One-dimensional conductance through an arbitrary potential
Tobias Stauber

arXiv:cond-mat/0301586v3 [cond-mat.str-el] 5 Apr 2004

Institut f¨r Theoretische Physik, Ruprecht-Karls-Universit¨t Heidelberg,
u
a
Philosophenweg 19, D-69120 Heidelberg, Germany
The finite-size Tomonaga-Luttinger Hamiltonian with an arbitrary potential is mapped onto a
non-interacting Fermi gas with renormalized potential. This is done by means of flow equations
for Hamiltonians and is valid for small electron-electron interaction. This method also yields an
alternative bosonization formula for the transformed field operator which makes no use of Klein
factors. The two-terminal conductance can then be evaluated using the Landauer formula. We
obtain similar results for infinite systems at finite temperature by identifying the flow parameter
with the inverse squared temperature in the asymptotic regime.
PACS numbers: 71.10.Pm, 73.22.-f, 73.63.-b, 05.10.Cc

The conductance of one-dimensional (1D) conductors at low temperatures is determined by a sophisticated interplay between impurity scattering and electronelectron interaction. The question was first raised about
thirty years ago by Luther and Peschel1 and Mattis2
in the context of enhanced conductivity, present in 1D
structures at low temperatures. Their results were later
confirmed and extended using renormalization group
techniques.3,4 Recently, evidence was found that the
conductance in single-walled carbon nanotubes5 and
GaAs/AlGaAs quantum wires6 behaves according to the
so-called Luttinger liquid theory.7
The above theoretical descriptions were only valid for
large systems at low temperatures due to the perturbative treatment in the impurity strength. This shortcoming was overcome by Monte-Carlo calculations8 and applying the thermodynamic Bethe ansatz.9 Nevertheless,
the latter method already fails for a double impurity.
To describe general external potentials, it thus seems
favorable to tackle the problem from the other end, i.e.,
to treat the impurity exactly but include the electronelectron interaction perturbatively. Since the interaction
of 1D systems is marginal in the renormalization group
sense,10 this has to be done with care. One approach is
to use scattering theory within the Born approximation
and renormalization group techniques.11,12 The interaction can also be included via exact renormalization group
equations based on the one-particle irreducible effective
action.13
Recently, we have employed a flow equation scheme
for Hamiltonians14 which is applicable for small electronelectron interaction but arbitrary impurity strength.15
For infinite systems at zero temperature, the approach
yielded a logarithmic expansion for the spectral weight
at the impurity site involving the exact boundary exponent. In this Brief Report, we will extend the analysis
and recover the exact power-law behavior. For this, we
will derive the exact Green’s function of the TomonagaLuttinger model without making use of the bosonization
formula. We will also demonstrate how the conductance
can be obtained as a function of the system size and the
temperature, respectively, using the Landauer formula.16
We start our discussion from the spinless Tomonaga-

Luttinger Hamiltonian17 of finite length L
HT L ≡ H0 + Hee

(1)

with
0
ωq b† bq , Hee =
q

H0 =
q

1
2

q

u0 (bq b−q + b† b† ) .
q
−q q

0
The initial conditions are given by ωq = vF |q|(1 +
vq /2πvF ) and u0 = |q|vq /2π with q = ±2πnq /L, nq ∈ N
q
where vF denotes the Fermi velocity and vq = v−q is
the Fourier transform of the electron-electron interaction.
The sum is over all non-zero wave numbers q = 0 and the
normalized operators of the electron density fluctuations
(†)
bq obey canonical commutation relations. They are
(†)
given by b±|q| = √1 q k c† ck±|q|,± where ck,± are the
k,±
n
fermionic annihilation (creation) operators of the plane
wave with wave number k of the right- and left-movers
and k = 2πn/L is unbounded (n ∈ Z).
We now perform a continuous unitary transformation,
U (ℓ), such that the interaction Hamiltonian Hee tends
to zero for ℓ → ∞. In differential form, this transformation can be characterized by the so-called flow equations
∂ℓ H(ℓ) = [η(ℓ), H(ℓ)] with the anti-Hermitian generator
η(ℓ) = −η † (ℓ).14 The parameters of the Hamiltonian (1)
thus become functions of the flow parameter ℓ. This will
be indicated by dropping the upper zero which was introduced in order to specify the initial conditions.
In this work, we choose the generator as η = [H0 , Hee ]
even in the presence of an external potential.18 This
yields

η(ℓ) =

1
2

q

ηq (ℓ)(bq b−q − b† b† )
−q q

(2)

with ηq (ℓ) = −2ωq (ℓ)uq (ℓ) = −sgn(vq )˜ q [sinh(4˜ q ℓ +
ω2
ω2
0 0
−1
2
˜
Cq )] where sinh(Cq ) = ωq /2ωq |uq | and ωq ≡ ωq (ℓ =
˜
∞). The fixed point dispersion ωq follows from the flow
˜
2
2
invariant ωq (ℓ) − u2 (ℓ) = vF q 2 (1 + vq /πvF ) as uq (ℓ) → 0
q
for ℓ → ∞. For attractive electron-electron interaction
we impose the lower bound vq > −πvF in order to avoid
phase separation.19

2
The ℓ-dependent parameters then read
0
0
ωq (ℓ) = cq (ℓ)ωq + sq (ℓ)u0 , uq (ℓ) = cq (ℓ)u0 + sq (ℓ)ωq
q
q
(3)

where cq (ℓ) ≡ cosh(Eq (ℓ)) and sq (ℓ) ≡ sinh(Eq (ℓ)) with
ℓ

Eq (ℓ) =

dℓ′ ηq (ℓ′ ) = ln

0

2
tanh(2˜ q ℓ)Kq + 1
ω2
−2
tanh(2˜ q ℓ)Kq + 1
ω2

1
4

(4)

2
0
0
and the Luttinger liquid parameter Kq = (ωq −u0 )/(ωq +
q
0
uq ).
In order to evaluate correlation functions, the observable has to be transformed by the same continuous unitary transformation, i.e., ∂ℓ ψ± (x, ℓ) = [η(ℓ), ψ± (x, ℓ)].
Starting from the fermionic representation of the field
operator, i.e., ψ± (x) = L−1/2 k eikx ck,± , the flow equations generate an infinite hierarchy of operators. Nevertheless, this expansion factorizes and only involves one
q-dependent function of ℓ, ϕ± (ℓ). The expansion can be
q
summed up to an exponential and is given by

ψ± (x, ℓ) = ψ± (x) exp

ϕ± (ℓ)
q
q

√

nq

φq (x)

,

(5)

where we defined φq (x) ≡ eiqx bq − e−iqx b† . Notice that
q
[ψ± (x), φq (x)] = 0. The flow equation for ϕ± (ℓ) reads
q
∂ℓ ϕ± (ℓ) = −ηq (ℓ)(Θ(∓q) + ϕ± (ℓ))
q
−q

(6)

where Θ(x) denotes the Heaviside step function. One
obtains the following solution
ϕ± (ℓ) = Θ(±q)(cq (ℓ) − 1) − Θ(∓q)sq (ℓ) ,
q

(7)

with cq (ℓ) and sq (ℓ) defined below Eq. (3). Notice that
no Klein factor is needed in the above representation, Eq.
(5).
We now define normal ordering : ... : such that
the bosonic operators b† (bq ) stand left (right) from the
q
(†)

fermionic operators ψ± (x). With the Baker-Hausdorff
formula eA+B = eA eB e−[A,B]/2 and the relation
−1/2 iqx
e )ψ± (x)
ψ± (x)f (b† ) = f (b† + Θ(±q)nq
q
q

,

(8)

which holds for an arbitrary function f , we obtain
ψ± (x, ℓ) = : ψ± (x, ℓ) : exp −

q>0

s2 (ℓ)
q
nq

.

(9)

With this normal ordering procedure, we can thus ex2
tract the anomalous scaling factor L−s (ℓ) , which follows
in the limit L → ∞ from a constant potential with finite cutoff qc , i.e., s(ℓ) = sq (ℓ) for q ≤ qc and zero otherwise. From the scaling behavior of the transformed
field ψ± (x, ℓ = ∞) - known from bosonization - we infer
that the normal ordered field operator scales like a free

fermionic field. This will enable us to substitute the normal ordered transformed field operator by the bare field
for weak electron-electron interactions.
To evaluate the Green’s function, we need to normal
order the operator
†
<
G± (x, x′ ; t) ≡ ψ± (x′ , 0, ℓ = ∞)ψ± (x, t, ℓ = ∞),

(10)

where the time dependence is defined in the Heisenberg picture with the fixed point Hamiltonian H ∗ =
˜ †
q ωq bq bq . We obtain
<
<
G± (x, x′ ; t) = : G± (x, x′ ; t) : g(x − x′ , t)

,

(11)

with (sq ≡ sq (ℓ = ∞))
g(x, t) ≡ exp −

q>0

2s2
q
ω
(1 − cos(qx)ei˜ q t ) .
nq

(12)

<
The Green’s function G< (x, x′ , t) ≡ −i G± (x, x′ ; t)
±
thus reads
†
G< (x, x′ , t) = −i ψ± (x′ , 0)ψ± (x, t) g(x − x′ , t)
±

(13)

where ... denotes the ground-state average of H ∗ =
˜ †
q ωq bq bq .
Up to now, we have assumed a finite interaction cutoff qc which yields well-defined expressions. For constant interaction potential vq = vΘ(qc − |q|) and qc →
∞, the remaining expectation value can be evaluated
using the Kronig relation which relates H ∗ to a free
fermionic Hamiltonian with linear dispersion.20 With
ωq = vc |q|, vc denoting the charge velocity, and ck,± (t) =
˜
ck,± e∓ivc (k∓kF )t , we obtain for L → ∞
′

†
ψ± (x′ , 0)ψ± (x, t) =

−i
e±ikF (x−x )
.
2π ±(x − x′ ) − vc t − i0

(14)

This yields the well-known result for the Green’s function of the Tomonaga-Luttinger model.
To complete the discussion on the observable flow,
we also transform the ladder operators ck,± according
to ∂ℓ ck,± (ℓ) = [η(ℓ), ck,± (ℓ)]. Again, an infinite hierarchy of operators is generated which is only characterized by one q-dependent function of ℓ, ϕ± (ℓ). Defining
q
φk′ ,q = bq δk′ ,k−q − b† δk′ ,k+q , the series is given by
q
k
ck,± (ℓ) =

∞

1
n!
n=0

n
i=1 ki ,qi

ϕ± (ℓ) ki−1
qi
ck ,±
φ
√
nqi ki ,qi n

(15)

with k0 = k. The q-dependent function of ℓ is again given
by Eq. (7). Notice that ψ± (ℓ) = L−1/2 k eikx ck,± (ℓ).
For a step function potential vq = vΘ(qc − q) in the limit
L → ∞, we therefore have
2

ck,± (ℓ) = : ck,± (ℓ) : L−s
where s(ℓ) as defined below Eq. (9).

(ℓ)

,

(16)

3
We will now add an arbitrary external potential to the
initial Hamiltonian, Eq. (1). The additional contribution He shall consist of a forward and a backward scattering part, i.e., He = HF + HB . The forward scattering contribution can be expressed by the bosonic density
fluctuations,20
n1/2 L−1
q

HF =

dxU F,0 (x)(eiqx bq + e−iqx b† ). (17)
q

q

UB ∝ L1−K

The backward scattering contribution reads
†
0
dxUB (x)(ψ+ (x)ψ− (x) + h.c.),

HB =

(18)

with the right- and left-moving field, ψ± (x).
The flow equations for HF close and the differential
equations for the forward scattering potential read
F
F
∂ℓ Uq (x, ℓ) = ηq (ℓ)U−q (x, ℓ)
F
Uq (x, ℓ

.

(19)

F,0

= ∞) = Kq U (x) such that
We thus have
the forward scattering amplitude is enhanced for attractive and reduced for repulsive electron-electron interaction.
The flow of the backward scattering operator
†
O±2kF (x, ℓ) ≡ ψ± (x, ℓ)ψ∓ (x, ℓ)

(20)

is determined by the flow equation of the field operator
ψ± (x, ℓ), outlined above. Defining the ℓ-dependent Luttinger liquid parameter Kq (ℓ) ≡ exp(2Eq (ℓ)), we have
O±2kF (x, ℓ) = O±2kF (x)
× exp ±

q

(21)

sgn(q)
( Kq (ℓ) − 1)φq (x)
√
nq

with φq (x) ≡ eiqx bq − e−iqx b† . Normal ordering the
q
above expression such that b† (bq ) stands left (right) from
q
O±2kF (x, ℓ = 0) yields
O±2kF (x, ℓ) = : O±2kF (x, ℓ) :
1 − Kq (ℓ)
.
× exp
nq
q>0

q>0

1 − Kq (ℓ)
nq

.

(24)

Notice that the renormalization of the external potential
depends linearly on the interaction strength v.
For an infinite system in the limit ℓ → ∞, the sum in
the exponent of Eq. (23) turns into an integral accordL
ing to q → 2π dq. For a constant electron-electron
interaction with sharp cutoff qc , we then obtain
UB ∝ ℓ(1−K)/2

.

(25)

Notice that the asymptotic limit ℓ → ∞ and the thermodynamic limit L → ∞ interchange. The commutation
of the two limits does usually not hold even if one considers impurity properties. But in contrary to the treatment
of, e.g., the spin-boson model of Ref. 21, here we do not
neglect finite-size corrections in the flow equations.
For repulsive electron-electron interaction, an initial
delta-impurity strength thus scales to infinity for L → ∞
and T → 0. The ground-state of the fixed point
Hamiltonian is then given by the ground-state of an
electron gas with fixed or open boundary condition.15
Concerning the operator flow, this change in boundary
conditions can be accounted for by the substitution
b−|q| → b|q| in the operator φq (x), defined below Eq.
(5), i.e., there are only right-movers. With the same
normal ordering procedure as outlined for the case of
periodic boundary conditions, this yields the well-known
Green’s function of the Tomonaga-Luttinger model with
open boundaries.22 We thus obtain the exact power law
behavior for the spectral weight at the impurity site.

(22)

Above, it is shown that the normal ordered transformed field operator scales like a free fermion field.
For small electron-electron interaction Kq (ℓ) − 1 → 0,
the normal ordered backscattering operator can thus be
approximated by the bare backscattering operator, i.e.,
: O±2kF (x, ℓ) : ≈ O±2kF (x).
This approximation provides a suitable truncation
scheme for the Hamiltonian flow. We can then define
an effective ℓ-dependent backscattering potential as
UB (x, ℓ) = UB (x) exp

It is interesting to analyze the asymptotic behavior of
UB (x, ℓ) for L → ∞ and ℓ → ∞. For this, we will choose
a step function potential vq = vΘ(qc − q) with finite
interaction cutoff qc and also drop the spatial component
in the following. Considering first the limit L → ∞,
the finite interaction cutoff qc = 2πnc /L shall remain
constant, such that nc is proportional to the system size
L. With nc 1/n → ln nc + C for nc → ∞, where C is
n=1
Euler’s constant, we readily find (K = Kq for q ≤ qc )

.

(23)

So far, we have mapped the TL model with external
potential onto a non-interacting bosonic bath with dispersion relation ωq = vF q 1 + vq /πvF and a renormal˜
ized external potential. The mapping holds in the limit
of weak electron-electron interaction. Linearizing the dispersion relation around q = 0, one can map the noninteracting bosonic bath onto a non-interacting Fermi
gas.20 Notice that, up to terms linear in vq and approximating : ck,± (ℓ) : ≈ ck,± , these are the same fermions we
started with, see Eq. (16).23
The conductance G is now given by the Landauer formula for a one-channel, spinless electron system at T = 0,
2
G = e T , where T denotes the transmission probabilh
ity through the renormalized external potential.16,24 By
this, we assume that the coupling of the interacting system to the non-interacting leads does not contribute to

4
the transmission probability. T is thus entirely determined by the renormalized potential which corresponds
to perfect coupling.25 Nonadiabatical coupling can be discussed by considering different boundary conditions of
the fermionic field operator.
One can also obtain results at finite temperatures if
one identifies the asymptotic flow parameter ℓ with the
inverse temperature squared, i.e.,
√
ℓ ∝ 1/T .
(26)
For this, it is crucial to choose the generator η “canonically”, i.e., based on the commutator of a diagonal
and a non-diagonal part. Then, in the case of a simple model, the flow parameter ℓ is related to the inverse squared energy uncertainty of the basis states of
the fixed point Hamiltonian.14 This interpretation is confirmed in case of the spin-boson model as the effective
tunnel-matrix element ∆∗ is asymptotically reached ac√
cording to ∆(ℓ)−∆∗ ∝ 1/ ℓ.21 Furthermore, it turns out
that spectral functions, initially depending on the energy
ω and on the flow parameter ℓ, can be characterized by
√
functions of the scale-independent variable y = ω ℓ for
ℓ → ∞.26 This also holds for the Tomonaga-Luttinger
model, see Eq. (4).
Physical quantities should not depend on energy shifts
of the spectrum which are smaller than the energy scale
given by the finite temperature. Therefore, the flow equations do not have to be integrated up to ℓ = ∞, but
it suffices to halt the integration at a finite ℓ. Since
the electron-electron interaction is exponentially small
for large ℓ and not marginally relevant at finite temperatures, we can neglect it. This leaves us with a one-particle

1
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3
4

5

6

7
8

9

10
11

12
13

14

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problem with ℓ-dependent parameters. The temperature
is introduced by substituting ℓ by T −2 .
The above procedure yields the same result for the
conductance as obtained by Ref. 11 in case of a delta
impurity, but is also applicable for general potentials.
From Eqs. (24), (25), and (26), we can also infer the
finite-size scaling relation L ∝ 1/T .
Finally, we want to mention that Refs. 11 and 13 obtain an effective non-interacting model with an oscillating
and bounded effective potential. But whereas these approaches depart from a non-interacting system with an
impurity and then include the electron-electron interaction, we start from an interacting system and add the
impurity later, which might explain the different fixedpoint Hamiltonians.27
To conclude, we have used Wegner’s flow equation
method to map the Tomonaga-Luttinger Hamiltonian
with an external potential onto a non-interacting Fermi
gas with an renormalized external potential. The mapping is valid for weak electron-electron interaction and
is based on an alternative bosonization formula for the
transformed field operator derived by the flow equation method. Calculating the transmission amplitude
from the non-interacting representation, the Landauer
formula then provides an analytic expression for the onedimensional two-terminal conductance for finite size systems. Interpreting the flow parameter in the asymptotic
regime as inverse temperature squared yieldes an analytic
expression of the conductance also at finite temperatures.
The formalism applies to arbitrary external potential.
The author wishes to thank V. Meden, A. Mielke, and
F. Wegner for helpful discussions.

15
16

17

18

19
20

21

22

23

24

25

26

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Notice that η does not contain contributions coming from
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We neglect terms involving the q = 0-mode, valid for large
systems.
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(1997).
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This approximation cannot account for bulk Luttinger liquid behavior and it is crucial that the external potential is
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5

27

T. Stauber, Phys. Rev. B 68, 125102 (2003).
The Friedel oscillations show up in correlation functions
after transforming the observables as well, but do not have

to appear in the fixed-point Hamiltonian which can - in
principal - be chosen freely.