Critical conductance of a one-dimensional doped Mott insulator
M. Garst,1 D. S. Novikov,2 Ady Stern,3 and L. I. Glazman2
1
Institut f¨r Theoretische Physik, Universit¨t zu K¨ln, 50938 K¨ln, Germany
u
a
o
o
W. I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA
3
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
(Dated: January 22, 2008)

arXiv:0708.0545v2 [cond-mat.mes-hall] 23 Jan 2008

2

We consider the two-terminal conductance of a one-dimensional Mott insulator undergoing the
commensurate-incommensurate quantum phase transition to a conducting state. We treat the leads
as Luttinger liquids. At a specific value of compressibility of the leads, corresponding to the LutherEmery point, the conductance can be described in terms of the free propagation of non-interacting
√
fermions with charge e/ 2. At that point, the temperature dependence of the conductance across
the quantum phase transition is described by a Fermi function. The deviation from the LutherEmery point in the leads changes the temperature dependence qualitatively. In the metallic state,
the low-temperature conductance is determined by the properties of the leads, and is described by
the conventional Luttinger-liquid theory. In the insulating state, conductance occurs via activation
√
of e/ 2 charges, and is independent of the Luttinger-liquid compressibility.
PACS numbers: 71.27.+a 71.10.-w 73.43.Nq 73.21.Hb

I.

INTRODUCTION AND OUTLINE

The dc conductance of a one-dimensional (1D) system
crucially depends on the properties of the leads attached
to it. Unlike in higher dimensions, this dependence is
not merely a technological, but rather a generic physics
problem.
In the case of the conventional Luttinger-liquid model
for a quantum wire,1 the role of the leads in the zerotemperature (T = 0) conductance is well studied.1,2,3,4,5
As long as the electrons do not backscatter off inhomogeneities, the conductance is not sensitive to the nature
of carriers in the wire, and is determined solely by the
leads. In the simplest case of the noninteracting leads,
the conductance is quantized in units of e2 /2π per channel (each of the two spin polarizations counts for a separate channel). This conclusion also remains true for the
incommensurate phase of a so-called Mott wire (doped
1D Mott insulator).6,7 Thus the charge fractionalization
characteristic of these systems1,8 is not observable in the
zero-temperature ballistic conductance; if the leads are
Fermi liquids, then the ballistic conductance of the wire
appears to coincide with that of the free electrons.9
The Tomonaga-Luttinger model is applicable to a relatively dense system of particles forming a compressible
liquid. The “distance” from the incompressible state may
be characterized by a parameter −r, having the meaning
of chemical potential measured from the point where the
system becomes incompressible. For example, for free
fermions filling up a conduction band
−r ≡ µ−∆

(1.1)

with µ and ∆ being the Fermi energy and band edge,
respectively. At T = 0 and µ < ∆ the system is incompressible.
The conventional Luttinger liquid model requires relatively high carrier density, −r/T ≫ 1. How robust is the

apparent free-electron picture of transport if the condition −r/T ≫ 1 is violated? One may expect deviations
from the apparent free-fermion form of the two-terminal
conductance in some intermediate region of parameters10
in a conventional quantum wire. However, for fully spinpolarized electrons (equivalent to spinless fermions) at
the point r = 0 of the quantum phase transition the interaction becomes irrelevant. The corresponding conductance equals 1 × e2 /h, i.e. a half of the unit conductance
2
quantum e2 /h for spinless fermions, which is again indistingushable from a free-fermion result.11
Charge carriers at the commensurate-incommensurate
quantum phase transition in a Mott wire can also be
mapped onto free spinless fermions.7 The distance to the
phase transition is characterized by a chemical potential −r, see Eq. (1.1), here measured with respect to the
Mott gap, ∆. However, these effective fermions have fractional charge, and are related to the original electrons in a
complicated way. It is therefore an important unresolved
question as to what the two-terminal critical conductance
of a Mott wire is.
The aim of this work is to investigate the finitetemperature conductance around the quantum phase
transition point in a Mott wire. Our main finding is
that such a setup may provide the way to experimentally
access the elusive character of the fractional quasiparticle
charge in an interacting 1D system.
To be specific, we consider the two-terminal conductance of a one-dimensional electron system close to the
Mott transition. The details of how the underlying commensuration is realized in practice, e.g. by a periodic potential, will not be important; the only parameter from
the Mott wire we will need in the end is the effective value
∆ of the charge gap. Next, we assume that this wire is
smoothly attached to uniform identical Luttinger liquid
leads characterized by the Luttinger parameter KL . We
evaluate the finite-temperature conductance close to the
Mott transition perturbatively in KL − 1 . The value of
2

2
1
KL = 2 is special (Luther-Emery point12 ), as it allows a
mapping of the entire system onto a free-fermion one. In
1
lowest order in KL − 2 , we find

1
1
G
= f (r/T ) + KL −
G0
2
2

f 2 (r/T ) ,

G0 =

2e2
,
h
(1.2)

where r is the tuning parameter (1.1) of the Mott transition, G0 is the conductance quantum, and
f (x) =

1
1 + ex

At low temperatures transport in the Mott insulator occurs via activation of the fractional charges (1.4), resulting in the announced prefactor 1 G0 ≡ 2e2 /h. Finally,
CI
2
in the extended quantum critical regime, |r| ≪ T , the
conductance is dominated by its critical value,
2KL + 1
1
G0 + O(KL − 2 )2 .
8

II.

(1.3)

is the Fermi function. On the metallic side, r < 0, the
conductance approaches the value KL G0 in the limit of
zero temperature, i.e., it is determined by the Luttinger
liquid parameter of the leads in agreement with Ref. 6.
On the other hand, on the insulating side of the transition, r > 0, transport at low T is thermally activated
as the Fermi function reduces to a Boltzmann factor,
f (r/T ) ≈ e−r/T . In lowest order in the fugacity e−r/T ,
we find that the conductance G = 1 G0 e−r/T is not af2
fected by the leads, i.e., is independent of KL . In particular, the prefactor of the exponential is determined by
the effective conductance quantum 1 G0 attributed to the
2
degrees of freedom of the critical Mott wire. As we will
explain in detail below, the critical degrees of freedom
are the fermions that carry fractional charge
√
(1.4)
eCI = e/ 2 .

Gcr ≡ G|r=0 =

finding its two-terminal dc conductance. The latter is
affected by the different nature of the charge carriers in
the leads and in the wire.
The paper is organized as follows. In Section II we
consider the properties of the uniform Mott wire. In Section III we consider the Mott wire with the leads, and
evaluate the modification of the critical conductance induced by the presence of the leads. Section IV is devoted
to the discussion of our results.

(1.5)

Note that the specific fraction of the conductance quantum here depends both on the fractional charge of the
carriers and on the properties of the leads. These results
differ from the considerations of Mori et al..7
Our study of the two-terminal conductance of the critical Mott insulator complements the theoretical investigation of ballistic transport in massive theories in general,
for a recent review see Ref. 13. The thermally activated
quasiparticles above a gapped ground state in a homogeneous system (no leads) may travel ballistically thus giving rise to a singular contribution to the optical conductivity, σsing. (ω) = 2πDδ(ω), with a characteristic Drude
weight D = D(T ). The low-temperature Drude weight of
the Mott insulator is believed to remain finite even away
from criticality.14 (In the exact commensurate case of an
undoped Mott insulator the fate of the Drude weight is
less clear and remains a subject of ongoing research.15,16 )
Remarkably, the knowledge of the Drude weight in the
conductivity of a uniform 1D system is insufficient for

UNIFORM MOTT WIRE

In this Section we consider the model of the uniform 1D
Mott wire. In Sec. II A we introduce the Hamiltonian of
our system, and in Sec. II B we sketch the critical theory
describing the Mott transition. In Sec. II C we outline
the application of the Kubo formula for the calculation
of conductance, and derive the critical conductance for
the Mott wire without leads in Sec. II D.
A.

Model of the one-dimensional Mott insulator

Consider an interacting one-dimensional electron system in the vicinity of the Mott transition. The latter can
be realized, e.g., in a Hubbard model close to half-filling.
The effective Hamiltonian is conventionally represented
in terms of the bosonic charge and spin excitations that
decouple at sufficiently low energies.8 In this work we
only focus on the charge sector which, near the transition, is described by the sine-Gordon model8
dx
u
Ku(∂x θc )2 + (∂x φc )2
2π
K
√
2VUmkl
cos( 8φc − q0 x) .
+ dx
(2πa)2

H[φc , θc ] =

(2.1)

Here K is the Luttinger parameter, u is the plasmon
velocity, VUmkl is the Umklapp scattering amplitude
(given by the Fourier component of the lattice potential at the reciprocal lattice vector) that corresponds
to g3 in the standard g-ology classification,8 and a is
a short distance cutoff. The conjugate fields φc and
1
θc , with [ π ∂x φc (x), θc (x′ )] = −iδ(x − x′ ), describe the
smooth components of the density and current fluctuations within the system,
√ ∂x φc
,
δn = − 2
π

and j =

√
∂xθc
,
2 Ku
π

(2.2)

√
respectively. The prefactor 2 in these expressions is attributed to the two spin polarizations of the electrons.
The parameter q0 measures the distance to half-filling.
In the presence of repulsive interaction between the electons, K < 1, the umklapp scattering process, VUmkl , becomes relevant and opens up a gap in the spectrum at
half filling q0 = 0, rendering the system into the Mott
insulating phase.

3
The elementary excitations of the Mott insulator are
massive quantum solitons (kinks) of the model (2.1).
Classically, the kink must connect two discrete values of
φc (x → ±∞), which realize the minimum of the cosine
term. This dictates the fractional value of the topological
charge
1
Q=
π

∞
−∞

dx ∂x φsoliton (x)
c

1
=√ .
2

(2.3)

(2.5)

The above Hamiltonian for the charge sector can now be
refermionized with the help of the standard bosonization
formula,
1
e−iκφ+iθ ,
ψκ = √
2πa

(2.6)

with κ = +1 and −1 for right (R) and left (L) movers,
respectively. The fermionic particles associated with the
operator ψκ are identified with the solitonic excitations
of the sine-Gordon theory; the fermionic density thereby
coincides with their topological charge density. The resulting fermionic theory is the massive Thirring model,
H[Ψ] =

dx Ψ† −ivσ 3 ∂x − µ + σ 1 ∆ Ψ

1
+ 4 (g4 + g2 ) Ψ† Ψ

2

+

1
4

(g4 − g2 ) Ψ† σ 3 Ψ

2

2

(1 + νg4 ) − (νg2 ) ,
1 + νg4 − νg2
,
1 + νg4 + νg2

(2.8)
(2.9)

where the density of states for the Ψ-fermions

In these variables the model (2.1) becomes
dx
u
2Ku(∂x θ)2 +
(∂x φ)2
2π
2K
2VUmkl
+ dx
cos(2φ − q0 x) .
(2πa)2

u=v
2K =

Below we will absorb the topological charge (2.3) carried
by the solitons into the fractional electric charge quantum, eCI = Qe, cf. Eq. (1.4).
Upon detuning the system from half-filling by increasing q0 across a critical value the system becomes
conducting via the commensurate-incommensurate (CI)
transition.8,17 The ground state of this conducting phase
is characterized by a finite topological charge density.
Consequently, this transition is described18,19 by exploiting the duality20,21 between the sine-Gordon model (2.1)
and the massive Thirring model. To establish this duality, we make a canonical transformation to the rescaled
fields
√
√
(2.4)
φ = 2φc , and θ = θc / 2.

H[φ, θ] =

of the model (2.7) are the chemical potential µ = −vq0 /2
and the gap ∆ = VUmkl /(2πa);22 the velocity v and the
interaction constants g4 and g2 are implicitly given by

(2.7)
2

where σ i are the Pauli matrices, and we used the spinor
†
†
notation Ψ† = (ψR , ψL ) for the new spinless fermions. As
the Fermi momentum for the fictitious Dirac fermions Ψ
is set to zero (at the Dirac point), there are no spatially
oscillating factors in Eq. (2.6). These fermions (or the
solitonic excitations of the sine-Gordon theory) describe
the density excitations smooth on the scale of the original
lattice responsible for the Mott phase. The parameters

ν=

1
.
2πv

(2.10)

Note the redundancy of the g4 interaction as it enters the
mapping only in combination with the velocity, 2πv + g4 .
We deliberately choose to keep both the g2 and g4 interaction constants, as it will help us later to classify
different perturbative contributions to the conductivity.
The Luttinger interaction parameter for the new fermions
(2.7) is 2K. As a result, the interaction among them
1
vanishes at the Luther-Emery point,12 K = 2 . Later,
in Section III we will apply the above formalism to the
wire with the leads, and develop a perturbative expansion around this special value for the Luttinger parameter
within the leads, KL .
In terms of the spinless fermionic degrees of freedom
the charge current density, J = ej, is given by
√
√
∂xθc
2eKu
(2.11)
= 2 eCI Ψ† vJ σ 3 Ψ.
π
√
Again, the overall multiplicative factor 2 can be traced
back to the spin degree of freedom of the original electrons. As announced, we absorbed the √
topological charge
associated with the fermions, Q = 1/ 2, into the fractional electric charge quantum, eCI = Qe, of Eq. (1.4).
Moreover, the charge velocity of the new fermions reads
J=

vJ = v (1 + νg4 − νg2 ) .

(2.12)

For the general case g2 = g4 , the velocity vJ depends on
the interactions (cf. Ref. 8, Sec. 7.2).
The quadratic part of the refermionized theory (2.7)
can be diagonalized by a Bogoliubov transformation resulting in the two bands with the spectrum ±ǫk , where
ǫk =
∆2 + (vk)2 . When the chemical potential lies
within the gap, |µ| < ∆, the ground state is a Mott insulator. When the chemical potential is tuned into one
of the two bands the ground state becomes a conductor
via the CI transition.
B. Critical theory of the
commensurate-incommensurate Mott transition

From now on in this work we will focus on the properties of the system close to the Mott transition and assume in the following that the chemical potential of the
Ψ fermions is near the upper band edge, µ ≈ ∆, i.e.,

4
r ≈ 0, where r is the tuning parameter for the quantum
phase transition, cf. Eq. (1.1). The generalizaton to the
situation when µ ≈ −∆ is straightforward and will not
be further discussed.
The effective Hamiltonian governing the transition is
given by nonrelativistic free spinless fermions,18,19
HCI =

dx c† (x) r −

2
∂x
c(x),
2m

1
4

δk+k′ ,p+p′ Γkk′ ;pp′ c† c† ′ cp cp′ .
k k

∆

quantum critical
regime

(2.13)

where the operator c† creates a fermionic excitation in the
upper band, the mass is m = ∆/v 2 , and r is defined in
Eq. (1.1). The scaling dimension of the tuning parameter
and the critical dynamical exponent are dim [r] = 1/νr =
2 and z = 2, respectively.
As the gas of fermions is very dilute at the transition,
their interaction is weak. As the CI transition is approached, r → 0, the interaction among the fermions decreases faster than their density, rendering the c fermions
free at the transition. Speaking crudely, the fermions are
free not because they do not interact per se but because
there are too few particles to interact with. Formally, this
can be seen from considering the residual interaction
CI
Hint =

T

(2.14)

T ∼ rνr z

TLL

LL
0
0

FIG. 1: Phase diagram of the CI transition for temperatures
smaller than the gap ∆, see text. A quantum phase transition
with exponents νr = 1/2 and z = 2 separates the commensurate, r > 0, and the incommensurate, r < 0, phases. The
CI Hamiltonian (2.13) controls the vicinity of the quantum
critical point. The quantum critical regime occurs within a
cone T ∼ |r|νr z = |r|.

the conductivity,23

kk′ pp′

G = lim σ(x, x′ ; ω),

The interaction amplitude Γkk′ ;pp′ is strongly constrained
as the Pauli principle requires the effective amplitude to
be antisymmetric upon exchange of momenta p ↔ p′ and
k ↔ k ′ . Near the CI transition we can expand the amplitude, and the lowest order contribution is suppressed
by two powers of momenta,
Γkk′ ;pp′ = γ(k − k ′ )(p − p′ ),

(2.15)

with the coefficient γ = −g2 v 2 /(4∆2 ). As a consequence,
at the quantum critical point the residual interaction, γ,
is irrelevant in the renormalization group sense.
The phase diagram of the CI Mott transition is shown
in Fig. 1. The tuning parameter r controls the distance
to the quantum critical point; for r > 0 the system is in
the commensurate phase with an insulating ground state.
In the metallic phase, r < 0, at lowest temperatures
T < TLL , with log(|r|/TLL ) ∝ 1/|r|, a crossover between
the critical behavior and the Luttinger liquid takes place;
the residual interaction γ at finite density induces Luttinger liquid correlations in the spectral function of the c
fermions with the Luttinger parameter Kc − 1 ∝ |r|.
C.

r

ω→0

(2.16)

at fixed positions x and x′ . The conductivity σ(x, x′ ; ω)
describes the linear response of the charge current at position x upon applying an external24 electric field E(x′ , ω)
at position x′ with frequency ω
dx′ σ(x, x′ ; ω)E(x′ , ω).

J(x, ω) =

(2.17)

The conductivity is derived with the help of the Kubo
formula that relates σ to the charge current autocorrelation function
σ(x, x′ ; ω) =

ret
ret
Kx,x′ ;ω − Kx,x′ ;ω=0
,
iω

(2.18)

where the retarded function is defined by
ret
Kx,x′ ;ω = i

dt eiωt [J(x, t), J(x′ , 0)] .

(2.19)

The subtraction of the ω = 0 mode in (2.18) ensures
gauge invariance. In the following, it will be convenient
to obtain the retarded response function from the temperature correlation function

Transport
1/T

We will consider electric transport through the critical Mott wire. Like the thermodynamic properties, the
transport has universal characteristics at the Mott transition, on which we focus in the following. The conductance can be obtained from the zero frequency limit of

Kx,x′ ;Ωn =

dτ eiΩn τ Tτ J(x, τ )J(x′ , 0)

(2.20)

0

by the analytic continuation from the upper half-plane,
ret
Kx,x′ ;ω = Kx,x′ ;Ωn |iΩn →ω+i0 , where Ωn = 2πnT is the
bosonic Matsubara frequency.

5
D.

Critical conductance of the homogeneous
system

Consider first the uniform system close to the metalinsulator transition. At the criticality, the transport is
determined by the free nonrelativistic c-fermions, i.e., by
the Hamiltonian (2.13). The resulting conductance
G(hom) =

2e2
1
CI
f (r/T ) = G0 f (r/T )
h
2

(2.21)

is simply the conductance quantum 2e2 /h for the efCI
fective fermions with charge (1.4), times the occupation
factor that is given by the Fermi distribution (1.3).
The critical conductance is universal and its temperature dependence is controlled by a Fermi function (1.3).
The non-universal corrections arising from the residual
interaction (2.14) vanish with the distance to the quantum critical point as the interaction γ is an irrelevant
perturbation. In particular, in the limit of low temperatures the conductance at criticality, r = 0, approaches
the value
G(hom)

r=0,T →0

=

1
G0 .
4

(2.22)

At zero temperature the conductance G(hom) |T =0,r→0+ ≡
0 in the insulating phase, while in the metallic phase
G(hom) |T =0,r→0− = 1 G0 , thus the conductance exhibits
2
a jump across the transition of the height
G(hom)

T =0,r→0−

− G(hom)

T =0,r→0+

=

1
G0 .
2

(2.23)

The factor 1 originates from the fractional nature of the
2
charge carried √ the critical fermionic degrees of freeby
dom, eCI = e/ 2. As was pointed out in Ref. 25, the
conductance jump across the transition (2.23) in fact corresponds to a full conductance quantum but with fractional charge, G0 /2 = 2e2 /h.
CI
In the following section the modifications to the critical
conductance (2.21) due to the presence of leads attached
to the Mott wire are discussed.

properties of the critical conductance are modified. Below we develop the perturbation theory in the deviation
from the Luther-Emery point, and evaluate the critical
1
conductance in the first order in KL − 2 .

A.

Leads with Luttinger parameter KL =

The Ψ fermions entering the Thirring model (2.7) are
the appropriate degrees of freedom describing the Mott
transition as the interaction, g2 and g4 , among them is
irrelevant. In the following we will also adopt a description of the leads, where the Mott gap is absent, ∆ = 0,
in terms of the same Ψ-fermions. The advantage of such
a description is that the Mott region of the wire can be
incorporated into an effective scattering approach for the
Ψ fermions which is outlined in the following. The caveat
is that, unfortunately, these effective fermions will in gen1
eral, i.e. for KL = 2 , interact within the leads. This is
what makes this problem difficult, as one has to solve the
strongly interacting fermionic problem which also lacks
translation invariance. The influence of the interaction
within the leads is considered in Section III B perturbatively.
Mori et al. in Ref. 7 considered a very similar setup
as we do here. They also calculated the current-current
correlation function with the help of the Ψ fermions at
the Luther-Emery point but only for the uniform system,
where the Mott gap extends over the full wire. They
proceeded by using the expression for the homogeneous
case to derive the conductance in the presence of the leads
where the original electrons do not interact. We think
that the last step has led them to the erroneous result for
the temperature dependence of the critical conductance.
We will further elaborate on the difference between the
treatment of Mori et al. and ours in Section IV.

1.
III.

1
2

Scattering states

CRITICAL CONDUCTANCE IN THE
PRESENCE OF LEADS

In a two-terminal conductance measurement the critical Mott wire will be connected to leads. In the presence of leads the conductance will differ from the expression (2.21) for the homogeneous system. We assume
here that the leads contain Luttinger liquids characterized by the Luttinger parameter KL . Here we argue that
the critical conductance is that of the homogeneous case,
Eq. (2.21), only if the Luttinger parameter KL has the
Luther-Emery value KL = 1/2. Note that this value of
KL corresponds to strong interaction between the original
electrons in the leads. If KL differs from 1 , the universal
2

Let us consider the situation where the Luttinger parameter of the leads KL = 1 , corresponding to g2 ≡ 0
2
in the Hamiltonian (2.7). Here we also set g4 = 0. The
absence of interactions allows us to describe the wire in
terms of a simple 1D scattering problem. The leads can
then be characterized by the states of the Ψ fermions that
scatter off the Mott region. The crucial point is that the
latter will be treated as an effective point scatterer. This
is a correct approximation for the dc Landauer transport, and physically it means that we attach infinitely
long leads to the Mott region.
The scattering wave incident from the left has the stan-

6
dard form
ψǫ,+ (x) =
(3.1a)

0
 iǫx/v 1
e
, x<0
+ e−iǫx/v rL (ǫ)
1 
1
0
√
2πv  eiǫx/v t (ǫ) 1 ,

x > 0.

L
0

The energy ǫ is measured here (and everywhere below)
with respect to the Fermi level. The phase velocity in the
R- and L- moving states (upper and lower spinor components) equals ±v correspondingly. The wave incident
from the right reads

ψǫ,− (x) =
(3.1b)

0
 −iǫx/v
e
,
x<0
tR (ǫ)
1 
1
√
2πv  e−iǫx/v 0 + eiǫx/v r (ǫ) 1 , x > 0.


R
0
1

The transmission and reflection coefficients are components of the scattering S-matrix,
S(ǫ) =

tL (ǫ) rR (ǫ)
,
rL (ǫ) tR (ǫ)

(3.2)

which is unitary, S† S = 1. The unitarity of the S-matrix
ensures that it has only three independent parameters,
which can be chosen in the form
tL = tR = t eiφt ,

rL = −r eiφ− ,

rR = r eiφ+

(3.3)

with φt = (φ+ + φ− )/2, and positive r and t, r2 + t2 = 1.
The relation tL = tR is a consequence of the time-reversal
symmetry.
The presence of the Mott gap at the center of the
wire makes the transmission and the reflection coefficients strongly energy-dependent: The propagation is
blocked for the fermionic states with energies within the
gap, ǫ < r. These states therefore have zero transmission coefficient as we assume the Mott gapped region
of the wire to be sufficiently long such that the tunneling across it can be neglected. On the other hand, the
fermionic states with energies above the gap, ǫ > r, are
able to traverse the Mott region. We will assume that
the leads are attached to the Mott-gapped region of the
wire in the adiabatic fashion (i.e. VUmkl = VUmkl (x) is a
smooth function), such that the backscattering of particles moving above the gap can be neglected. The tuning
parameter, r, of the transition thus divides the fermionic
scattering states into the propagating waves with unit
transmission for energies ǫ > r, and into the standing
waves for energies ǫ < r. As a result, we model the Mott
region of the wire by utilizing simple unit step-function
transmission and reflection amplitudes
t(ǫ) = Θ(ǫ − r) ,

r(ǫ) = Θ(r − ǫ) .

(3.4)

Summarizing,
S(ǫ) =

Θ(ǫ − r)eiφt (ǫ) Θ(r − ǫ)eiφ+ (ǫ)
.
−Θ(r − ǫ)eiφ− (ǫ) Θ(ǫ − r)eiφt (ǫ)

(3.5)

The scattering phases φ± (ǫ) are accumulated while
traversing or reflecting from the Mott barrier. The barrier shape is encoded in their smooth energy dependencies,
φ± (ǫ) ≃ φ0 + ℓ± ǫ/v,
±

(3.6)

The scattering lengths ℓ± are a measure of the effective
length of the region of the wire where the Mott gap is
present.
2.

Green’s Function

The single-particle Green’s function in the Matsubara representation is a 2 × 2 matrix in the spinor space
[a, b = 1, 2 are the spinor components of the wave functions (3.1)]
ab
Gx,x′ ;τ = − Tτ ψ a (x, τ )ψ †b (x′ , 0) .

(3.7a)

In what follows we will be using its Lehmann representation
Ax,x′ ;ǫ
,
(3.7b)
dǫ
Gx,x′ ;ωn =
iωn − ǫ
Aab ′ ;ǫ =
x,x

where

a
†b
ψǫ,σ (x)ψǫ,σ (x′ ) ,

(3.7c)

σ=±

and ωn = π(2n+1)T is a fermionic Matsubara frequency.
As we show now, the Green’s function (3.7) has qualitatively different properties when its arguments x and x′
are located on the same and on the opposite sides of the
impermeable potential barrier. Introducing the labels
s = sign x and s′ = sign x′ ,

(3.8)

one finds:
s = s′ :
′

Ax,x′ ;ǫ = νΘǫ−r

eiǫ(x−x )/v+isφt (ǫ)
0
′
0
e−iǫ(x−x )/v−isφt (ǫ)
(3.9a)

s = s′ :
′

A

x,x′ ;ǫ

eiǫ(x−x )/v
0
=ν
′
0
e−iǫ(x−x )/v

+ νs Θr−ǫ

0
′

e−iǫ(x+x )/v−isφs (ǫ)

′

eiǫ(x+x )/v+isφs (ǫ)
.
0
(3.9b)

The spectral function (3.9a), as well as the first term in
the spectral function (3.9b) become translation-invariant
in the limit |x − x′ | → ∞, since they both depend only
on the coordinate difference x − x′ . They originate from
the propagating states and from the corresponding comoving parts of the standing waves. The second term
in Eq. (3.9b) is due to the counter-moving parts of the
standing waves and comes only from the states with energies below the barrier, ǫ < r. It lacks translational
invariance, and is affected by the barrier shape through
the energy-dependent reflection phase shifts φ± .

7
3.

Polarization operator

A basic ingredient in the following calculation is the
polarization operator that is a convolution of the temperature Green functions (3.7),
Πij ′ ;Ωn = −T
x,x

ωn

tr τ i G(x, x′ ; ωn + Ωn )τ j G(x′ , x; ωn )
(3.10)

where i, j = 0, 1 and
τi =

1 0
,
0 βi

with

βi =

+1 ,
−1 ,

i = 0;
i = 1.

(3.11)

The polarization matrix Πij ′ ;Ωn depends on the bosonic
x,x
Matsubara frequency Ωn = 2πnT . A detailed calculation
of the above expression can be found in the Appendix A.
In the dc limit (iΩn → ω + i0, ω → 0) it reduces to
(0) ij

Πij ′ ;Ωn = Πx,x′ ;Ωn + [f (r/T ) − 1]
x,x
′

¯
= 2νδij δ(x − x ) + Πij ′ ;Ωn ,
x,x
′

′
¯
Πx,x′ ;Ωn = −2πν 2 e−|Ωn ||x−x |/v

×

(0)ij
Πx,−x′ ;Ωn

,

=

¯
Πij n e−|Ωn x |/v
x,0;Ω
′

,

s = s′ ,
′

s=s.

(3.15)
(3.16)

Using the above simplification the expression Eq. (3.12)
can be cast into a form characteristic for a scattering
problem,

(3.12)

where f is the Fermi function (1.3), and we used the
abbreviation (3.8). The polarization operator matrix in
the absence of the Mott barrier is
(0) ij
Πx,x′ ;Ωn

′
(0)ij
¯
Πx,x′ ;Ωn = Πij n e−|Ωn x |/v ,
x,0;Ω

(0)
¯
¯
Πx,x′ ;Ωn = Πx,x′ ;Ωn + Πx,0;Ωn T Π0,x′ ;Ωn .

′

1 + ss (0) ij
1 − ss (0) ij
Πx,x′ ;Ωn − β j
Πx,−x′ ;Ωn
2
2

×

vanishes as sin(ωx′ /v) in the dc limit: Given enough
time, the system finds out about the barrier. For finite
f , the part of the expression (3.12) responsible for the
transmission of the particles is proportional to f (r/T ).
Technically, the presence of the barrier leads to an additional term in the expression (3.12) that is multiplied
by the factor f (r/T ) − 1. This term can be further simplified by noting that, in the dc limit, the point-scatterer
approximation for the Mott region allows us to substitute
|x + x′ | = |x| + |x′ | for s = s′ , and |x − x′ | = |x| + |x′ | for
s = s′ , yielding

(3.17)

In the limit of small Ωn (in the sense of analytic continuation iΩn → ω → 0 specified above), the scattering
T-matrix is given by
T=

(3.13)

0 0
,
0 T

T=

1 − f (r/T )
.
2πν 2 |Ωn |

(3.18)

(3.14)

|Ωn |
Ωn sign (x − x′ )
,
Ωn sign (x − x′ )
|Ωn |

with the density of states (2.10). The rows and columns
of the matrix (3.14) are numbered by the indices i, j =
¯
0, 1, such that Π00 corresponds to the upper left corner. The corrections to the result (3.12) are subleading
in powers of frequency Ωn , see Appendix A for details.
Let us discuss the meaning of the expression (3.12).
The trivial observation is that, in the absence of the
barrier, f = 1, and Π ≡ Π(0) becomes translation invariant. In the opposite case of an infinitely high barrier, f = 0, and Πij ′ ;Ωn ≡ 0 identically vanishes when
x,x
the points x and x′ are on the opposite sides of the
barrier (no transmission). We now consider the polarization operator when x and x′ are on the same side,
(0)ij
(0)ij
Πij ′ ;Ωn = Πx,x′ ;Ωn + β j Πx,−x′ ;Ωn . This expression has
x,x
a static part (that remains finite in the limit Ωn → 0),
and the dynamic part ∝ Ωn . The static part of the compressibility Π00 ′ ;Ωn =0 = 2νδ(x − x′ ) is not affected by
x,x
the barrier at all (this in fact is true for any value of
f ), as the density of states in the leads is independent of
the barrier. The dynamic part is more interesting. For
¯
¯
f = 0, it is given by Πij ′ ;Ωn + β j Πij ′ ;Ωn . This exx,−x
x,x
pression can be understood by looking at the dynamic
part of the current-current correlator, corresponding to
i = j = 1. After the analytic continuation it behaves
∝ ωeiωx/v sin(ωx′ /v). Therefore, the conductivity (2.18)

4.

Critical conductance with leads at the Luther-Emery
point KL = 1
2

From the evaluated polarization matrix (3.17) we can
derive the conductance in the presence of leads that have
a Luttinger parameter of the Luther-Emery value KL =
1
2 . The current autocorrelator (2.20) is then given by
LE
Kx,x′ ;Ωn = 2e2 v 2 Π11 ′ ;Ωn .
CI
x,x

(3.19)

The conductance (2.16) follows from the Kubo formula
(2.18),
GLE =

1
G0 f (r/T ),
2

(3.20)

with the conductance quantum G0 = 2e2 /h. We thus
reproduce the critical conductance of the homogeneous
system (2.21). Below, we show how the critical properties of the conductance are modified in first order in the
deviation KL − 1 .
2
B.

Critical conductance: Perturbation theory
around the Luther-Emery point

Here we evaluate the correction to the conductance
arising from the deviation of the Luttinger parameter in
the leads from its Luther-Emery value KL = 1 . The
2

8
1.
(a)

(b)

(c)

(d)

(e)

(f)

FIG. 2: First order corrections to the current autocorrelator.
The straight line represent fermionic Ψ propagators, the wiggled line is the interaction and the dot is the current vertex
√
2eCI vσ 3 .

deviation from KL = 1 implies according to Eq. (2.9) a
2
finite g2 interaction between the scattering Ψ fermions.
In the following we calculate the conductance up to the
first order in the interactions g2 and g4 . As in the first
order the g4 interaction only modifies the velocity, it is
expected that it does not enter the conductance correction. The verification of this expectation serves as a check
of our calculation and is actually the only reason to introduce at all the redundant g4 interaction in the mapping
(2.8) and (2.9).
There are various contributions to conductance arising
from finite g2 and g4 . First of all, these interactions modify the expression of the charge current velocity vJ , see
Eq. (2.12). As a consequence, the conductance obtains
the modification,
δGA /GLE = 2(g4 ν − g2 ν).

(3.21)

In addition, there are in total six first order diagrams
that contribute to the current autocorrelator, see Fig. 2.
Diagrams (a) and (b) cancel each other exactly due to the
Fermi statistics. Diagram (c) does not modify the conductance because its contribution to σ(x, x′ ; ω) vanishes
in the low frequency limit. In the absence of the barrier
this is self-evident, as the small frequencies correspond to
small incoming momenta that cannot change an R-mover
into a L-mover. In the presence of the barrier this argument essentially goes through, since only the propagating parts (3.9a) of the Green’s functions contribute; the
latter have the same diagonal form as in the translationinvariant problem. Diagram (d) corresponds to the conductance correction coming from fermion scattering off
a Friedel oscillation;26,27,28 we will see that in our case
this contribution is absent. The only modifications to the
conductance arise from the self-energy diagram (e) and
from the random-phase approximation (RPA) diagram
(f). They are discussed in detail below.

Scattering off Friedel oscillations: Diagram (d)

The self-energy correction in diagram (d) embodies the
scattering off a Friedel oscillation. In the general case of
a semitransparent barrier, this process leads to a logarithmic renormalization of the scattering coefficients and
of the conductance.26,27,28 However, the S-matrix in our
case has a very specific energy dependence (3.5) as the
transmission at a given energy is either zero or unity. In
such a situation the renormalization e.g. of the transmission coefficient vanishes as it is proportional to the
product of transmission and reflection coefficients at a
given energy. As a consequence, diagram (d) does not
contribute to the conductance here.
The fact that there is no contribution from the Friedel
oscillation in the first order in g2 can be undestood already from the following simple argument. Consider the
propagating state, for which the transmission is unity in
the absence of interactions between fermions. Due to the
interaction g2 with the counterpropagating states, the reflection amplitude will appear, being proportional to g2 ,
2
thus the reflection probability is O(g2 ). This means that
the transmission amplitude (and the probability) is of
2
the form 1 − O(g2 ), i.e. the correction to the transmission (and hence, to the conductance) vanishes to the first
order in g2 .
2.

Density of states renormalization: Diagram (e)

The other fermionic self-energy that appears in diagram (e) in Fig. 2 arises from the g4 interaction. The left
and right moving fermions interact only among themselves via g4 which ensures that the resulting self-energy
is translationally invariant,
Σ(e) (x, x′ ) = −

g4 (x, x′ ) 3
σ ,
2πi(x − x′ )

(3.22)

as g4 (x, x′ ) = g4 (x − x′ ). This self-energy causes a perturbative correction to the velocity of the scattering particles. This is seen from the analysis of the Schr¨dinger
o
equation for the scattering states,
−ivσ 3 ∂x ψǫ (x) +

dx′ Σ(e) (x, x′ )ψǫ (x′ ) = ǫψǫ (x).
(3.23)

The Schr¨dinger equation can be solved perturbatively
o
with the help of the Wentzel-Kramers-Brillouin ansatz
ψǫ (x) ∝ exp iσ 3

x

dy ϕ(y)

(3.24)

where in zeroth order ϕ(0) = ǫ/v. The first order correction to the eikonal phase is determined by
ϕ(1) (x) =

dx′

′
g4 (x, x′ )
eiǫ(x −x)/v .
2πvi(x − x′ )

(3.25)

9
In particular, the energy derivative of the eikonal phase
up to first order is given by
∂ϕ
1 − νg4
=
∂ǫ
v
′

(3.27)

attributed to diagram (e).

3.

Random-phase approximation contribution: Diagram (f )

The remaining RPA diagram (f) in Fig. 2 can be reduced to the spatial convolutions of the two components
of the polarization matrix (3.10). The resulting correction to the current autocorrelator (2.20) reads
δK(f ) (x, x′ ; Ωn ) =
− g2 e2 v 2
CI

(3.28)

dx1 Π10 1 ;Ωn Π01 ,x′ ;Ωn − Π11 1 ;Ωn Π11 ,x′ ;Ωn
x1
x,x
x1
x,x

As we are only interested in the behavior at small frequency Ωn , we can use the simplified expressions (3.17)
for the polarization operators. The integral over x1 can
then be performed with the help of the convolution (no
summation over index i implied)
¯
¯
dx1 Π1i 1 ;Ωn Πi1 ,x′ ;Ωn
x1
x,x
¯
= ν Π11 ′ ;Ωn
x,x

βi −

|Ωn |
|x − x′ |
v

(3.29)

¯
where βi was defined in Eq. (3.11) and Πij in Eq. (3.14).
After the evaluation of the x1 integral, we obtain for the
RPA contribution
δK

(f )

′

(x, x ; Ωn )

= −2g2 νe2 v 2 [−2νδ(x − x′ )
(3.30)
CI
¯ 0,x
¯ 0,0;Ω T Π11 ′ ;Ω
¯ x,0;Ω T Π11
¯ 11 ′ ;Ω + Π11
−Πx,x n
n
n
n

The presence of the product of bosonic T-matrices, T,
see Eq. (3.18), leads to the correction to conductance
that contains an additional Fermi function,
δG(f ) /GLE = νg2 [2 − f (r/T )] .

KLG0

(3.26)

where g4 =
dx′ g4 (x − x′ )eiǫ(x −x)/v is the Fourier
transform of the g4 interaction at small wavevector ǫ/v.
This is equivalent to the renormalization of the velocity
v → v + δv, where δv/v = g4 ν.
The velocity renormalization enters the normalization
of the scattering states (3.1a) and (3.1b) and thus contributes to the conductivity by changing the density of
states, ν → ν(1−νg4 ). The density of states on the other
hand enters the polarization matrix (3.17) which finally
leads to the change in the conductance
δG(e) /GLE = −2g4 ν

G

(3.31)

r<0
r=0

Gcr

r>0
0

∼ GCI e−r/T
0

T

FIG. 3: Temperature dependence of two-terminal conductance close to the Mott transition for different values of the
tuning parameter r. On the metallic side, r < 0, at T = 0
the conductance is determined by the Luttinger parameter of
the leads, KL ; G0 = 2e2 /h is the conductance quantum. In
the insulating phase, r > 0, at lowest temperature the conductance is thermally activated with a universal exponential
√
prefactor involving the fractional charge, eCI = e/ 2, of the
critical degrees of freedom, GCI = 2e2 /h. In the quantum
CI
critical regime, |r| < T , of the phase diagram, see Fig. 1,
the conductance is determined by its critical value, Gcr , see
Eq. (1.5), that depends both on the Luttinger parameter, KL ,
and on the fractional charge, eCI .

4.

Full perturbative correction

Collecting all the perturbative contributions to conductance, (3.21), (3.27) and (3.31), we obtain the follow1
ing for the first order correction in the parameter KL − 2 ,
δG/G0 = −

νg2 2
f (r/T ) = KL −
2

1
2

f 2 (r/T ),

(3.32)

where we used the identification νg2 = 1 − 2KL valid
to the first order in g2 , cf. Eq. (2.9). As expected, the
first order correction to conductance is independent of
g4 . Note that the correction starts with the square of
the Fermi function whose origin can be traced back to
the RPA diagram (f) in Fig. 2. Adding to this the zeroth order contribution (3.20) we finally obtain the result
announced in Eq. (1.2).
IV.

SUMMARY AND DISCUSSION

In the present work we considered the finitetemperature dc conductance close to the commensurateincommensurate Mott transition in a one-dimensional
wire as a function of the tuning parameter, r, of the
transition. For a homogeneous system in the absence of
leads the result is universal and is given by Eq. (2.21).
It is determined by the critical degrees of freedom of the
transition — the fermionic√
charge carriers that have a
fractional charge eCI = e/ 2, — and the temperature
dependence is simply a Fermi function.

10
Furthermore, we addressed the question on how the
conductance is modified when the leads are attached to
the Mott wire. The result is summarized in Fig. 3. In
the incommensurate phase, r < 0, when the system is a
metal, the dc conductance at zero temperature does not
contain any information about the universal signatures
of the Mott transition because it is solely determined by
the compressibility of the electron liquid in the leads,
i.e., by the Luttinger parameter KL in agreement with
Refs. 6 and 7. However, in the insulating phase, r > 0,
we found evidence that the critical degress of freedom
of the quantum phase transition become manifest in the
dc transport. In particular, in the lowest order in the
fugacity e−r/T , the conductance is independent of the
properties of the leads, KL , while the exponential prefactor is determined by the universal characteristic of the
Mott transition as it is given by an effective conductance
quantum of the fractional charge carriers, GCI = 2e2 /h.
CI
The conductance in this regime originates from thermal
activation of solitonic excitations that travel ballistically
across the Mott-gapped region of the wire. Finally, in the
quantum critical regime, |r| < T , of the phase diagram
in Fig. 1 the conductance is dominated by its critical
value (1.5) that depends on the Luttinger parameter of
the leads, KL , as well as on the fractional charge eCI .
Our main finding is that the dc transport shows signatures of charge fractionalization (1.4) in the Mott wire.
The underlying reason is that the electrons, irrespective
of their interactions in the leads, must “transform” into
Mott solitons with charge (1.4) in order to overcome the
Mott barrier ∆. The fractional charge can in principle
be detected with the help of the finite temperature behavior of the critical two-terminal conductance of a Mott
wire. We contrast this fractionalization with that in a
Luttinger-liquid wire, where only the forward scattering
is included;1 there, the phenomenon of fractionalization
is not manifest in the dc conductance.3,4,5
The regime discussed above is restricted to the immediate vicinity of the quantum critical point of Fig. 1, i.e.
to the temperatures and to the tuning parameter much
smaller than the Mott gap: |r|, T ≪ ∆. In particular, we
neglected corrections arising from the thermal activation
of particles in the other Mott band which are exponentially small in e−(2∆−r)/T . Furthermore, we disregarded
effects of tunneling of solitons through the Mott barrier,
which gives rise to a finite conductance in the insulating regime of the form δG ∼ e−S , where S ∝ rL/v with
the length of the barrier L and the velocity of solitons v.
The approximation of neglecting this tunneling correction breaks down at sufficiently low temperatures when
the length L becomes comparable to the thermal wavelength ξT = v/T .
We arrived at our conclusions by performing a controlled perturbative expansion around the special value
KL = 1/2 of the Luttinger parameter in the leads. At
this specific Luther-Emery value, the solitonic excitations
are free, and the problem of the Mott wire with leads reduces to a 1D scattering problem as descibed in Section

III A. For KL different from the Luther-Emery value,
the interaction between the scattering states in the leads
yields the result (1.2).
Our conclusions concerning the temperature dependence of the critical conductance differ from the results
of the work of Mori et al..7 There, a similar setup was
considered, and the current autocorrelation function was
also calculated in terms of the critical degrees of freedom
of the Mott transition. However, the current correlator
was only considered for a homogeneous system when the
Mott gap extends across the full wire length resulting
in a translationally invariant expression. The result was
then used to match the current autocorrelator within the
Mott wire with the one of the Fermi liquid leads in order
to derive the conductance. We believe, however, that the
use of the current autocorrelator for the homogeneous
Mott wire is inconsistent with the matching conditions
applied in Ref. 7, which explicitly break translationary
invariance. Instead of using the homogeneous current autocorrelation function one may consider the result (3.17)
for the case of a gapped region inside the wire as a starting point for a matching procedure. Preliminary considerations along these lines have led us to the conjecture
described below.
Let us speculate about the functional form of dc conductance beyond the leading KL − 1 correction. We con2
jecture that, in the limit |r|, T ≪ ∆ and T ≫ v/L, the
corresponding resistance is a sum of the two contributions
R = RL + RM with
(4.1)
h 1 − f (r/T )
1 h
, and RM = 2
,
RL =
KL 2e2
2eCI f (r/T )
√
where eCI = e/ 2 and f is the Fermi function (1.3). The
resistance RL is attributed to the leads and can be interpreted as a contact resistance. In particular, only RL depends on the Luttinger parameter within the leads, KL .
The contribution RM has the form of a four-terminal
resistance with an effective transmission given by the
Fermi function, f (r/T ). It originates from the critical
Mott part of the wire and involves the charge quantum
of the critical degrees of freedom, eCI . The expression
(4.1) reproduces the known limits. In the incommensurate phase, r < 0, at T = 0 the resistance reduces to
the contact resistance R = RL . In the insulating state,
r > 0, at lowest temperature the resistance is dominated
by the contribution of the Mott region RM and becomes
independent of the lead properties. Expanding (4.1) in
1
KL − 2 one also reproduces our hard-to-guess perturbative result (1.2).
The Mott region in a quantum wire may be realized
by employing an external periodic potential created by
gating,29 or an acoustic field.30
Our approach applies also to the problem of drag between two quantum wires. At equal electron densities,
the inter-wire interaction locks the charges in both wires
such that the drag resistance at T = 0 is infinite in the

11
commensurate state. This system bears resemblance to
the Mott insulator: if opposite biases are applied to the
two wires, there is no current at T = 0.31,32 Density imbalance drives the system into an incommensurate phase,
for which one recovers the ballistic conductance for each
of the wires; drag is absent at T = 0. The problem of a
finite-T drag resistance may be mapped onto that of the
critical conductance in a Mott wire. Technically, such a
mapping is established by introducing the even and odd
√
modes, ρ± = (ρ1 ± ρ2 )/ 2, where ρ1,2 (x) are the electron densities in the two wires. The dynamics of the odd
mode is described by the sine-Gordon model (2.1), while
the even mode is a conventional Luttinger liquid. The
drag resistance is determined by the conductance in the
odd sector. The present analysis of the critical behavior
of the transport at the commensurate-incommensurate
transition is fully applicable for the evaluation of the drag
resistance.

Acknowledgments

M.G. was supported by SFB 608 of the DFG. D.N. and
L.G. were supported by NSF grants DMR 02-37296 and
DMR 04-39026. A.S. acknowledges financial support by
the Israel Science Foundation, administered by the Israeli
Academy of Sciences.

APPENDIX A: POLARIZATION OPERATOR IN
THE COORDINATE-FREQUENCY
REPRESENTATION

We present a detailed calculation of the polarization operators (3.10). After performing the sum over
fermionic Matsubara frequencies we obtain
f (ǫ) − f (ǫ′ )
tr τ i Ax,x′ ;ǫ τ j Ax′ ,x;ǫ′
iΩn − (ǫ − ǫ′ )
(A1)
with the spectral function A given in (3.9). In this section, we use a temperature dependent definition of the
Fermi function, f (ω) = 1/(eω/T + 1). As a first step we
evaluate the trace. We will use the abbreviations (3.8).
Consider first a situation with s = s′ , i.e. with x and x′
on opposite sides of the barrier.
Πij ′ ;Ωn =
x,x

dǫdǫ′

s = s′ : tr τ i Ax,x′ ;ǫ τ j Ax′ ,x;ǫ′
=ν

2

Θǫ−r Θǫ′ −r Aij ′ (x
ǫ,ǫ

(A2)
′

− x + sℓt ) .

Here
Aij ′ (x) = ei(ǫ−ǫ )x/v + β i β j e−i(ǫ−ǫ )x/v ,
ǫ,ǫ
′

′

(A3)

where the β i were defined in (3.11); and we made use
of the approximations (3.6) for the phase shifts with
ℓt = (ℓ+ + ℓ− )/2. The case s = s′ is done explicitly

[in Eq. (A4) set v = 1 for brevity]:
s = s′ :

tr τ i Ax,x′ ;ǫ τ j Ax′ ,x;ǫ′ =

iǫ(x+x′ )+isφs (ǫ)

e
sΘr−ǫ e
′
′
sβ i Θr−ǫ e−iǫ(x+x )−isφs (ǫ)
β i e−iǫ(x−x )

= ν 2 tr

′

×

(A4)

iǫ(x−x′ )

′

′

′

′

e−iǫ (x−x )
sΘr−ǫ′ eiǫ (x+x )+isφs (ǫ )
′
′
′
′
′
β j eiǫ (x−x )
sβ j Θr−ǫ′ e−iǫ (x+x )−isφs (ǫ )

= ν 2 Aij ′ (x − x′ ) + β j Θr−ǫ Θr−ǫ′ Aij ′ (x + x′ + sℓs )
ǫ,ǫ
ǫ,ǫ
where we used the approximation (3.6) for the reflection
phase shifts.
When substituting the above expressions for the trace
into Eq. (A1), it is convenient to define
∞

Fr (x) = ν 2

dǫdǫ′

r

f (ǫ) − f (ǫ′ ) i(ǫ−ǫ′ )x/v
e
.
iΩn − (ǫ − ǫ′ )

(A5)

Note that in both cases, after evaluating the trace, the
conditions ǫ > r and ǫ′ > r, or ǫ < r and ǫ′ < r, enter simultaneously. Hence the polarization operator (A1) can
be expressed via (A5) as
Πij ′ ;Ωn =
x,x

(A6)

′

=

1 − ss
Fr (x − x′ + sℓt ) + β i β j Fr (x′ − x − sℓt )
2
1 + ss′
(0)ij
Πx,x′ ;Ωn
+
2
+ β j F−r (x + x′ + sℓs ) + β i β j F−r (−x′ − x − sℓs )

.

The last term is obtained from Eq. (A5) via the interchange of the dummy variables ǫ → −ǫ′ and ǫ′ → −ǫ,
and using f (−ǫ) = 1 − f (ǫ). We also define the polarization operator in the translation-invariant case
(0)ij

Πx,x′ ;Ωn = Fr (x − x′ ) + β i β j Fr (x′ − x)

r→−∞

. (A7)

We now proceed to evaluating (A5). Introducing the
variable ε = ǫ − ǫ′ , write
∞

Fr (x) = ν 2

ǫ−r

dǫ
r
0

= ν2

dε
−∞
iεx/v

e
dε
iΩn − ε

f (ǫ) − f (ǫ − ε) iεx/v
e
(A8)
iΩn − ε
∞

r
∞

0

+ ν2

−∞
∞ iεx/v

r+ε

e
dε
iΩn − ε

dǫ [f (ǫ) − f (ǫ − ε)]
dǫ [f (ǫ) − f (ǫ − ε)] .

The integration with respect to ǫ and ε is over a domain
that can be split into an infinite rectangle, ǫ > r and
ε < 0, and an infinite triangle, ǫ > r and 0 < ε <
ǫ − r. After changing the order of integration in each of
the two sub-domains separately, such that the integral
over ǫ is performed first, one obtains the last equation in
(A8). Shifting variables in the energy integrations and
introducing the function
r+ε

ϕ(ε) =

dǫ f (ǫ) = ε + T ln
r

f (r + ε)
,
f (r)

(A9)

12
we get
∞

Fr (x) = ν 2

dε ϕ(ε)
0

e−iεx/v
eiεx/v
.
+
ε − iΩn
ε + iΩn

(A10)

stat
Consider first the static part, Fr (x) ≡ Fr (x)|iΩn =0 ; it
can be represented as
stat
Fr (x) = νf (r)δB (x)

(A11)

where δB (x) is a function with a peak at x = 0 that
carries unit weight, dx δB (x) ≡ 1. In the limit r →
−∞, the function δB reduces to a delta function; in the
classical limit, 0 < r/T ≪ 1, the peak has a width of
order ξT = v/T .
dyn
The dynamic part of the Fr function, Fr
≡ Fr −
stat
Fr ,
dyn
Fr (x) = ν 2 iΩn

∞

dε
0

e−iεx/v
ϕ(ε) eiεx/v
−
ε
ε − iΩn
ε + iΩn
(A12)

is dominated by the poles, ε = ±iΩn . The leading behavior
dyn
Fr (x) ≃ −πν 2 Ωn f (r) [ sign x + sign Ωn ] e−|Ωn x|/v
(A13)

1

2

3

4
5
6

7

8

9

10
11

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13
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1
originates from the odd part ϕ(ε) → ϕ− = 2 [ϕ(ε) −
3
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to the real axis and evaluating the Taylor-expanded
1
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n
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the analytic continuation.

We now have all the ingredients to construct the polarization operator (A6). First, the absolute values of the
arguments of the Fr functions in (A6) are all larger than
the scattering lengths ℓ± that characterize the length of
the Mott region of the wire. In the limit of a long Mott
wire, ℓ± T /v ≫ 1 we can neglect the tails of the static part
(A11) of the Fr functions in (A6). The static part then
only enters the polarization via the contribution Π(0)ij
which was defined in (A7). The Fr functions in (A6) thus
only contribute with their dynamic part (A13). Moreover, for the dynamic part the scattering lengths ℓ± in
the arguments of the Fr functions in (A6) can be effectively set to zero as they give only subleading corrections
in the dc limit ωℓ± /v ≪ 1. After these simplifications we
finally obtain the expression (3.12) for the polarization
operator.

15

16

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23
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13
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32

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