Transport properties of a two-lead Luttinger liquid junction out of equilibrium:
fermionic representation
D.N. Aristov1, 2, 3 and P. W¨lfle3, 4
o
1

arXiv:1408.4914v2 [cond-mat.str-el] 8 Dec 2014

Department of Physics, St.Petersburg State University, Ulianovskaya 1, St.Petersburg 198504, Russia
2
NRC ”Kurchatov Institute”, Petersburg Nuclear Physics Institute, Gatchina 188300, Russia
3
Institute for Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
4
Institute for Condensed Matter Theory, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
(Dated: December 9, 2014)
The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached
to two reservoirs at chemical potentials differing by the applied voltage bias. The relevant scaledependent contributions in perturbation theory in the interaction up to infinite order are evaluated
and summed up. The result allows to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which
the conductance follows a power law in terms of a scaling variable Λ, which equals the bias voltage
V , if V is the largest energy scale compared to temperature T and inverse wire length L−1 , and
interpolates between these quantities in the crossover regimes.

I.

INTRODUCTION

In the past few years, exactly one-dimensional quantum wires have become available for experimental investigation in the form of carbon nanotubes, chains of metal
atoms or weakly side-coupled molecular chains in solids.
The new data emerging from these experiments1–3 , in
particular in non-equilibrium situations, require a more
detailed and more general theoretical description than
presently available. Electron transport in nanowires has
been studied theoretically for more than two decades. In
the first papers it was found that electron-electron interaction affects even the conductance of a clean wire4,5 . In
the case of realistic boundary conditions, namely adiabatically attaching ideal leads to the interacting quantum wire, the two-point conductance of a clean wire
is that of the leads, equal to one conductance quantum per channel, irrespective of the (forward scattering) interaction6 . The work of Kane and Fisher5 and
Furusaki and Nagaosa7 showed, that interaction has a
dramatic effect on the conductance in the presence of
a potential barrier. Namely, for repulsive interaction
these authors found that the conductance tends to zero
as the temperature T , or more generally, the excitation
energy of the electrons approaches zero, while for attractive interaction the conductance scales to its maximum value. This behavior has been shown to carry
over to the dependence on bias voltage, at sufficiently
low temperatures (and for long wires). There exists a
large body of theoretical work addressing different aspects of transport through Luttinger liquids without or
with impurities, such as the effect of the leads on a finite length wire,8,9 the response to an a.c. electric field10 ,
the appearance of oscillatory behavior in the nonlinear
conductance11,12 and the emergence of bistability for the
very strong interaction and bias voltage13 . The transport through Luttinger liquid junctions at not too strong
interaction has also been calculated using the Functional

Renormalization Group method as reviewed in14 . The results mentioned above have been mostly obtained within
the bosonization method, which needs to be amended by
a correction taking care of the physical situation of a wire
of finite length attached to reservoirs (see above). Experimentally, the predictions of theory have been found to
be observed, at least qualitatively15–19 .
A proper treatment of the two-point conductance in
the limit of weak interaction, taking into account the
gradual build-up of the Friedel oscillations around the
barrier as the infrared cutoff is lowered has been given by
Yue, Glazman and Matveev20 . These authors used the
perturbative RG for fermions to derive the conductance
for an arbitrary (but short) potential barrier (”fermionic
representation”). In this paper we extend the approach of
Yue et al. to transport under stationary non-equilibrium
conditions. Following our extensive work on transport
in the linear regime through junctions of Luttinger liquids at arbitrary strength of interaction21–28 we derive in
the following RG-equations for the conductance at finite
bias voltage and for any interaction strength. We use
the fact that the β- function of the RG-equation for the
conductance can be obtained in very good approximation by summing a class of contributions in perturbation
theory in all orders of the interaction22 . A comparison of
our previous results on the linear response conductances
of two-22 and three-lead junctions23,25,28 with or without
additional symmetries, or an applied magnetic flux27,28 ,
with the results of the bosonization method, of conformal
field theory methods, of Bethe ansatz, where available,
are in full agreement provided those results were wellfounded. In a few cases where the conformal field theory
result was based on an additional assumption we found
disagreement, which we interpret as saying that the assumption was not justified.
In this paper we consider the transport of spinless
fermions, which begs the question of how our results may
be applied to experiment. The spinless Luttinger liquid model has actually been used to describe transport

2
through spin-polarized quantum wires, as considered in
Ref.2 A generalization of our theory to spinful fermions
is in progress.
II.

THE MODEL

We consider a system of spin-less fermions in one dimension, interacting in the region a < |x| < L (the
”wire”), adiabatically connected to reservoirs at |x| > L.
There is a barrier in the narrow regime |x| < a , which
scatters the fermions as described by the S-matrix (up
to overall phase factors in the individual wires)
r t
t r

S=

=

sin θ
i cos θe−iϕ
iϕ
i cos θe
sin θ

(1)

We choose this parametrization in terms of the transmission and reflection amplitudes t, r , since it is readily
generalizable to the case of multi-wire junctions (n wires,
n > 2 ). The above form of the S-matrix is completely
general.
In the continuum limit, linearizing the spectrum at the
Fermi energy and adding forward scattering interaction
of strength gj in wire j , we may write the Hamiltonian
H in the representation of incoming and outgoing waves
as
2

∞

H=
0
Hj

=

0
int
[Hj + Hj Θ(a < x < L)] ,

dx
0
†
vj ψj,in i

j=1
†
ψj,in − vj ψj,out i ψj,out ,

(2)

where vj is the group velocity of the fermions. We use
units where electrical charge e = 1, also = 1 and Boltzmann’s constant kB = 1.
We work with the Green’s functions in this chiral basis and in Keldysh formulation (we denote matrices in
Keldysh space by an underbar),
G=

GR (l, y|j, x) = − √
ω
GA (l, y|j, x) = √
ω

CHARGE CURRENT OF FREE FERMIONS

The net current flowing outward through the point z in
wire j is composed out of two chiral components, moving
towards (η = −1) and from (η = 1) the junction,
Jj (z) = vj

ρj,+ (z) − ρj,− (z)

(3)

i
δ
0
θ(τ )eiωτ lj
Slj δlj
vl vj

i
δ S∗
θ(−τ )eiωτ lj jl
0 δlj
vl v j

GK (l, y|j, x) = − √
ω

(5)

∗
Sjl hl

i
δlj hl
eiωτ
∗
Slj hj Sjm Slm hm
vl vj

τ = ηl y/vl − ηj x/vj
where hj (ω) = tanh[(ω − µj )/2T ] is the equilibrium distribution function in the reservoir of lead j , characterized by the chemical potential µj . We shall assume the
temperatures T in the leads to be equal.
The average density of the chiral current at point z ,
ρj,η (z) , is represented by the diagram in Fig. 1.

z

Ω

FIG. 1: The diagram showing the zeroth order contribution
to the current.

In terms of the Green’s function matrix and defining
the external vertex by the Keldysh matrix γ ext
i
2

1 1
−1 −1

(6)

dΩ
TrK γ ext · GΩ (j, η, z|j, η, z)
2π

(7)

γ ext =
we have
ρj,η (z) =

III.

(4)

Here retarded, advanced and Keldysh components of
the Green’s functions, in matrix form in the chiral basis
are given by (2×2 matrices in the chiral basis are denoted
by a hat, Gηl ηj (l, y|j, x) = G(l, ηl , y|j, ηj , x))

†
†
int
Hj = 2πvj gj ψj,in ψj,in ψj,out ψj,out .

We are using the chiral representation, labeling electrons
in lead j by (j, η) ≡ jη where η = +1 for outgoing and
η = −1 for incoming electrons and all position arguments
x are on the positive semi-axis. The range of the interaction lies within the interval (a, L), where a > 0 serves
as an ultraviolet cutoff (at energy scale vj /a) and separates the domains of interaction and potential scattering
on the junction; non-interacting leads are attached adiabatically at large x beyond L. In terms of the doublet
of incoming fermions Ψ− = (ψ1,− , ψ2,− ) the outgoing
fermion operators may be expressed with the aid of the
S-matrix as Ψ+ (x) = S · Ψ− (x) . For later use we define
†
density operators ρj,η=−1 = ψj,− ψj,− = Ψ+ ρj Ψ− , and
−
†
ρj,η=1 = ψj,+ ψj,+ = Ψ+ ρj Ψ− , where ρj = S + · ρj · S .
−
The 2 × 2 matrices are defined by (ρj )αβ = δαβ δαj and
+
(ρj )αβ = Sαj Sjβ .

GR GK
0 GA

with the trace TrK taken over the Keldysh indices.
Using the expressions (5), we obtain
vj ρj,− (z) =

1
2

vj ρj,+ (z) =

1
2

dΩ
(1 − hj (Ω)),
2π
dΩ
(1 −
|Sjm |2 hm (Ω))
2π
m

(8)

3
Notice here that the incoming current in the jth wire
is characterized by the distribution function referring to
the same wire. The outgoing current in the jth wire is
characterized by the distribution functions referring to
the remaining wires. The dependence on z vanishes in
the d.c. limit considered here.
Using the unitarity property (i.e. charge conservation),
2
m |Sjm | = 1, we may represent the net current in the
form
(0)

Jj (z) =

1
2

dΩ
2π

m

|Sjm |2 (hj (Ω) − hm (Ω))

(0)

1
2π

m

|Sjm |2 (µm − µj )

(10)

(11)

where V = µ1 − µ2 is the applied bias voltage. In the following we will find it convenient to introduce the quantity
Y = 1 − 2G0 characterizing the conductance.
IV.

CURRENT TO FIRST ORDER IN THE
INTERACTION

The first order correction to the current in the nonequilibrium case is represented as the diagram depicted
in Fig. 2. Here the wavy line stands for the electronic
interaction, taking place at the point x in the wire l. The
contribution to the current of chirality ηn in the n-th wire
can be expressed as

y

TrK [γ ext
µ=1,2 l,ηl

× GΩ (jη , z|lη , y)¯ µ GΩ+ω (lη , y|lη , x)γ µ
γ

(12)

× GΩ (lη , x|jη , z)](2πigl vl )δ(x − y)

The trace TrK is over the lower (fermionic) Keldysh
µ
¯µ
¯
indices; the fermion-boson vertices, γij → γ µ , γij → γ µ ,
tensors of rank 3 defined in Keldysh space, are given by
1
γij = γij =
¯2

The conductance (in units of the conductance quantum
e2 /2π ) of a two-lead junction is in lowest order given
by
G0 = J/V = |S12 |2 = t2

dx dy

(9)

which is a well-known expression. For the above choice
of the weight function hj (Ω) , the remaining integration
can be easily done with the result
Jj (z) =

dΩ dω
(2π)2

(1)

Jjη (z) = vj

γ
˜

1
√ δij ,
2

2
γij = γij =
¯1

1
√ τ1 ,
2 ij

(13)

with τ 1 the first Pauli matrix.
Notice that, similarly to the case of zeroth order in the
interaction, the factor vj at the external point z is compensated by the prefactor coming from the Green’s function, Eq. (5). If the point of the observation z lies outside
the interacting region, z > L, then the dependence on z
(1)
(1)
disappears in the outgoing current, Jj,+ (z > L) = Jj ,
whereas the corrections to the incoming current are all(1)
together absent, Jj,− (z > L) = 0. In what follows we
discuss the corrections to the outgoing current. In view
of the later generalization involving an infinite summation of higher order terms it is useful to represent the
above first order expression as
(1)

Jj

=i

dω
2π

dx dy
lη ,mη

× TrK [ T ω (mη , y|lη , x; j, +, z)L0,ω (lη , x|mη , y)]
(14)
where we recall the definitions
lη = (l, ηl )
etc. Here we defined a ”boson propagator” representing
the interaction line
1 0
0 1
(15)
and the quantity T representing the triangle of Green’s
functions in Fig. 2
1
L0,ω (l, ηl , x|m, ηm , y) = (2πgl vl )δ(x − y)δlm τηl ,ηm

Ω
z

ω

ω+Ω

νµ
Tω (mη , y|lη , x; jη , z)
dΩ
= vj
TrK γ ext GΩ (jη , z|mη , y)¯ ν
γ
2π

(16)

× GΩ+ω (mη , y|lη , x)γ µ GΩ (lη , x|jη , z) ,

Ω
x

γ

FIG. 2: The diagram providing first-order correction to the
current due to interaction.

this diagram should be combined with the one, where the
arrows on the fermonic lines are reverted.
The triangle diagram is characterized by two Keldysh
indices and thus is subdivided into four blocks. Symbolically, we write
TrK [ T L] = T 11 LR + T 22 LA

4
anticipating that T 21 = 0, L21 = 0 (to be shown later) .
When integrating over Ω in (16) we find two generic
integrals. One of them is
dΩ (hj (Ω + ω) − hj (Ω)) = 2ω

TABLE I: Convention for the indices
α=
before/after
wire #

1

2

3

4

B
1

A
2

1

2

and the other is
dΩ [1 − hj (Ω + ω)hm (Ω)] = 2F (ω + µm − µj ). (17)
For the above form of hj (Ω), we have F (x) =
x coth(x/2T ).
As mentioned above there are no corrections to the
incoming currents. In addition to this observation we
should recall Kirchhoff’s law, stating the conservation of
charge. Given that the total incoming current is equal to
the total outgoing current, we should have J1 + J2 = 0,
which is indeed confirmed by direct calculation.
Taking these facts into account, only the difference of
(1)
(1)
1
the currents, J (1) = 2 (J2 − J1 ), is of interest. This
involves the difference of the components of T belonging
to different leads. Accordingly, for the case of two leads,
we define the weighted difference (denoted by the same
symbol, T , but dependent on fewer variables),

ωc = W , the band width. The conductance as a function
of voltage V , temperature T , wire length L, is found from
there as
G(1) = −2g G0 (1 − G0 ) Λ(V, T, L) .
Here we introduced the scaling variable Λ
ωc

dω F (ω + V ) − F (ω − V )
ωL
sin2
.
ω
V
v
0
(21)
The factor sin2 (ωL/v)] guarantees convergence of the integral at ω < 1/t0 = πv/L. At ω > 1/t0 we may average this rapidly oscillating function and approximate
sin2 (ωL/v)]
1/2. Employing this and analogous procedures for the cases of small V, L−1 or small V, T we
may approximate Λ as
Λ(V, T, L) =

1 µν
[T (mη , y|lη , x; 1, +, z > L)
2 ω
(18)
µν
− Tω (mη , y|lη , x; 2, +, z > L)]

µν
Tω (mη , y|lη , x) =

The 4×4 matrices appearing here are now direct products of 2 × 2 matrices in chiral space (outer block structure) and 2 × 2 matrices in lead space (inner block structure, see Table I )
r 2 t2
[F (ω + V ) − F (ω − V )]
8π


0 0 0 0
 0 0 0 0 ∗
× Φω (y) 
Φ (x) ,
1 −1 0 0 ω
−1 1 0 0

(20)

Λ(V, T, L)

ln

ωc
max{V, T, v/L}

.

(22)

V. SCALE-DEPENDENT PART OF THE
CURRENT: SUMMATION TO INFINITE ORDER
IN THE INTERACTION

T 11 =

T 22 = −(T 11 )† |x↔y ,

e−iωx/v1 e−iωx/v2 eiωx/v1 eiωx/v2
,
,
,
v1
v2
v1
v2
(19)
The vanishing of T 21 implies that the Keldysh component of the renormalized interaction, LK , does not enter.
Inserting the components of T µν and L0 into the expression (14) for the current for two equal wires, gj = g,
vj = v, we find
gt2 r2
π

ωc
0

−x1 −x1

−x2 −x1

+

+

=
x2

x1

x1

x2

−z

−x2 −x1

−z

−x2
+ ···

+
x1

z

x2

x1

z

x2

T 21 = 0 .

Φω (x) = diag

J (1) = −

−x1

dω
ωL
[F (ω + V ) − F (ω − V )] sin2
ω
v

Here we apply an upper cut-off ωc given in the microscopic model either as ωc = v/a as mentioned above or

FIG. 3: A series of diagrams showing the screening. The
negative sign of the coordinate corresponds to the incoming
(B) electrons.

As shown in our previous work, the perturbative treatment may be extended into the strong coupling regime by
summing up an infinite series of relevant scale-dependent
contributions to the conductance in all orders (“ladder
approximation”). These represent a self-energy renormalization of the “boson propagator” L0 introduced
above. They thus technically constitute a renormalized
one-loop contribution to the RG equation. This series
can still be represented by the generic diagram of Fig.
2, but the wavy line of electronic interaction should be
dressed by screening effects, as discussed below.

5
As a result of this summation the interaction line, gl ,
acquires non-locality and retardation effects. Moreover,
if we have initially only diagonal components in Keldysh
space, after the summation we generate a Keldysh component and in general a rather complicated structure.
Schematically, we replace L0 by L in Eq. (14):

L0 → L =

LR (lη , x|mη , y) LK (lη , x|mη , y)
ω
ω
0
LA (lη , x|mη , y)
ω

(23)

We now embark on the calculation of L . Introducing
compact notation, we express the lowest order result L0
1
in the form Lµν (lη , x|mη , y) = δµν τηl ,ηm δlm gδ(x − y) =
0
1
1K ⊗ τC ⊗ 1w gδ(x − y).
The integral equation describing the summation of the
relevant infinite class of diagrams (see Fig. 3) takes the
form

and hence,
LK = (1 − L0 ∗ ΠR )−1 ∗ L0 ∗ ΠK ∗ LA
= LR ∗ ΠK ∗ LA

where we used (26) to obtain the second equality. This
means that once we have LR , we can easily find the two
remaining components, LA and LK . We recall, however,
that as pointed out above the component LK does not
enter the calculation of the current.
The solution for LR in the linear response case was presented previously in our work26 . We follow that derivation but reformulate it somewhat for the present purposes. First we define the integral (scalar) kernel
Pω (j, x|l, z) = (2πvj vl )−1 (δ(τ ) + iωθ(τ )eiωτ )
τ = x/vj − z/vl

(27)

and the matrix quantity
L = L0 + L0 ∗ Π ∗ L0 + L0 ∗ Π ∗ L0 ∗ Π ∗ L0 + . . .
(24)
= L0 + L0 ∗ Π ∗ L

ΠR =

where Π represents a fermion bubble
dΩ
Πµν (lη , x|jη , y) = i
TrK [¯ µ GΩ+ω (lη , x|jη , y)
γ
ω
2π
(25)
× γ ν GΩ (jη , y|lη , x)]
The multiplication ∗ is defined as
(Π ∗ L)µν (jη , y|nη , x) =
ω

Πµλ (jη , y|lη , z)
ω

dz
lη λ=1,2

× Lλν (lη , z|nη , x)
ω

LR LK
0 LA

=

L0 0
0 L0

+

L0 0
0 L0

∗

ΠR ΠK
0 ΠA

LR (x|y) = 2πδ(x − y)
− 2π

+
∗

LR LK
0 LA

which means that we can solve the integral equation in
three steps.
First, we solve the coupled integral equations in the
retarded sector
LR = L0 + L0 ∗ ΠR ∗ LR
Second, considering that
LA = L0 + L0 ∗ ΠA ∗ LA
if we are using the relation between ΠA and ΠR , we need
not solve this equation separately. Third, we notice for
completeness that
LK = L0 ∗ ΠR ∗ LK + L0 ∗ ΠK ∗ LA

0 g
g 0
(29)

dz
a

gYΠ(x| − z) gΠ(x|z)
LR (z|y) ,
gΠ(−x| − z)
0

Here LR is a 4×4 (in the general case of n leads 2n×2n )
matrix. The elements of the 2×2 matrices in chiral space
(denoted by a hat) are 2 × 2 matrices in the space of the
2 leads (denoted by bold letters). The matrix of interaction constants is then given by g = diag[g1 v1 , g2 v2 ]. The
scattering properties of the junction are encoded in the
2 × 2 matrix Y. The equation for LA is similar to the
above, but
ΠA = ΠR

†

=
x↔z

(26)

(28)

where Πjl (x|z) = δjl Pω (j, x|l, z), YΠ(x|z) =
Yjl Pω (j, x|l, z) and YT Π(x|z) = Ylj Pω (j, x|l, z) with
Yjl = |Sjl |2 . In the case of two leads we have Y = YT .
Notice that YT Π(−x|z) = 0 for x, z > 0, and we use the
full form (28) for future reference.
Then we express the integral equation for LR as a 2×2
matrix equation in the chiral space

L

At the level of Keldysh structure we have

Π(−x| − z), YT Π(−x|z)
YΠ(x| − z),
Π(x|z)

0 1
ΠR
1 0

0 1
1 0

ω→−ω, Y→Y T

Because L0 does not contain ω, Y , it follows that
LA = LR

†

(30)
x↔y

The Keldysh part of the kernel takes the form presented in the appendix C. We show there, that ΠK is an
even function of V , and therefore LK does not contribute
to the current.

6

y

Following the method of solution of the integral equation described in26 we first solve the equation for the
case Yjl = 0 , to give a partial summation resulting in a
auxiliary quantity C

z

0 g
C(x|y) = 2πδ(x − y)
g 0

L(ω)

T

(31)

L

− 2π

γ
˜

dz
a

0
gΠ(x|z)
C(z|y) ,
gΠ(−x| − z)
0

γ

x

In terms of C the integral equation for LR may be expressed as

FIG. 4: The schematic diagram, with already algebraic quantity T (ω) and dressed interaction line L(ω).

L

LR (x|y) = C(x|y) − 2π

dz1 dz2 C(x|z1 )
a

0
0
LR (z2 |y) ,
YΠ(z1 | − z2 ) 0

×

(32)

LR
jk,simple =

+Υjk

The solution of the integral equation for C, which is of
the Wiener-Hopf type, may be found by an appropriate
ansatz described in26 . By construction, C(x|y) is diagonal in wire space, Cjl (x|y) = δjl Cj (x|y). The explicit
expressions for Cj (x|y) are presented in the Appendix D.
Returning to the integral equation (32) for LR in terms
of C we observe that its kernel is separable and thus the
solution may be readily obtained. The explicit expressions and some details of the derivation of this result are
given in Appendix D.
An inspection of Eqs. (14), (19) shows that the x, y
dependence of T µν comes only from the matrices Φ∗ (x),
ω
Φω (y). We combine these matrices with LR and integrate
over the position
dx dy Φ∗ (x) LR (x|y) Φω (y) .
ω

Cj,simple =

e

dω
11
Tr [Tcore LR
simple ]
2π

(34)

µν
with Tcore obtained by putting Φω (x) = Φω (y) = 1 in
Eq. (19). We show the algebraic relation (34) diagrammatically in Fig. 4.
Introducing the quantities dj and qj
2
d2 = 1 − gj ,
j

−1
qj =

gj
,
1 + idj cot( vωLj )
jd

(35)

we present the simpler expressions of C, L integrated
over position.

,

V1,j = (Cj,simple )12 ,
−1

Υjk = Yjl (1 − Q

1
idj e−iLω/v
gj sin(ωL/vj dj )


 ,

V2,j = (Cj,simple )11 ,

· Y )−1 ,
lk

−1
Q−1 = δjk qj ,
jk

(37)
Combining the above results, (19), (34), (36), (37), we
find the current for two equal wires, gj = g, vj = v, as
J (L) =

a

J (L) = −2 Im

−1, 0
0, −1

−iLω/v

idj e
,
−1
+qj  gj sin(ωL/vj dj )
−2iLω/v

(33)

Making now use of relations (19), (30), we arrive at a
much simpler algebraic expression for the current. Instead of (14) we have

(36)

V1,j V2,k , V1,j V1,k
V2,j V2,k , V2,j V1,k

where

L

LR
simple =

2πi
δjk Cj,simple
ω

g
8π

ωc

dω
0

× Re

F (ω + V ) − F (ω − V )
ω

2(1 − Y 2 )
1 − gY + id cot

(38)
Lω
dv

1
Again the ω −singularity of the integrand leads to a logarithmically divergent contribution, which we identify as
a scaling contribution. The singularity is controlled by
the largest of the three energy scales, (i) energy scale
ωL = v/L controlled by the length L, (ii) temperature ωT = T , (iii) bias voltage ωV = V . In the limit
V → 0, T → 0 we have F (ω+V )−F (ω−V ) = 2V sign(ω).
1
The ω −singularity is in this case cut off at the scale
ωL = dv/L by the cot Lω −term in the denominator.
dv
Above this scale we may average the rapidly oscillating
function in the curly brackets in (38) over one oscillation
period, ω0 < ω < ω0 + (π/t0 ) , with t0 = L/dv :

t0
π

ω1 +π/t0
ω1

dω
= (1−gY +d)−1 , (39)
1 − gY ± id cot ωt0

7
such that the correction to the conductance is obtained
as
G(L) = −g

L
(1 − Y 2 )
ln
1 − gY + d a

in agreement with22 . In the general case we find accordingly

G(L) = −g

(1 − Y 2 )
Λ
1 − gY + d

(40)

where Λ = ln(ωc / max{V, T, v/L} . In the limit of long
wires, L → ∞, a closed expression is found in Appendix
F in the form
Λ = ln

ωc
2πT

− Re ψ 1 +

iV
2πT

,

(41)

with ψ(x) digamma function. This function shows
a smooth interpolation between the regimes with
ωc
ln 2πT + 0.577 at V
T and ln (ωc /V ) at V
T.
Further corrections not considered here are generated
by the Hartree diagrams of the self energy: in the
nonequilibrium situation the local chemical potential is
renormalized by a molecular field term involving the bare
interaction and the local particle density. In Ref.13 this
effect is included by applying a corresponding boundary
condition to the thermodynamic Bethe ansatz fields. As
a consequence a bistability of the current has been found
at very strong interaction. We have not included this correction term into our analysis, as it would require a separate calculation of the single particle Green?s function,
especially of the local chemical potential shift, which is
beyond the scope of the present paper. We are therefore
confining our considerations from weak up to moderately
strong interaction, such that K > 0.2.

VI.

RENORMALIZATION GROUP EQUATION
FOR THE CONDUCTANCE

The above calculation of the leading scale-dependent
contribution to the current allows us to derive a renormalization group (RG) equation for the conductance
G = I/V as a function of the scaling variable Λ =
ln(ωc / max[V, T, v/L]) , G = G(Λ). We thereby use the
scaling property of G , G(V, T, v/L, G0 ; g) = G(Λ; g) . In
our previous works22,24 we explicitly checked this property in the equilibrium situation. We directly calculated
all the contributions to the conductance up to third order
in the interaction, which involves about 104 diagrams.
It was shown that the principal contribution near the
fixed points (FPs) of this equation is obtained in one-loop
order, with the interaction being dressed as described
above, g → L. The scaling exponents obtained this
way are identical to those found earlier by the method
of bosonization.

Away from the FPs one finds in general additional nonuniversal contributions, appearing first in the third order.
These determine the prefactor in the scaling law near the
FP and also fine details of the conductance in the intermediate regime. In the present study focused on the
transport far out of equilibrium it would be too costly
to perform a similar direct computation of all contributions up to third order. Instead we assume that even
out of equilibrium we have the scaling property and the
scaling exponents are fully determined by the contribution provided by the approximation of fully dressing the
interaction line of the one-loop calculation.
This assumption may be justified by at least two facts.
First, in the renormalized one-loop calculation presented
here the bias voltage V appears as an infrared cutoff
in the scale dependent terms, replacing the cutoff energy scales temperature T or level splitting v/L present
in equilibrium. It may therefore be expected that the
structure of the scale-dependent terms generated by the
cutoff V is analogous to that of the terms generated
by the cutoff T or v/L. No additional scale dependent
terms are found in non-equilibrium and none of the scaledependent terms present in equilibrium disappears in
non-equilibrium. This suggests that the structure of the
scale-dependent terms is preserved and therefore the scaling property is preserved even out of equilibrium. Secondly, as will be shown below, the results of our theory
are in agreement with exact results obtained by other
methods.
We now briefly review the logic by which the RGequation is derived from the perturbative result. We start
from the result for the renormalized conductance G as a
power series expansion in the interaction, and dependent
on the scattering properties of the junction (encapsulated
in the conductance G0 in the absence of interaction) obtained above, which takes the general form ,
G = G0 − gf (g, G0 )Λ + O(g 2 Λ2 )

(42)

In the approximation of summing up the leading terms
in each order, considered above, a very good approximation fapp of the function f (g, G) has been obtained, see
Eq.(40). We do not calculate the terms of order g 2 Λ2
and higher in this paper. The relation Eq. (42) is valid
in the asymptotic regime gΛ → 0 . With the aid of the
scaling property we may find the analytic continuation to
finite values of gΛ. To this end we first invert the relation
(making use of G = G0 + O(gΛ) ) and write
G0 (g, G; Λ) = G + gf (g, G)Λ + O(g 2 Λ2 )
Formally G0 here is a function of G, g and Λ . We now
employ the crucial property that the value of the bare
conductance, G0 , should not depend on the scaling variable Λ , which means
0=

∂G0 dG
∂G0
+
∂Λ
∂G dΛ

(43)

8
and hence

VII.

gf (g, G) + O(g 2 Λ)
dG
=−
dΛ
1 + gΛ[∂f (g, G)/∂G] + O(g 2 Λ2 )

SOLUTION OF RG EQUATION

(44)

Inverting Eq. (48) we write

The scaling property of G implies that the explicit Λdependence in (44) cancels. This leads to the definition
of the RG β-function

2(1 − K)dΛ = −dG

dG
= β(g, G) = −gf (g, G) .
dΛ

d
dΛ G

= −gfapp (g, G) + c3 g 3 G2 (1 − G)2 + O(g 4 )

(46)

The second term here of order g 3 originates from terms
not contained in the perturbation series for L considered
above. This term is subleading in the sense that it vanishes more rapidly on approach to the fixed points at
G = 0, 1 than the first term and does therefore not influence the critical properties. There are indications that
this is also the case with the higher order contributions
not captured by the ladder summation.
A similar conclusion regarding the relative unimportance of corrections beyond the ladder summation g → L
was reached in24 for the more general case of the threelead Y-junction. In the symmetrical setup the Y-junction
was characterized by two conductances, and after extensive computer analysis of perturbative corrections we arrived at a set of two coupled RG equations. We found
that the three-loop corrections, not contained in the ladder series of diagrams, did not contribute to the scaling
exponents.
We expect that non-universal contributions to the
β−function will also exist in the case of non-equilibrium,
but those terms will again be unimportant when it comes
to determine the critical behavior at the fixed points.
We will therefore approximate the exact function f (g, G)
by the one determined in the ladder approximation and
given through eq.(42), which gives rise to the β−function
d
G(1 − G)
G = −4g
dΛ
1 − g(1 − 2G) + d
Introducing the
(1 − g)/(1 + g),
as

which is explicitly solved in the next section.

(49)

1 − G0 2(1−K)Λ
1−G
=
e
.
K
G
GK
0

(50)

It is more instructive to exclude here the bare conductance, G0 , and to represent our result as (cf.3 )
e2(1−K)Λ(V1 ,T1 )
(1 − G)/GK |V1 ,T1
= 2(1−K)Λ(V ,T ) .
K|
2
2
(1 − G)/G V2 ,T2
e

(51)

The latter exponential can be written as
e2(1−K)Λ =

ωc
max{V, T, v/L}

2(1−K)

.

We see that near the two fixed points of the RG equation, G = 0, G = 1, we have the well-established scaling
behavior5
G
1−G

(V /ω0 )

2(K −1 −1)

c∗ (ω0 /V )

G → 0,

,

2(1−K)

,

(52)

G → 1.

with an appropriate ω0 and where V should be replaced
by exp(Λ(V, T, vF /l)) in the more general situation. At
the same time, (50) provides a smooth crossover between
the fixed points, i.e. for those values of G which, strictly
speaking, are inaccessible by the bosonization approach.
We notice further, that if the overall energy scale ω0
is fixed near one fixed point, then the constant c∗ is entirely defined by the three-loop and higher-loop terms
in the RG equation. In the approximation of neglecting
the three-loop terms, as in Eqs. (48), (41), the coefficient c∗ = 1. Keeping the three-loop terms, Eq. (50) is
approximately given by22
GK
1 + G 1−K
K
1−G

4c3 (1−K)

=

max{V, T, v/L}
ω0

2(1−K)

,
(53)

which implies c∗ = K −4c3 (1−K) .
(47)
A.

Luttinger parameter K
=
Eq. (47) may be re-expressed

d
G(1 − G)
G = −2(1 − K)
,
dΛ
K + (1 − K)G

,

which is integrated with the result

(45)

Our earlier direct third order calculation in22,24 showed
that the above ratio was indeed independent of Λ to the
considered accuracy g 3 . The function gf (g, G) has been
calculated beyond the ladder approximation in22 for the
present case of a two-lead junction with the result

K
1
+
G
1−G

(48)

Comparison with the exact solution at K = 1/2

To understand better the limitations of our formula
(50), we compare it with the exact result at K = 1/2.
Explicitly, our expression in this case reads as
√
T
1 + 4x2 − 1
iV
G=1−
, x = ∗ exp Re ψ 1 + 2πT .
2
2x
T
(54)

9

G1/2 = 1 −

1 T∗
V
4πT ∗
Im ψ
+
+i
V
2
T
4πT

,

(55)

with T ∗ depending on the impurity backscattering amplitude and the ultraviolet cutoff. In two important limiting
cases we have for the linear conductance

20.0
10.0

G1/2/(1-G)

The exact formula, obtained with the aid of the Bethe
ansatz29 is

G1/2 (T ) = G1/2 (V → 0, T ) ,
=1−

T∗
T ψ

1
2

+

T∗
T

T
, T
T∗
π2 T ∗
1−
, T
2 T

T∗ ,

(56)

G1/2 (V ) = G1/2 (V, T → 0) ,
1
12

4πT ∗
V

1.0
0.5
0.2

0.1 0.2

0.5 1.0 2.0

x

5.0 10.0 20.0

√
FIG. 5:
Comparison of the ratios G/(1 − G) for (i)
∗
G = G1/2 (T = xT1 ), Eq. (56), (black dashed line), (ii)
∗
G = G1/2 (V = xT2 ), Eq. (57), (blue dotted line) and the
√
linear dependence, G/(1 − G) = x, (red solid line) expected
for the expression (54). See text for additional explanations.

T∗ .

And for the nonlinear conductance

=1−

2.0

0.1

,

2

1
12

5.0

V
arctan 4πT ∗ ,
2

V
, V
T∗ ,
2πT ∗
2πT ∗
, V
T∗ .
1−π
V

(57)

These expressions indicate the existence of non-universal
three-loop terms in the RG β-function. Indeed, fixing
∗
the overall scale at small T by G1/2 (T ) = (T /T1 )2 with
√
√
∗
T1 = 12T ∗ gives the above constant c∗ = π 2 /(4 3)
1.424. At the same time, fixing the scale at small V
√
∗
∗
by G1/2 (V ) = (V /T2 )2 with T2 = 2π 12T ∗ produces
√
c∗ = π/(2 3) 0.91.
This means, firstly, that the three-loop term ∼
c3 g 3 G2 (1 − G)2 in the RG equation (46) has a different
prefactor c3 , depending on whether the choice of lowenergy cutoff is T or V . This fact was noted in22 on the
basis of direct computation of perturbative corrections.
From the above estimate c∗ = K −4c3 (1−K) we retrieve
c3
0.255 and c3
−0.070 for G1/2 (T ) and G1/2 (V ),
respectively.
Secondly, in the absence of three-loop RG terms (c∗ =
1) the ratio GK /(1 − G), appearing in (51), should be
a linear function of V , T at K = 1/2. Plotting this
ratio for the functions (56), (57), we compare it with the
straight line corresponding to Eq. (54). We confirm much
better agreement with the straight line in the case of the
non-linear conductance G1/2 (V, T → 0), see Fig. 5.
In practical terms these observations mean the following. When fitting experimental data with one universal
curve for the whole range of conductances, one should
use slightly different expressions for G(V = 0, T ) and
G(V, T = 0). The generic formula is (53), where the
value c3 = 1/4 is appropriate for G(V = 0, T ), while
c3 −0.07 is better suited for G(V, T = 0).

B.

Oscillatory nonlinear conductance

Let us also discuss the case T = 0 and V L finite. The
expression (21) is reduced in this case to
ωc e
+ f (2V L/v) ,
V
f (x) = Ci (x) − sin x/x ,
Λ = ln

(58)

− cos x/x2 , x
1,
γE − 1 + ln x , x
1,

with γE
0.577 . . .. We see the appearance of oscillations in 2V L, discussed in11 for the case of weak impurity.
In our treatment it corresponds to G 1, and from (50)
may may represent the conductance as follows
2(1−K)

G = 1 − c∗ (ω0 /V )

exp (2(1 − K)f (2V L/v)) (59)

cf. (52).
In the limit of large 2V L/v we have
|f (2V L/v)|
1 and
exp (2(1 − K)f (2V L/v))

1 + 2(1 − K)f (2V L/v) (60)

This is in agreement with11 where the corresponding expression in this limit and in our notation reads as
osc
1 + fBS

1 − 2(1 − K)

VIII.

cos(2V L/v)
.
(2V L/v)2

CONCLUSION

Electron transport through one-dimensional quantum
wires of various types has been studied experimentally

10
in several recent works. In a typical set-up stationary
charge transport is measured in a two-point geometry of
a system of one or several wires connected by a junction.
The quantum wires are adiabatically connected to reservoirs kept at a fixed chemical potential and temperature.
These systems are described by modeling the quantum
wires as Luttinger liquids (spinless, or spinful) of fermions
with linear dispersion subject to point-like interaction
and treating the reservoirs as non-interacting. A useful
picture of the transport process is to think of individual
electrons entering the interaction region (quantum wires
plus junction) from an initial reservoir and leaving as individual electrons into the final reservoir. If we model the
reservoirs as non-interacting systems there is no room for
collective excitations such as fractional quasiparticles or
multiple quasiparticles in the final state.
Conventionally this problem has been addressed by the
bosonization method, which takes advantage of the fact
that the exact excitations of a clean Luttinger wire are
bosons, at least in the infinite wire. The problem of
including the transformation of incoming electrons into
bosons has been addressed for the clean wire, and is believed to be solved. For the case of semi-wires connected
by a junction there is no convincing calculation of the
above transformation available. In order to avoid this
difficulty we are using a fermionic representation.
Our approach starts with determining the leading
scale-dependent contributions to the conductances in all
orders of perturbation theory. We have demonstrated in
the linear response case that by summing up these terms
one arrives at a description of the critical properties near
the fixed points (i.e. the location of the FPs and the critical exponents describing the power laws followed by the
conductances). For this it is necessary to establish the
scaling property of the conductances (or else to assume
its validity, which is usually done), allowing to derive a
set of renormalization group equations out of the perturbative result.
In the present paper we followed this approach for the
case of stationary non-equilibrium transport. We first derived a general result for the scale-dependent terms in the
conductances of an n-lead junction in first order of the
interaction. Then we presented the infinite order summation for the dressed interaction. At this point we specialized our considerations to the case of two symmetric semiwires. We derived the corresponding RG-equation for
the conductance. In general the scaling is dependent on
three energy scales, bias voltage V , temperature T , and
infrared cut-off provided by the wire length v/L. Whenever one of these energy scales dominates, the scaling
v/a
variable is varying logarithmically, Λ = ln( max{V,T,v/L} ).
In the case V, T
v/L we were able to determine the
form of the scaling variable describing the crossover from
the regime characterized by V
T to V
T , as well.
The intermediate results presented for the general case
of an n-lead junction should be a good starting point
for analyzing the behavior of non-equilibrium transport
through Y -junctions or even four-lead junctions. Work

in this direction is in progress.

Acknowledgments

We are grateful to D. A. Bagrets, I. V. Gornyi and D.
G. Polyakov for useful discussions. This work was partly
supported by the German-Israeli Foundation (GIF) and
by a BMBF grant. The research of D.A. was supported
by the Russian Scientific Foundation grant (project 1422-00281).

Appendix A: Normalization of wave functions

The usual summation over the quantum states in the
infinite medium is done as an integration over the momentum dk/(2π) or the summation, n over the quasimomentum k = 2πn/L in a ring geometry with a finite
length L. In our situation with a broken translational
symmetry we should resort to the integration over the
energy, then the correct normalization factor is given by
the density of states, which is the inverse Fermi velocity
in the simplest situation, n → dE/vF . In case with
several Fermi velocities, vj , in different wires we shall
keep the integration over the energy, and the normalization factor enters the definition of the wave functions.
Thus in the formula for the retarded Green’s function,
GR (l, y|j, x) =
E

dE

φE,l (y)φ∗ (x)
E,j
,
ω − E + i0

(A1)

we adopt the wave functions in the jth wire in the form
φE,j (y) = eiE y/vj /

2πvj

(A2)

and come to the formula (5). Notice also that the integration in (A1) should be restricted by the electronic
bandwidth |E| < W = EF , which can be modeled by introducing the density of states function, Nj (E), with the
−1
property Nj (0) = vj . So strictly speaking the formulas
(5) are defined at |ω|
W , which justifies the upper cutoff in energy in the calculation of logarithmic corrections
and the RG procedure.

Appendix B: Keldysh structure of the triangle T

The straightforward calculation shows that only a few
terms in the complicated expression for T contribute to
the final result. Let us sketch here the derivation and
present arguments showing the selection of the relevant
terms.
To condense our writing, we use the position dependent notation, GR → R, and position in the product
ω
denotes the position in the initial expression, (16). So
that GR (z|x)GL (x|y)GA (y|z) ↔ RLA etc. Up to a
Ω
ω+Ω
Ω
numerical factor we have

11
Appendix D: Full form of the solution for LR (x|y)
jk

T11 = RAA + (K + A)(RK − RR + KA),
T22 = RRA + (RK + KA + AA)(K − R),
T21 = RKA + KAA + RRK − RRR + AAA.

(B1)

The combinations RRR and AAA are necessarily zero for
the point z outside the interacting region. We may suggest (and it is confirmed by the direct calculation), that
the contributing terms in (B1) are those which contain
two Keldysh components, K. In this sense, we may keep
only the terms
T11
T22

KRK + KKA,
RKK + KAK,

T21

0

(B2)

Note that the notation “ ” here also means that the
combination KK should be regularized at Ω → ±∞ by
subtracting 1 from the product of distribution factors
hj (ω + Ω)hl (Ω). This regularization is suggested by inspection of the corresponding expressions in the direct
calculation. A closer inspection shows that the combinations hj (Ω)hl (Ω), not containing ω do not contribute to
the corrections, when multiplied by L(ω).
Thus the expressions for Tij can be simplified even
further :
T11

KKA,

T22

RKK,

T21

0

The solution of (31) can be found as follows. We iterate
the right-hand side of the equation once, to arrive at the
diagonal kernel with components of the form
g2
δ
v

x−z
v

ω
+ i (eiω(x+z)/v + eiω|x−z|/v )
2

and another component obtained from here by changing
x → L−x, y → L−y. We pick first the easier part of this
iterated kernel, ∝ δ(x − z), and arrive at the equation for
Cj (x|y) with more complicated inhomogeneity instead of
L0 and non-singular kernel. This latter kernel shows a
jump in its derivative at x = z, which we use by twice
differentiating Cj (x|y) with respect to x. We thus arrive
at a second-order differential equation, similarly to what
was done in26 . The difference now is that we deal with a
2 × 2 matrix for Cj (x|y) for each wire j. We determine
the solution to this differential equation dependent on
the x variable up to terms proportional to e±iωx/(vj dj )
ˆ
which are multiplied by as yet unknown matrices A(y),
ˆ
B(y), respectively. Considering the initial Eq. (31) for
Cj (x|y) in the simpler cases x = 0, x = L, we form a set
ˆ
ˆ
of two coupled (matrix) equations for A(y), B(y), which
is eventually solved. As a result we obtain the quantity
C diagonal in wire space, with its diagonal elements Cj
of the form

(B3)

The last expression means that the corrections to the
incoming current are absent, because GA (y|z) = 0 and
Ω
GR (z|x) = 0 in this case, due to the step functions in (5).
Ω

Appendix C: Keldysh kernel of integral equation

Cj (x|y) =

2
iπωgj
2πvj gj
−gj , 1
δ(x − y)
+
2
3
dj
dj
1, −gj

× eiω|x−y|/vj dj
+

dj sgn(y − x) − 1,
gj
gj ,
dj sgn(x − y) − 1

2
iπωgj iωx/vj dj
dj eiω(2L−x)/vj dj ˆ
ˆ
e
Aj (y) +
Bj (y)
4q
dj j
sin( vωL )
d
j j

(D1)
with dj , qj defined in (35) and
ΠK =

i ∗
1F (ω) YF (ω)
Φω (y)
Φ (x)
2π ω
YF (ω) K(ω)

(C1)

with
∞

Kjl (ω) =
−∞

dΩ
∗
S ∗ Slm Sln Sjn (1−hm (Ω)hn (Ω+ω))
2 m,n jm

ˆ
Aj (y) =

−1
−1
gj (qj − gj ),
gj (1 − gj qj )
−1
−1
(dj − 1)(qj − gj ), (dj − 1)(1 − gj qj )

× cos(

−1
ωy
ωy
−qj , 1
) + idj sin(
)
−1
vj d j
vj dj −qj , 1

ωy
ˆ
)
Bj (y) = i cos(
vj dj

The latter quantity may be cast in the form
+ dj sin(
K(ω) = F (ω)

1 0
0 1

+ r2 t2 F2 (ω, V )

1 −1
−1 1

F2 (ω, V ) = F (ω + V ) + F (ω − V ) − 2F (ω) .

,

(C2)
with F (ω) defined in (17). Importantly, K(ω) is an even
function of ω.

ωy
)
vj dj

(dj −1)
gj ,

1,

(1 − dj )
−gj

(dj −1)
gj ,

0

1,

0

(D2)
We next use these expressions in Eq. (32), which can
be schematically represented as
L = C + C ∗ Y ∗ L = C + C ∗ Υ ∗ C,

Υ = Y + Y ∗ C ∗ Y + . . . = Y ∗ (1 − C ∗ Y )−1

(D3)

12
and obtain finally
LR (x|y) = δjk Cj (x|y) − iω
jk

2πgj gk
Υjk
d2 d2
j k

V1,j (x)V2,k (y) V1,j (x)V1,k (y)
V2,j (x)V2,k (y) V2,j (x)V1,k (y)
ωy
ωy
−1
V1,j (x) = (1 − gj qj ) cos(
) + idj sin(
)
v j dj
vj dj
ωy
ωy
−1
−1
V2,j (x) = (qj − gj ) cos(
) − idj qj sin(
)
vj dj
vj d j
(D4)
with Υ given in Eq. (37)
×

Appendix E: Analytic properties of L(ω).

In contrast to our previous studies22,26 , we see now the
appearance of poles in the ω-plane of the quantities qj ,
Eq. (35), which we further integrate over ω. Given the
arbitrariness of the S-matrix, reflected in Yik = |Sik |2 ,
we check here the absence of singularities in LR (ω) in
the upper semiplane of complex ω.
The poles of qj correspond to the solution of

n = 0, ±1, ±2, . . .

−1
hence the poles of qj are always in the lower semiplane
of complex ω, as it should be for a retarded function.
Less trivial is the question about the position of the
poles of the above expression (1−Q−1 ·Y)−1 . We consider
it for a simpler situation with identical wires, gj = g,
dj = d = 1 − g 2 , vj = v.
The poles are defined by

gY
= 0.
1 + id cot ω
¯

Since the denominator in the last expression cannot modify the location of poles, and sin ω = 0 is not a solution,
¯
we can rewrite
det [id cot ω + 1 − gY] = 0.
¯
Defining the eigenvalues of Y as yj , with j = 1, . . . N (for
a junction connecting N wires), the poles are are defined
d
by conditions (cf. above) tan ω = −i 1−gyj , or
¯
ω = −i arctanh
¯

d
+ πn,
1 − gyj

In the limit L → ∞ we have to evaluate the integral
P (T, V ) =

where we introduced ω = ωL/vj dj . The last equation
¯
means that we have an infinite sequence of roots

det(1 − Q−1 · Y) = det 1 −

Appendix F: Scaling variable Λ in the crossover
regime.

W

tan ω = −idj ,
¯

ω = −i arctanh dj + πn,
¯

1 in the jth row and 0 otherwise. This is a set of N
generators of a Cartan subalgebra of the algebra U (N ),
normalized according to Tr(λj λk ) = δjk . A rotation of
˜
these generators is defined as λj = S † λj S where S is the
˜
unitary matrix. Obviously the new set λj remains or˜ j λk ) = δjk . The operator P , defined by
˜
thonormal Tr(λ
˜
˜
P A = j λj Tr(λj A), is a projection operator, P 2 = P .
˜
We have P λk = j λj Yjk because Y can be written as
2
˜
Yjk = |Sjk | = Tr(λj λk ). Let {cj } be an eigenvector of
Y, i.e. k Yjk ck = ycj . Introducing the diagonal matrix
˜
˜
λ∗ = j cj λj , we obtain P λ∗ = y λ∗ , with λ∗ is the ro˜
tated vector λ∗ . Since P λ∗ ≤ λ∗ and λ∗ = λ∗ ,
we conclude that |y| ≤ 1.
d
It follows that the above ratio 1−gyj is always positive.
−1
As a result, the poles of (1 − Q · Y)−1 lie in the lower
semiplane of ω.

−W

Generally we have |yj | ≤ 1 for all j. This is evident for N = 2, and can be easily extended to any
N . The proof of this statement is as follows.24 Consider a set of diagonal N × N matrices λj with values

(F1)

with the upper cutoff being W = vF /a or the bandwidth.
After the rescaling ω = 2T x, V = 2T y, W = 2T w we
have
w

P (T, V ) = 2T
−w

x+y
x−y
dx
−
.
x tanh(x + y) tanh(x − y)

We can make a shift x → x ± y in two terms here; the
contributions from the limits ±w do not cancel upon this
shift, but add a constant in the limit w → ∞. After
simple calculation we get in this limit
w

P (T, V ) = 4T y 2 +
−w

dx x coth x
x2 − y 2

,

(F2)

where the principal value of the integral should be taken.
Next we close the contour of integration in the upper
semiplane of complex x, by adding a semicircle of radius w. The contribution from this semicircle vanishes
as O(w−2 ) and we reduce the remaining integral to the
sum over the residues of coth x at x = iπn as follows :
w
−w

j = 1, . . . N.

dω
[F (ω + V ) − F (ω − V )] ,
ω

w/π

2n
dx x coth x
→
,
x2 − y 2
n2 + (y/π)2
n=1

(F3)

The last sum is easily evaluated with the final result of
the form P (T, V ) = 4V Λ with Λ given by
Λ = ln

We
2πT

−

1
2

ψ 1+

iV
2πT

+ψ 1−

iV
2πT

with e = 2.718 . . . and ψ(x) digamma function.

. (F4)

13

1

2

3

4
5

6

7
8
9

10

11

12

13

14

15

M. G. Prokudina, S. Ludwig, V. Pellegrini, L. Sorba, G. Biasiol, and V. S. Khrapai, Physical Review Letters 112,
216402 (2014).
H. T. Mebrahtu, I. V. Borzenets, H. Zheng, Y. V. Bomze,
A. I. Smirnov, S. Florens, H. U. Baranger, and G. Finkelstein, Nature Physics 9, 732 (2013).
S. Jezouin, M. Albert, F. D. Parmentier, A. Anthore,
U. Gennser, A. Cavanna, I. Safi, and F. Pierre, Nature
Communications 4, 1802 (2013).
W. Apel and T. M. Rice, Phys. Rev. B 26, 7063 (1982).
C. L. Kane and M. P. A. Fisher, Phys. Rev. B 46, 15233
(1992).
D. L. Maslov and M. Stone, Phys. Rev. B 52, R5539
(1995).
A. Furusaki and N. Nagaosa, Phys. Rev. B 47, 4631 (1993).
I. Safi and H. J. Schulz, Phys. Rev. B 52, R17040 (1995).
A. Furusaki and N. Nagaosa, Phys. Rev. B 54, R5239
(1996).
M. Sassetti and B. Kramer, Phys. Rev. B 54, R5203
(1996).
F. Dolcini, H. Grabert, I. Safi, and B. Trauzettel, Phys.
Rev. Lett. 91, 266402 (2003).
F. Dolcini, B. Trauzettel, I. Safi, and H. Grabert, Phys.
Rev. B 71, 165309 (2005).
R. Egger, H. Grabert, A. Koutouza, H. Saleur, and
F. Siano, Phys. Rev. Lett. 84, 3682 (2000).
W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, and
K. Sch¨nhammer, Rev. Mod. Phys. 84, 299 (2012).
o
M. Bockrath, Science 275, 1922 (1997).

16

17

18

19

20

21

22

23

24
25

26
27

28

29

F. Milliken, C. Umbach, and R. Webb, Solid State Communications 97, 309 (1996).
A. Yacoby, H. L. Stormer, N. S. Wingreen, L. N. Pfeiffer,
K. W. Baldwin, and K. W. West, Phys. Rev. Lett. 77,
4612 (1996).
S. Tarucha, T. Honda, and T. Saku, Solid State Communications 94, 413 (1995).
S. J. Tans, M. H. Devoret, H. Dai, A. Thess, R. E. Smalley,
L. J. Geerligs, and C. Dekker, Nature 386, 474 (1997).
D. Yue, L. I. Glazman, and K. A. Matveev, Phys. Rev. B
49, 1966 (1994).
D. N. Aristov and P. W¨lfle, Europhysics Letters 82, 27001
o
(2008).
D. N. Aristov and P. W¨lfle, Phys. Rev. B 80, 045109
o
(pages 22) (2009).
D. N. Aristov, A. P. Dmitriev, I. V. Gornyi, V. Y. Kachorovskii, D. G. Polyakov, and P. W¨lfle, Phys. Rev. Lett.
o
105, 266404 (2010).
D. N. Aristov, Phys. Rev. B 83, 115446 (2011).
D. N. Aristov and P. W¨lfle, Phys. Rev. B 84, 155426
o
(2011).
D. N. Aristov and P. W¨lfle, Lith. J. Phys. 52, 89 (2012).
o
D. N. Aristov and P. W¨lfle, Phys. Rev. B 86, 035137
o
(2012).
D. N. Aristov and P. W¨lfle, Phys. Rev. B 88, 075131
o
(2013).
U. Weiss, R. Egger, and M. Sassetti, Phys. Rev. B 52,
16707 (1995).