2
and and fix the low T behavior in the form G ≃ τ 2 , with
τ = T /T∗ and T∗ ∼ T0 |r|2 . We then plot G(τ ), obtained
from (66) for several values of c3 in Fig. 5, together with
the exact solution, known from4,5,10
1
G1/2 = 1 − √ ψ ′
2 3τ

1
1
+ √
2 2 3τ

(68)

where ψ ′ (x) the derivative of digamma function. For K =
1/2, and the implied CFT value c3 = 1/4 we can invert

17
(66) and write
G=

2τ 2
√
1 + 2τ 2 + 1 + 8τ 2

(69)

The Fig. 5 shows that the CFT solution lies between
the curve corresponding to pure ladder summation, c3 =
0, and the one with the above c3 ≃ 0.60. The proposed
value c3 = 1/4 fits the CFT solution rather precisely.
This tells us that the approximation leading to (66) works
rather well at the interaction strength K = 1/2 (g = 3/5)
, provided the coefficient c3 appropriate for the regularization scheme used in the CFT solution is taken. In fact,
we show in the Appendix H, that the overall agreement
criterion using the coefficient C, Eq. (67), is about 1 % in
this case. For a further case reported (K = 1/3 , g = 4/5
)11,12 the agreement is worse, about 10 %, which might
be expected in this more strongly interacting situation.
Finally, we emphasize again that in any comparison
with experiment the correct regularization scheme is determined by the details of the corresponding experimental setup. Indications of the possible inadequacy of the
regularization scheme used in the CFT solutions exist.
For instance, we tend to interpret the systematic deviation of the experimental data on the conductance between the quantum Hall edge states from the CFT prediction at low T in12 not as a shortcoming of the experiment, but rather as a consequence of choosing an
inappropriate regularization. The correct regularization
might lead to a different value of c3 in (66), providing
better agreement with experiment.

Acknowledgments

We are grateful to I.V. Gornyi, D.G. Polyakov, P.M.
Ostrovsky, K.A. Matveev, D.A. Bagrets, O.M. Yevtushenko, M.N. Kiselev, A.M. Finkel’stein, A.A. Nersesyan, L.I. Glazman and N. Andrei for various useful
discussions.

APPENDIX A: FROM THE HAMILTONIAN TO
THE S-MATRIX

The continuum Hamiltonian (3) corresponds to a lattice model
∞

Hlat =

(c† ci+1 + h.c.) +
i
i=−∞

1
2

2

(u1 + (−1)j u2 )c† cj ,
j
j=1

at half-filling (2kF = π) and with potential present at
two adjacent sites.
In Eq. (3) we let u1,2 (x) = u1,2 δa(1,2) (x). Here u1,2
are dimensionless quantities and a1,2 is the range of the
corresponding regularization of the δ-function; at the end
the limit a1,2 → 0 will be taken.

Combining the two chiral fermions into a spinor quan†
†
tity Ψ† = (ψR , ψL ) we can write the equation of motion
as
i∂t Ψ = vF

i∂x + u1 (x)
u∗ (x)
2
Ψ,
u2 (x)
−i∂x + u1 (x)

(A1)

= vF (iσ3 ∂x + u1 (x) + u′ (x)σ1 + u′′ (x)σ2 )Ψ,
2
2
with real and imaginary parts u2 = u′ + iu′′ . The formal
2
2
solution of (A1) is
Ψ(x, t) = eiωt T (x, a)Ψ(a, 0),
x

T (x, a) = Tx exp i

a

(A2)

−1
dy σ3 [vF ω

+u1 (y) + u′ (y)σ1 + u′′ (y)σ2 ],
2
2

(A3)

where Tx stands for the x-ordering operator. In the long
wavelength limit, ω → 0, we obtain
−1
T (x, a) = exp[iωvF σ3 (x − a)],
−1
exp[iωvF σ3 x]T0

=

x < 0,

(A4)

−1
exp[−iωvF σ3 a],

x > 0,

with T0 depending on u1,2 (x). We assume first that the
regularized δ-functions in the definitons of u1,2 (x) are
identical, which means a1 = a2 . In this case
T0 = exp(iu1 σ3 −u′ σ2 +u′′ σ1 ) = cosh b+bσ
2
2

sinh b
, (A5)
b

with b = (u′′ , −u′ , iu1 ) and b2 = |u2 |2 − u2 . This trans2
2
1
fer matrix simplifies in two important cases : i) point-like
impurity, when u1 = |u2 | and ii) purely backward scattering impurity, u1 = 0, u2 = 0. We have
T0 =
T0 =

1 + i|u2 |
iu∗
2
,
−iu2 1 − i|u2 |

u1 = |u2 |,

cosh |u2 |
ie−iφ sinh |u2 |
,
iφ
−ie sinh |u2 |
cosh |u2 |

(A6)
u1 = 0

with eiφ = u2 /|u2 |.
Let us also consider two further limiting cases, when
the range of the forward scattering potential u1 (x) is
much longer (shorter) than the range of the backward
scattering potential u2 (x), i.e. a1 ≫ a2 (a1 ≪ a2 ). In
these two cases we get
T0 =

eiu1 cosh |u2 | ie−iφ sinh |u2 |
−ieiφ sinh |u2 | e−iu1 cosh |u2 |

(A7)

for a1 ≫ a2 and

cosh |u2 | + i tan u1
ie−iφ sinh |u2 |
iφ
−ie sinh |u2 |
cosh |u2 | − i tan u1
(A8)
for a1 ≪ a2 . By construction, the determinant of the
transfer matrix is unity for arbitrary u1,2 (x).
The elements of the scattering matrix for the incident
waves ∼ eikx and ∼ e−ikx at x < 0 and x > 0, respectively, are defined by the equation
T0 = cos u1

1 0
t r
˜
= T0 .
˜
0 1
r t

(A9)

18
Hence the scattering matrix is determined by the elements of the transfer-matrix T as
S≡

t r
˜
1 T12
−1
˜ = T22 −T21 1
r t

u2 (x) cos(2ϕ(x)) = u2 (x) cos(2ϕ(x) − u1 sgna1 (x)),
˜
→ u2 δ(x) cos(2ϕ(x)),
˜
→ u2 δ(x) cos(2ϕ(x)) cos u1 , (A14)
˜

In the above two cases with a1 = a2 we have
S =
S =

1
1 − i|u2 |

1 iu∗
2 ,
iu2 1

u1 = |u2 |,

1/ cosh |u2 | ie−iφ tanh |u2 |
,
ieiφ tanh |u2 | 1/ cosh |u2 |

with u1 from Eq. (5), where the regularized sign function
x
sgna1 (x) = −1 + 2 −∞ dy δa1 (y) is used. We have the
backward scattering in the form

(A10)
u1 = 0.

for the above cases of regularization with a1 ≫ a2 and
a1 ≪ a2 , respectively.
APPENDIX B: UNITARY TRANSFORMATION

Whereas for model potentials of different range we obtain
S = eiu1

1/ cosh |u2 | ie−iφ tanh |u2 |
ieiφ tanh |u2 | 1/ cosh |u2 |

We consider a transformation of the operators Jj
(A11)

for a1 ≫ a2 and
1/ cos u1 ie−iφ sinh |u2 |
,
iφ
ie sinh |u2 |
1/ cos u1
(A12)
for a1 ≪ a2 . We see that if the forward scattering u1
has wider range, than u2 , then Eq. (A11) differs from
the second Eq. (A10) only by the overall phase which is
unimportant in physical observables. At the same time,
if the model regularization of u1 is narrower than u2 , the
resulting Eq. (A12) closely resembles the S-matrix in the
double-barrier situation.14
Expanding the reflection coefficient in powers of u1,2
and using (A5), (A11), (A12) we have
1
S=
cosh |u2 | − i tan u1

1
|r| ≃ |u2 | − |u2 |3 , a1 ≫ a2 ,
3
1
|r| ≃ |u2 | − |u2 |(u2 + 2|u2 |2 ), a1 = a2 , (A13)
1
6
1
|r| ≃ |u2 | − |u2 |(3u2 + 2|u2 |2 ), a1 ≪ a2 ,
1
6
which shows that the forward scattering amplitude u1
enters the reflection coefficient in the third order, i.e.
beyond the Born approximation. Moreover the contribution of u1 to the S-matrix depends also on the details
of the microscopic Hamiltonian and the regularization of
the model δ-function potentials. This complication happens at the step (A2); however, once the transfer matrix
T0 in (A4) is known, the connection to the scattering
state basis is straightforward.
Returning to (2), (4), we notice that ψ1 (x) obeys
the equation of motion for right movers, and the spinor
+
+
Ψ† = (ψ1 (x), rψ1 (−x)) is an eigenstate (A2) of (A1)
R
with the above property (A9). We thus clarified the scattering states representation in the framework of the chiral
Hamiltonian (1).
In the bosonization formulation, (5), the assumption
is made that the forward scattering u1 (x) can be completely removed from the Hamiltonian. This statement
is not obvious, in general. Consider the redefinition
ϕ(x) = ϕ(x) − u1 sgna1 (x)/2, which eliminates the term
˜

Oj (x) = U † Jj (x)U
with
∞

U = exp i
−∞
∞

dx dy θ(x − y)Bj (y)Jj (x), (B1)
dy Bj (y)Θj (y),

= exp i

(B2)

−∞
∞

Φj (y) =

dx Jj (x).

(B3)

y

In our case we have Bj (x) = Bj δ(x) and we may denote
the direction of B as ”1”, i.e. B e1 . Then we have the
ˆ
following set of equalities from the Kac-Moody relations
(20)
i
δ(x − y)
4π
∞
1
[J1 (y), i
dx B1 (x)Θ1 (x)] = − B1 (y)
4π
−∞
[J1 (y), Φ1 (x)] =

J1 (y)U = U (J1 (y) −

(B4)

1
B1 (y))
4π

And for J + (x) = J3 (x) + iJ2 (x) one has

[J + (x), i

∞

[J + (x), J1 (y)] = J + (x)δ(x − y)
[J + (x), Θ1 (y)] = J + (x)θ(x − y)
dy B1 (y)Θ1 (y)] = iJ + (x)

−∞

(B5)

x

dy B1 (y)
−∞

So that the relations between the operators are
O1 (x) = U † J1 (x)U = J1 (x) −
O3 (x) + iO2 (x) = U † J + (x)U

1
B1 (x) (B6)
4π
(B7)
x

= (J3 (x) + iJ2 (x)) exp i

dy B1 (y)
−∞

The transformation U shifts the value of O1 (x) by a
c−number, so that strictly speaking, the operators Oj (x)
do not obey the same Kac-Moody algebra in the region
where B1 (x) = 0. It is clear, though that this might affect only the physically non-observable component of the
fermionic density and is therefore not important.

19
APPENDIX C: EQUATIONS OF MOTION IN
CURRENT OPERATOR FORMALISM

In this section we focus on the time evolution of the
observables J3 (−x), J3 (x), Eqs. (22), (23), using the relation i∂t A = [A, H]. It is more convenient to use the representation (27), with O(x) = J3 (x)Θ(−x) + J3 (x)Θ(x).
In terms of operators Oj (x) one has
ρs (x) = (O3 (x) − O3 (−x))
js (x) = vF (O3 (x) + O3 (−x))

(C2)

where we let vF = 1. Since O3 (x) is the observable density, the relation (C2) looks deceptively close to the desired answer. However, in the region of the barrier |x| < a
(with B O1 ) we have
−∂t O3 (x) = ∂x O3 (x) + B(x)O2 (x),
and the O2 component comes into play. (In other words,
for the partially reflecting barrier, the observable of our
problem, J3 (x), includes the eigenmode operator J2 (x).)
Therefore it is necessary to consider also the time evolution of operator O2 , which reads
− ∂t O2 (x) = ∂x O2 (x) − 4πgO1 (x)O3 (−x),

O2 (¯) =
x

(C3)

at a < |x| < L and −∂t O2 (x) = ∂x O2 (x)+ B(x)O3 (x), at
|x| < a. The four-fermion object O1 (x)O3 (−x) is not reduced to O2 (−x) and therefore we do not obtain a closure
of the equations of motion.
Our situation is somewhat similar to the well-known
Heisenberg model of spins with pairwise exchange interaction. Due to the non-commutative character of spin
operators, the attempt to write the exact time evolution
of spin leads to an infinite series of coupled equations,
including ever increasing numbers of spins. The approximate solutions are obtained by truncating this series at
some step, according to a certain criterion, and discarding higher terms as inessential.22 In magnetism, these
criteria might be (i) low temperatures in the magnetically ordered phase, T ≪ Tc , (ii) large value of spin S or
(iii) wide range of interaction. In our case, we choose the
criterion of relevancy of operators, i.e. their importance
in the logarithmically divergent theory.
We now briefly discuss the dynamics of O4 (x) ≡
4πO1 (x)O3 (−x). The compound operator O4 (x) consists of four fermion operators and may be decomposed
into the normally ordered part and a part, representing
internal contractions. To clarify the notation we write
ψ1 (x) = a↑ (x), ψ2 (−x) = a↓ (x), then we have
O4 (x) = π(a† (x)a↓ (x) + a† (x)a↑ (x))
↑
↓
×(a† (−x)a↑ (−x) − a† (−x)a↓ (−x)) (C4)
↑
↓

1 †
(a (x)a↓ (−x) + a† (−x)a↓ (x)) + h.c. (C5)
↑
4i ↑

which does not coincide with O2 (x) = (2i)−1 a† (x)a↓ (x)+
↑
h.c. We may write
O2 (¯) ≃ O2 (−x) − x∂x O2 (−x) + . . .
x

(C1)

Using (20) and formulas of Appendix B, it can be easily
shown that we have −∂t O3 (x) = ∂x O3 (x) in the leads,
|x| > L, whereas in the interacting region a < |x| < L
we obtain
− ∂t O3 (x) = ∂x O3 (x) − g∂x O3 (−x),

Contracting here, we get O4 (x)− : O4 (x):= −x−1 O2 (¯)
x
with (cf. Eq. (29))

(C6)

and, dropping the less relevant higher derivatives, obtain
from (C3)
− ∂t O2 (x) ≃ ∂x O2 (x) + g(x−1 − ∂x )O2 (−x) − g : O4 (x) :
which means that the most relevant part of the operator
O4 (x) has the form
O4 (x) ≃ (−1/x + ∂x )O2 (−x).

(C7)

Regarding the normally ordered part : O4 (x) :, its time
evolution is given by
−∂t : O4 (x) : = 4π : O3 (−x)∂x O1 (x) − O1 (x)∂x O3 (−x) :
+4πg : O5 (x) :
2
O5 (x) = 4πO1 (x)O3 (−x) + O1 (x)∂x O3 (x)
i.e. described by new operators, not reduced to simple
spatial derivatives of : O4 (x) :. Such derivative objects (i)
obey an infinite set of coupled equations and (ii) possess
formal scaling dimension greater than two, and hence are
irrelevant in the RG sense.
Our strategy of such truncation (C7) is successful only
in part, as we explain in the next Appendix D.
APPENDIX D: SOLUTION OF THE EQUATION
OF MOTION

Taking a Fourier transform in time t, and introducing
the vector
Ψ(x, ω) = [O3 (x, ω), O3 (−x, ω), O2 (x, ω), O2 (−x, ω)]
the above truncated set of equations can be written as
ˆ
∂x Ψ(x, ω) = D0 (ω, x)|g=0 Ψ(x, ω), |x| > L
ˆ
ˆ
D1 ∂x Ψ(x, ω) = D0 (ω, x)Ψ(x, ω), a < |x| < L

(D1)

with

iω,
 0,
ˆ
D0 (ω, x) = 
0,
0,

1,
−g,
ˆ
D1 = 
0,
0,


0,
0,
0
−iω,
0,
0 
(D2)
0,
iω,
−gf (x)
0, −gf (x), −iω

−g, 0, 0
1, 0, 0 
(D3)
0, 1, −g 
0, −g, 1

20
and the fermionic correlation function at finite temperature
2πT
f (x) =
sinh 2πT x

(D4)

The solution to (D1) is given by the x−ordered exponent
ˆ
Ψ(x, ω) = U (x, y)Ψ(y, ω)
x

ˆ
U (x, y) = Tx exp
y

(D5)

ˆ −1 ˆ
dz D1 D0 (ω, z),

The overall transfer-matrix over the interaction region is
ˆ
ˆ
ˆ
ˆ
U (L, −L) = U (L, a).U (a, −a).U (−a, −L)

Finally we observe that the definition of Ψ(x) implies that
the transfer matrix connecting opposite points should
satisfy the boundary condition:
ˆ
ˆ
U (x, −x)Ψ(−x, ω) = I.Ψ(−x, ω)

(D6)

ˆ
similarly to (A2), but now with transfer-matrix U (x, y)
for densities. The transfer-matrix in the region of the
barrier |x| < a is already known and given by


cos 2θ,
0,
sin 2θ,
0
cos 2θ,
0,
− sin 2θ
 0,
ˆ
(D7)
U (a, −a) = 
− sin 2θ,
0,
cos 2θ,
0 
0,
sin 2θ,
0,
cos 2θ


(D8)


0,
ˆ = 1,
I 
0,
0,

1,
0,
0,
0,

0,
0,
0,
1,


0
0
1
0

(D9)

(D10)

and we should choose a solution, by demanding that the
scattered component, O2 , is absent in the incoming wave,
i.e. Ψ3 (−∞, ω) = 0.

ˆ
ˆ
We consider here only the technically feasible limit, ω = 0. In this case U (x, L) = U (−L, −x) = 1 in the leads,
x > L. In the interacting region


1, 0,
0,
0

0, 1,
0,
0

ˆ
U (L, a) = 
(D11)
0, 0, e−g2 Λ cosh gΛ, −e−g2 Λ sinh gΛ
−g2 Λ
−g2 Λ
0, 0, −e
sinh gΛ, e
cosh gΛ
Λ = (1 − g 2 )−1

L

a

dx f (x) = (1 − g 2 )−1 ln

tanh πT L
tanh πT a

(D12)

ˆ
A similar contribution on the negative semiaxis, U (−a, −L), is obtained from (D11) by the change Λ → −Λ. Overall
we have


2
2
cos 2θ,
0,
−eg Λ sin 2θ cosh gΛ, −eg Λ sin 2θ sinh gΛ
2
2


0,
cos 2θ,
eg Λ sin 2θ sinh gΛ, eg Λ sin 2θ cosh gΛ 

ˆ
U (∞, −∞) =  −g2 Λ

2

e
sin 2θ cosh gΛ, e−g Λ sin 2θ sinh gΛ,
cos 2θ,
0
2
2
−e−g Λ sin 2θ sinh gΛ, −e−g Λ sin 2θ cosh gΛ,
0,
cos 2θ
Solving Eq. (D9) together with the above condition Ψ3 (−∞) = 0, we find
Ψ(−∞) = (cos2 θ + e2gΛ sin2 θ, cos2 θ − e2gΛ sin2 θ, 0, −eg(1−g)Λ sin 2θ)

(D13)

which is the desired solution. Let us briefly discuss it.
We observe that the d.c. current j(x) = O3 (x) +
O3 (−x) is independent of x, as it should be. This property follows from the form of matrix (D11) and condition
(D9). The relation of the current to the voltage bias defines the conductance of the system. In our definitions,
the voltage bias is proportional to the difference in number of incoming left- and right-going electrons, which is

2O3 (−∞) = 2Ψ1 (−∞), so that the conductance is
j(x)
O3 (∞)
1
=
+1
2O3 (−∞)
2 O3 (−∞)
cos2 θ
=
cos2 θ + e2gΛ sin2 θ

G =

(D14)

which coincides with the above result (11).
At first glance, the result (D13) should thus be viewed
as satisfactory. However there are details, involved in
its derivation, which spoil the significance of this approach. First we notice that, by keeping less relevant

21
gradient term in the above relation, (C7), we get the appearance of g 2 terms in the definition of Λ, eq. (D12),
2
and also in the exponents, eg Λ , in (D11) – (D13). Further, trying a different approximation instead of (C7),
e.g. O4 (x) ≃ (−1/x + ∂x )O2 (x), would not produce the
above renormalized value of conductance.
Therefore we conclude that using the equations of motion method and persisting with the current operator algebra leads to ambiguities, connected with the non-local
character of the interaction.

processor), we obtain
δY {4,2,2} = −g 3 (1 − Y 2 )

dxi

3Y 2
1
+
3
(x1 + x2 + x3 )
(x1 + x2 − x3 )3
Λ0
= −g 3 (1 − Y 2 )(1 + Y 2 ) ,
(E4)
4
×

δY {4,4} = −g 3 (1 − Y 2 )2

1
+ (symm.)
x1 (x2 + x3 )(x1 + x2 + x3 )
1
= −g 3 (1 − Y 2 )2 (Λ2 + 2Λ0 (1 − ln 2)), (E5)
4 0

×

APPENDIX E: CALCULATION OF
CORRECTIONS: ZERO TEMPERATURE

Proceeding as described in Sec. V A, we find corrections to the conductance, or to Y in (31). In the first
order of g the correction from the diagrams of the type
{4} has the form
L

δY {4} = −g(1 − Y 2 )

a

dx
= −g(1 − Y 2 )Λ0
x

L
a

dx1 dx2
2(x1 + x2 )2

1
= g 2 Y (1 − Y 2 )(Λ0 − 2 ln 2)
2

(E2)

where x1,2 are two points of fermionic interaction.
All diagrams consisting of only one loop, {6} in above
notation, produce the correction

×

dxi

1
+ (symm.)
x1 (x2 + x3 )2

= g 3 (1 − Y 2 )Y 2 Λ0 (Λ0 − 2 ln 2),
δY {8} = 2g 3 (1 − Y 2 )

dxi

(E6)

1 − 3Y 2
x1 x2 x3

1−Y2
(E7)
(x1 + x2 )(x2 + x3 )(x3 + x1 )
1 − 3Y 2 3
π2
= g 3 (1 − Y 2 )
Λ0 + (1 − Y 2 ) Λ0 .
3
12

+

In the final expressions (E4), (E5), (E7) we omitted finite
parts, O(Λ0 ), which are unimportant in the order g 3 .
These expressions lead to Eq. (43) above.

APPENDIX F: CALCULATION OF
CORRECTIONS: FINITE TEMPERATURE

L

dx1 dx2
= −g 2 Y (1 − Y 2 )Λ2
0
x1 x2
a
(E3)
We should mention that individual diagrams in the set
{6} (and higher orders) may contain singularities of the
form (x1 − x2 )−1 etc. However these singularities cancel
each other in the resulting expressions. For example, one
finds two particular contributions x−1 (x1 − x2 )−1 and
1
x−1 (x2 − x1 )−1 whose sum leads to the above “regular”
2
form, (x1 x2 )−1 , in (E3).
In the third order, we have to analyze the following
types of the diagrams : {4, 2, 2}, {4, 4}, {6, 2}, {8}. We
assume the ordered sequence, L > x1 > x2 > x3 > a,
in all integrals below. Performing rather long computer
calculations (the stable routine in Mathematica requires
about ten hours of computation time for dual-core 3 GHz
δY {6} = −g 2 Y (1 − Y 2 )

δY {6,2} = 2g 3 (1 − Y 2 )Y 2

(E1)

In the second order of g we have two sets of diagrams
for corrections to the conductance. The diagrams containing two fermionic loops lead to the following expression :
δY {4,2} = g 2 Y (1 − Y 2 )

dxi

In case of finite temperature we proceed in the same
way as for T = 0 , only with the integration replaced by a
summation over Matsubara frequencies. Introducing the
scaling variable Λ according to (47) we have for a ≪ ξ =
vF /πT ≪ L
δY {4} = −g(1 − Y 2 )Λ,
δY {4,2} = g 2 Y (1 − Y 2 )

a/ξ

2d¯1 d¯2
x x
(F1)
2
sinh 2(¯1 + x2 )
x
¯

1
= g 2 Y (1 − Y 2 )(Λ − 3 ln 2),
2
where x1,2 are two points of fermionic interaction, and
xi = xi /ξ. In the last equality in the right-hand side of
¯
Eq. (F1), we shall neglect the terms ∼ O(1).

22
Instead of Λ2 in the expression (E3) for δY {6} we now
0
have the following integral
L/ξ

d¯1 d¯2
x x
a/ξ
τL

=
τ0
2

dτ1 dτ2
τ1 τ2

3 cosh(¯1 + x2 ) + cosh(¯1 − x2 )
x
¯
x
¯
sinh(2¯1 ) sinh(2¯2 ) cosh(¯1 + x2 )
x
x
x
¯
1−

1 τ1 τ2
2 1 + τ1 τ2

(F2)

= Λ − F [τL , τ0 ]
π2
→ Λ2 −
, T ≫ vF /L
24

APPENDIX G: ”NON-UNIVERSALITY OF
HIGHER TERMS” IN β−FUNCTION

with τi = tanh xi . The integration limits in (F2)
¯
are τ0 = tanh(a/ξ) and τL = tanh(L/ξ).
Here
2
2
F [τL , τ0 ] = L2 (−τ0 τL ) − (L2 (−τL ) + L2 (−τ0 ))/2 and
∞
L2 (x) = 1 xn /n2 is the dilogarithm function. We have
F [τL , τ0 ] = 0 in the limit T ≪ vF /L.
In contrast to the T = 0 case, the expressions now become extremely cumbersome in the third order of g. The
summation over Matsubara frequencies lead to expressions involving sums and products of exponents exp(¯i ).
x
A general criterion for the correctness of intermediate
expressions is the property a change of sign of each individual diagram upon performing the reflection transformation zi → −zi .
An additional check of the validity of calculation is the
observation that we should get zero correction to G =
1 in the absence of the barrier, θ = 0, and to G = 0
for the fully reflecting barrier, θ = π/2, for arbitrary T .
This is not a trivial statement, because each diagram in
third order contains a prefactor, either cos 8θ, or cos 4θ
or cos 0θ = 1. We verified that the algebraic expressions
canceled each other in this case.
The actual expressions are not shown here, and we
describe our method of analysis below.
Let us consider an expression of the form of a triple
integral, relevant to our discussion in Sec. V
L

f (a) =

L

dz1
a

L

dz2
z1

¯
dz3 f [z1 , z2 , z3 ].

(F3)

z2

We assume that f (a) ≃ −c ln a+. . . at a → 0 and wish to
determine the coefficient c in this asymptotic behavior.
We have
df (a)
= lim
a→0 d ln a
a→0

c = − lim

L

L

dz2
a

Therefore the recipe for the determination of c is first
¯
to determine the limiting form of f2 , taking z2 ∼ a ≪ ξ
; in this case the expressions are drastically simplified.
This will determine c at T → 0. To this term we should
¯
add the contribution lima→0 af [a, z2 , z3 ], integrated over
z2 in the interval (0, ∞) ; the sum of these two contributions gives the value of c at finite temperatures, L ≫ ξ.

¯
dz3 af [a, z2 , z3 ]

z2

(F4)
Here taking the limit in the integrand may lead to incorrect results. We have two possibilities for the behavior of
L
¯
¯
the quantity f2 [a, z2 ] = z2 dz3 af [a, z2 , z3 ], for small a,
¯ [a → 0, z2 ] ∼ a/(z2 + a)2 and f2 [a → 0, z2 ] ∼ 1.
¯
namely f2
¯ ∼ a/(z2 + a)2 , is alThe first type of contribution, f2
ways important and survives in the limit of T → 0.
¯
Recalling that f [a, z2 , z3 ] is exponentially suppressed at
¯
zi
ξ, we see that the case f2 [a, z2 ] ∼ 1 contributes to
c only at finite temperatures, L ≫ ξ ; typically, we find
¯
f2 [a → 0, z2 ] ∼ (ξ cosh(2z2 /ξ))−1 .

In a very general sense the β-function of the RGequation is non-universal: it depends on the precise definition of the scaling quantity g . Let the RG equation
read
2
3
4
dgr /dΛ = β(gr ) = b2 gr + b3 gr + b4 gr + . . .

(G1)

and consider a change of variable gr = g + c2 g 2 + c3 g 3 .
¯
¯
¯
With the same accuracy we have
d¯/dΛ = β(¯) = β(g)/(dg/d¯) = b2 g 2 + b3 g 3
g
g
g
¯
¯
2
4
+(b4 + b3 c2 + b2 (c2 − c3 ))¯ + . . . (G2)
g
which shows that only the two first two coefficients b2 and
b3 are invariant. Since b4 is associated with the three-loop
contribution, one may speak about a non-universality of
the β-function beyond two-loop order.
Let us elucidate the origin of the Eq. (G2), and
compare it with the equation obtained in the CallanSymanzik approach. In the Gell-Mann–Low approach
we perform our calculations with the running coupling
constant gr and add counterterms to the Hamiltonian
in order to achieve the finite value of the bare constant
g0 . One can show that the above value of dgr /dΛ =
−(∂g0 /∂Λ)/(∂g0 /∂gr ) is obtained from the expression
2
3
g0 = gr − b2 Λgr − (b3 Λ − b2 Λ2 )gr
2
5
4
−(b4 Λ − b2 b3 Λ2 + b3 Λ3 )gr + . . .
2
2

(G3)

If we make the above change gr → g in the right-hand
¯
side of (G3) and use the same recipe for d¯/dΛ, we return
g
to (G2).
In the CS approach we fix the bare value g0 and calculate the renormalized quantity gr (g0 , Λ). Inverting the
equation g0 (gr , Λ) and using the same recipe we find
dgr /dΛ. One can check that
2
3
gr = g0 + b2 Λg0 + (b3 Λ + b2 Λ2 )g0
2
5
4
+(b4 Λ + b2 b3 Λ2 + b3 Λ3 )g0 + . . .
2
2

(G4)

Assume now that we have changed the starting value
g0 = g0 +c2 g0 +c3 g0 +. . ., according to above prescription.
¯
¯2
¯3
Evidently, the expansion for gr (¯0 , Λ) will be different
g
from (G4), as well as the inverse function g0 (gr , Λ) will
¯
not coincide with (G3). One can verify however that the

23
β-function, −(∂¯0 /∂Λ)/(∂¯0 /∂gr ), is given by the same
g
g
Eq. (G1) and not by Eq. (G2). This fact becomes rather
obvious, when we notice that only the left-hand side of
(G3) is changed and it corresponds to a change in the
boundary condition for gr (Λ) rather than to a change in
the differential equation for it.

we seek a solution in perturbation theory as Ψ(z) =
−2K0 (z)+ sR(z)+ O(s2 ). After some calculation we find
the conductance G(T ) ≃ 1−KsF (K) with F (K) ≡ R(0)
given by
F (K) = Γ2 [K]

2
[−∂z − z −1 ∂z + 1 + s1 (z/2)2(K

and Λ = ln(T0 /T ). For given s one finds a solution Ψ(y)
characterized by a unique value y0 . Then, according to
Lukyanov,28 the conductance can be determined as

(H3)

At s = 0 (i.e. T → ∞) the solution is a modified
Bessel function Ψ(z) = −2K0 (z). At infinitesimal s

2

3
4

5

6

7
8

9
10

−1)

]Ψ(z) = 0,

(H5)

with s1 = s−1/K K 2(K −1) ≪ 1. The solution is found
similarly to (H3) and we get G(T ) ≃ −s∂s s1 F (K −1 ).
Comparing the two solutions we obtain the prefactor
C = K 3 F (K)

1/K

K −3 F (K −1 )

(H6)

(H2)

We now fix the overall temperature scale τ = T /T0
by the high-T behavior, G ≃ 1 − τ 2(K−1) , and ask what
is the coefficient C in the asymptotic low temperature
−1
expression G ≃ Cτ 2(K −1) . To answer this question, we
slightly rewrite the differential equation. Introducing a
new variable z = 2ey/2 we have

1

−1

−1

s = K 2 exp(2(1 − K)Λ)

∗

(H4)

(H1)

such that Ψ(y → ∞) = 0 and Ψ(y → −∞) = y − y0 .
Here

2
[−∂z − z −1 ∂z + 1 + s(z/2)2(K−1) ]Ψ(z) = 0

dκ
1+κ

In the opposite limit, s → ∞ (T = 0), we choose a
√
variable z = (2 s/K)eKy/2 and arrive at an equation

Consider the differential equation

∂y0
K
∂y0
=1+
G(T ) = 1 + Ks
∂s
2(1 − K) ∂Λ

0

2 F1 (K, K, 1; −κ)

= Γ4 [K]/Γ[2K] .

APPENDIX H: COMPARISON TO CFT
SOLUTION

2
[−∂y + seKy + ey ]Ψ(y) = 0

∞

On leave from Petersburg Nuclear Physics Institute,
Gatchina 188300, Russia
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