Conductance of quantum wires: a numerical study of the effects of an
impurity and interactions
Amit Agarwal and Diptiman Sen

arXiv:cond-mat/0507097v2 [cond-mat.str-el] 2 Mar 2006

Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India
(June 24, 2018)
We use the non-equilibrium Green’s function formalism and a self-consistent Hartree-Fock approximation to numerically study the effects of a single impurity and interactions between the electrons
(with and without spin) on the conductance of a quantum wire. We study how the conductance
varies with the wire length, the temperature, and the strengths of the impurity and interactions.
The numerical results for the dependence of the conductance on the wire length and temperature
are compared with the results obtained from a renormalization group analysis based on the HartreeFock approximation. For the spin-1/2 model with a repulsive on-site interaction or the spinless
model with an attractive nearest neighbor interaction, we find that the conductance increases with
increasing wire length or decreasing temperature. This can be explained using the Born approximation in scattering theory. For a strong impurity, the conductance is significantly different for
a repulsive and an attractive impurity; this is due to the existence of a bound state in the latter
case. In general, the large density deviations close to the impurity have an appreciable effect on the
conductance at short distances which is not captured by the renormalization group equations.

PACS number: 73.23.-b, 73.63.Nm, 71.10.Pm

with length scale is governed by a renormalization group
(RG) equation. (In this paper, we will only consider a
non-magnetic impurity which scatters electrons in a spin
independent way).
There are three length scales which are of interest in
the problem. The smallest of them is λ ≡ π/kF (where
kF is the Fermi wavenumber); this is the wavelength of
the oscillations in the electronic density near impurities
as we will see. The other two length scales are the length
of the wire L, and the thermal coherence length LT which
is equal to ¯ vF /(kB T ) (where vF is the Fermi velocity).
h
LT gives an idea of the distance beyond which an electron
wave function loses its phase coherence. At a temperature T , the conductance typically receives contributions
from a number of states near the Fermi energy whose energies have a spread of the order of ∆E = kB T . Hence
the spread in momentum ∆p = ∆E/vF is of the order
of kB T /vF . A superposition of waves with such a spread
of momenta loses phase coherence in a distance of the
order of LT = hvF /(kB T ). Electronic transport is there¯
fore thermally incoherent if the wire length L >> LT ,
coherent if L << LT , and partially coherent in the intermediate range. This will become clearer when we discuss
our numerical results.
The RG equations for the transmission coefficient |t|2
has been derived from continuum theories in several ways
[7,8,13–15]. We will discuss these equations for two models in Sec. II. (The conductance is related to the transmission as G = (2e2 /h)|t|2 , where the factor of 2 is due
to the electron spin). The RG equation is used to analytically follow the evolution of |t|2 starting from a short
distance scale and going up to a length scale which is the
smaller of the two quantities L and LT . If L >> LT ,

I. INTRODUCTION

The conductance of electrons in a quantum wire has
been the subject of intensive study in recent years, both
experimentally [1–6] and theoretically [7–9]. For a wire in
which only one channel is available to the electrons and
the transport is ballistic (i.e., there are no impurities inside the wire, and there is no scattering from phonons or
from the contacts between the wire and its two leads),
the conductance is given by G = 2e2 /h for infinitesimal bias [10,11]. This result is expected to hold even
if one takes into account the interactions between the
electrons since such two-body scatterings conserve the
momentum. However, if there is an impurity inside the
wire which scatters the electrons, then the conductance
is reduced because such a scattering does not conserve
the momentum of the electron. For a one-dimensional
system containing a δ-function impurity with strength
V , we obtain
G =

2e2
(1 − c V 2 ) ,
h

(1)

to lowest order in V , where c is a constant related to the
Fermi velocity of the electrons (see the discussion below
Eq. (3)). (The situation is different if the wire is only
quasi-one-dimensional, and the impurity potential has a
finite range [12]). In the absence of interactions, G does
not depend on the wire length L or the temperature T
(as long as kB T is much less than the Fermi energy).
But in the presence of interactions, it turns out that V
effectively becomes a function of the length scale (which
is related to either L or T as will be explained below),
and G therefore varies with L and T . The variation of V
1

the conductance is governed by LT and not L, and vice
versa if LT >> L. The RG equation has two fixed points
which lie at |t|2 = 0 and 1; the system approaches one of
these fixed points if L and LT are both very large.
The above statements are only valid for length scales
much longer than λ because only then can one use a
continuum description from which the RG equation is derived. It would therefore be useful to consider an alternative method for computing the conductance which works
even at short length scales where the continuum description is not valid. An interesting thing which occurs at
short distances is that if the impurity provides an attractive potential to the electrons, there is a bound state
whose wave function decays exponentially away from the
impurity. We may wonder what effect a bound state has
on the conductance; the RG equations mentioned above
do not take this state into account. One may think that
a bound state (whose energy lies outside the bandwidth
of the leads) cannot directly affect the conductance, because electrons coming in from or going out to the leads
cannot enter or leave such states in the absence of any
inelastic scattering. However, such a state contributes
to the electronic density near the impurity, and that can
affect the transport in the presence of interactions.
In a series of papers, the technique of functional RG
has been used to numerically study the conductance and
other properties of interacting electron systems in one dimension [16–19]. The authors of those papers have gone
up to very large system sizes and have found excellent
agreement at those length scales between their numerical results and the asymptotic scaling forms given by the
RG equations.
In this paper, we will use the non-equilibrium Green’s
function (NEGF) formalism to numerically study the
conductance of a quantum wire [10,20–23]. Since we will
use lattice models and the Hartree-Fock (HF) approximation for dealing with interactions in our numerical studies, we will first discuss those topics in Sec. III. In that
section, we will also show how the Born approximation
for scattering can qualitatively explain the dependence
of the conductance on the wire length which we obtain
from the RG equations in Sec. II.
The NEGF formalism will be briefly described in Sec.
IV. The advantage of this method is that it treats the infinitely extended leads (reservoirs) in an exact way, and
it can be used for all values of the wire length and the
impurity strength. However, it is accurate only for weak
interactions between the electrons because we are forced
to use a HF approximation for dealing with the interactions (for reasons which will be explained below). We will
use a lattice model for both the wire and its leads. The
Hamiltonian will have hopping terms in the wire and in
the leads, and a density-density interaction (on-site for
spin-1/2 electrons, and between nearest neighbor sites for
spinless electrons) only in the wire.
In Sec. V, we will describe our results for spin-1/2 and
spinless electrons for different values of the impurity po-

tential and the interaction strength, and we will compare
our results with those obtained by the RG analysis. For
the case of spinless electrons, we find that the agreement
between the RG and numerical results is excellent if we
follow a certain procedure. In Sec. VI, we will make
some concluding remarks.
II. RENORMALIZATION GROUP EQUATION
FOR SCATTERING FROM A POINT

There are several ways of studying the renormalization
group (RG) evolution of the scattering from one or more
points in a one-dimensional system of interacting electrons. One can use the technique of bosonization [7,8,24],
a fermionic RG method [13–15], and the functional RG
method [16–19]. Since our numerical calculations use a
HF approximation, the method of Refs. [13,14] will be
the most useful for us. Before considering the HF approach, however, we will briefly discuss the RG equations
obtained by bosonization for spinless electrons.
A. Spinless electrons

Let us first consider the Hamiltonian for noninteracting electrons in the presence of a δ-function impurity placed at the origin,
H =

p2
+ V δ(x) .
2m

(2)

It is easy to check that for plane waves incident either
from the left or from the right with wavenumber k, the
reflection and transmission amplitudes are given by [25]
imV
,
h
¯ 2 k + imV
h
¯ 2k
.
and t(k) = 2
h
¯ k + imV
r(k) = −

(3)

If the Fermi energy of the electrons is given by EF =
h 2
¯ 2 kF /(2m) (where kF is the Fermi wavenumber, and
vF = ¯ kF /m is the Fermi velocity), then the conduch
tance G for spinless electrons at zero temperature is given
by e2 /h times |t(kF )|2 . For |V | << ¯ vF , we see that
h
|t(kF )|2 = 1 − (V /¯ vF )2 up to order V 2 . (We will usuh
ally set Planck’s constant ¯ = 1). Eq. (2) has a bound
h
state if V < 0, but this does not play any role in the RG
analysis described below.
Now let us introduce interactions between the electrons. We assume a density-density interaction between
spinless electrons of the form
Hint =

1
2

dxdy ρ(x) U (x − y) ρ(y) ,

(4)

where the density ρ is given in terms of the secondquantized electron field Ψ(x) as ρ = Ψ† Ψ. The electron
2

field can be written in terms of the right and left moving
fields ΨR and ΨL (whose variations in space are governed
by wavenumbers much smaller than kF ) as
Ψ(x) = ΨR eikF x + ΨL e−ikF x .

For a given value of the wire length L and temperature
T , the RG equations can be used to compute the conductance analytically as follows. We begin at a short length
scale (of order λ) with an initial value of V ; we then
use Eq. (9) or (10), depending on whether V is small or
large, to follow the evolution of V with the length scale
l. The RG flow stops when l reaches a distance of the order of L or LT , whichever is smaller. When the RG flow
stops, we take the value of V obtained at that point, and
compute the conductance G in terms of the transmission
amplitude given in Eq. (3). [We will see later that the
RG flow of the conductance does not stop abruptly at one
particular length scale. There is an intermediate range of
length scales where the conductance continues to evolve
slowly.]
Let us now discuss the second method for obtaining the
RG equations, namely, using a HF approximation for the
case of weak interactions [13–15]. This method directly
gives an RG equation for the scattering matrix which
is produced by the impurity. The idea is that reflection from the impurity leads to an interference between
the incoming and outgoing electron waves; this leads to
Friedel oscillations in the density with an amplitude proportional to r and a wavelength given by λ = π/kF .
In the presence of interactions, these density oscillations
cause the electrons to scatter; these scatterings therefore
renormalize the scattering caused by the impurity. The
RG equations obtained by this method are given by

(5)

If the range of the interaction U (x) is short (of the order
of λ), such as that of a screened Coulomb repulsion, the
Hamiltonian in (4) can be written in the form
Hint = g2

dx Ψ† ΨR Ψ† ΨL ,
R
L

(6)

where g2 is related to the Fourier transform of U (x) as
˜
˜
g2 = U (0) − U (2kF ). It is convenient to define the dimensionless constant
α =

g2
.
2πvF

(7)

A system of interacting electrons in one dimension such
as the one introduced above is described by TomonagaLuttinger liquid (TLL) theory. The low energy excitations of a TLL are particle-hole pairs which are bosonic
in nature and have a linear relation between energy and
momentum. For spinless electrons, the low energy and
long distance properties of the TLL are governed by three
quantities, namely, the velocity v of the low energy excitations, a dimensionless parameter K which is related to
the interaction strength, and the Fermi wavenumber kF .
For the model described above, we find that [24]
2 1/2

v = vF (1 − α )
1 − α 1/2
and K =
.
1+α

,
and

(8)

(11)

Thus K = 1 for non-interacting fermions. For weak interactions, v = vF and K = 1 − α to first order in α. In
this paper, we will be interested in the case in which the
interaction is weak, i.e., |α| << 1.
It turns out that in the presence of interactions, the
impurity strength V effectively becomes a function of a
length scale l, and satisfies a RG equation. On bosonizing
the TLL theory [7,8,24], we obtain the equation
dV
= (1 − K) V ,
d ln l

to first order in the interaction parameter α. (Eq. (11)
is consistent with the unitarity of the scattering matrix,
namely, |r|2 + |t|2 = 1 and r∗ t + t∗ r = 0). The solution
of Eq. (11) is given by
|t(l)|2 =

|t(d)|2 (d/l)2α
,
1 − |t(d)|2 + |t(d)|2 (d/l)2α

(12)

where d is a short distance scale which is of the order of
λ. However, we will see at the end of Sec. V C that it is
necessary to choose d to be substantially larger than λ in
order to obtain a good fit between the numerical results
and the expression in Eq. (12). [At l → ∞, Eq. (12) has
fixed points at |t| = 0 and 1, depending on the sign of α;
there is no fixed point at intermediate values of t.]
We can check that Eq. (11) is equivalent to Eqs. (9)
and (10) if α is small and |t| is close to 1 and 0 respectively. If α is small, the Luttinger parameter K ≃ 1 − α.
We note that Eq. (11) is complementary to Eqs. (9)
and (10) obtained from bosonization in the following
sense. Eq. (11) is valid for all values of t but only small
values of α. On the other hand, Eqs. (9) and (10) are
valid for arbitrary values of the interaction parameter α

(9)

to first order in V , i.e., this is valid in the weak barrier
limit. In the strong barrier limit, which implies |V | >>
vF , bosonization leads to a different RG equation
d(1/V )
1
= (1 −
) (1/V ) ,
d ln l
K

dt
= − α t |r|2 ,
d ln l
dr
= α r |t|2 ,
d ln l

(10)

which is valid to first order in 1/V . We see that V =
0 (V = ∞) is a stable fixed point if K > 1 (K < 1
respectively). Since the above RG equations are not valid
if |V |/vF is of order 1, we cannot conclude from them
whether or not there is an intermediate fixed point.

3

It is interesting to note that unlike in the spinless case,
Eq. (16) allows the possibility of |t| increasing for a while
and then decreasing (or vice versa) [13]. This is due to
˜
the flow of the couplings g1 and g2 ; it happens if U (0) −
˜ (2kF ) and U (0) − (1/2)U(2kF ) have opposite signs.
˜
˜
2U
This is precisely the situation for the Hubbard model in
˜
˜
which U (0) and U (2kF ) are both equal to the Hubbard
parameter U .

but only for small values of V or 1/V , i.e., only when
either the reflection amplitude r or the transmission amplitude t is small.
B. Spin-1/2 electrons

We now turn to the more realistic case of electrons
with spin. For this situation, we will only discuss the
RG equations obtained from the HF approximation for
the case of weak interactions [13,14]. We again consider
a δ-function impurity at one point in the TLL, and a
density-density interaction of the form given in (4); the
density is now given by
ρ = Ψ† Ψ↑ + Ψ† Ψ↓ ,
↑
↓

III. LATTICE MODELS AND THE
HARTREE-FOCK APPROXIMATION

Although the system of interest may be defined in the
continuum, it is convenient to approximate it by a lattice
model in order to do numerical calculations. (We should
of course ensure that physical quantities like the wire
length are much larger than the lattice spacing, so that
the lattice approximation does not introduce significant
errors). Let us discuss the form of the lattice Hamiltonians that we will consider. The Hamiltonian has a
hopping term

(13)

where ↑ and ↓ denote spin-up and spin-down respectively.
Introducing right and left moving fields ΨR,σ and ΨL,σ
as in Eq. (5), with σ =↑ or ↓, we obtain the interaction
Hamiltonian
Hint =

1
+ g4 (Ψ† Ψ† ′ ΨRσ′ ΨRσ + Ψ† Ψ† ′ ΨLσ′ ΨLσ )],
Rσ Rσ
Lσ Lσ
2
(14)

where cn,σ annihilates an electron with spin σ at site n,
and the hopping amplitude γ will be taken to be positive
for convenience. Next, we will place an impurity at one
site, say, n = 0, so that

˜
˜
where g1 = U (2kF ), and g2 = g4 = U (0). (We have
ignored umklapp scattering terms here; they only arise if
the model is defined on a lattice and we are at half-filling).
It is known that g1 , g2 and g4 satisfy some RG equations
[26]; the solutions of the lowest order RG equations are
given by [13]
˜
U (2kF )
1 +

˜
U (2kF )
πvF

(18)

σ

Finally, we will introduce an interaction between the electrons in the wire (but not in the leads which will be considered in the next section). For the spin-1/2 case, the
interaction will be taken to be of the Hubbard form,

,

c† cn,↑ c† cn,↓ .
n,↑
n,↓

Hint = U

˜
1 ˜
U (2kF )
1
˜
g2 (l) = U (0) −
U (2kF ) +
,
˜
2
2 1 + U (2kF ) ln l
πvF

(19)

n

For the case of spinless electrons, the spin index σ will be
dropped in Eqs. (17) and (18), and the interaction will
be taken to be between nearest neighbor sites,

(15)

Next, we can use the HF approximation and the existence of Friedel oscillations in the density to derive the
RG equation of the transmission amplitude t. We find
that [13,14]
g2 (l) − 2g1 (l)
dt
t |r|2 .
= −
d ln l
2πvF

c† c0,σ .
0,σ

HV = V

ln l

˜
g4 (l) = U (0) .

(17)

n,σ

σ,σ′

g1 (l) =

[ c† cn+1,σ + c†
n,σ
n+1,σ cn,σ ] ,

H0 = − γ

[g1 Ψ† Ψ† ′ ΨRσ′ ΨLσ + g2 Ψ† Ψ† ′ ΨLσ′ ΨRσ
Rσ Lσ
Rσ Lσ

dx

Hint =

U
2

c† cn c† cn+1 .
n
n+1

(20)

n

(For spinless electrons, we cannot have an on-site interaction since (c† cn )2 = c† cn ).
n
n
In the absence of the impurity and interactions, the
energy is related to the wavenumber as

(16)

Due to the RG flow of the couplings g1 and g2 , the solution of Eq. (16) is more complicated than the expression
for spinless fermions given in Eq. (12) [13]. We will
therefore not attempt to make a quantitative comparison between our numerical results and the RG equations
for the case of spin-1/2 electrons.

E(k) = − 2γ cos k ,

(21)

where −π < k ≤ π. (We have set the lattice spacing
equal to 1). The velocity is given by v = dE/dk =
2γ sin k. If the chemical potential is given by µ, with
4

−2γ < µ < 2γ, the ground state is the one in which
the states with momenta −kF to kF are filled, where
µ = −2γ cos kF , with 0 < kF < π. It is convenient
to define the Fermi energy as the difference between the
chemical potential and the bottom of the band, namely,
EF = µ + 2γ = 2γ(1 − cos kF ).
Let us now consider the effect of the impurity placed
at n = 0 (we are still ignoring the interactions). The
reflection and transmission amplitudes are given by
iV
,
2γ sin k + iV
2γ sin k
and t(k) =
.
2γ sin k + iV

an unrestricted HF approximation in which one allows
< c† cn,↑ > to differ from < c† cn,↓ >.]
n,↑
n,↓
At this point, we would like to impose the condition
that the interaction should have no effect on the conductance if there is no impurity, i.e., if ρn is equal to the
constant ρ at all sites. The reason for imposing this con¯
dition is that we do not want the interactions by themselves to lead to any scattering; that would change the
conductance from the ideal value of 2e2 /h in the absence
of any impurities which is an undesirable effect. (This
will become clearer in the next section when we describe
our model for the wires and the leads). We will therefore
modify Eq. (26) to the form

r(k) = −

(22)

In addition, there is a bound state if V < 0; the bound
state energy is given by Eb = −2γ cosh kb , where V =
−2γ sinh kb , with kb > 0. The normalized bound state
wave function is given by
ψb,n = (tanh kb )1/2 e−kb |n|

n

(27)
where we have dropped the constant ρ2 which turns out
n
to have no effect on the conductance. (The terms proportional to ρ in Eq. (27) are equivalent to adding a
¯
chemical potential).
To gain an insight into the effect of Eq. (27), let us
consider the Born approximation. In order to use this
approximation, we will assume here that both V and U
are small. The total potential seen by either spin-up or
spin-down electrons at site n is

(23)

for all values of n.
It is interesting to compute the electron density ρn ≡
ρn,↑ = ρn,↓ , as a function of the site index n. The contribution from the scattering states is given by
kF

ρn =
0

dk
[ |e−ik|n| + r(k)eik|n| |2 + |t(k)|2 ]
2π
kF

=ρ +
¯
0

Vn = V δn,0 + U (ρn − ρ) .
¯

dk
[ r(k)ei2k|n| + r∗ (k)e−i2k|n| ] ,
2π
(24)

Vn = V δn,0 −

VU
(ei2kF |n| + e−i2kF |n| ) , (29)
4πvF |n|

where the second term is valid only for |n| >> π/kF .
The Born approximation for the reflection amplitude
in one dimension for a lattice model is given by

ρn − ρ
¯
i
[ r(kF )ei2kF |n| − r∗ (kF )e−i2kF |n| ] . (25)
4π|n|

rB (kF ) = −

Now we will introduce interactions and consider the
HF approximation for the cases of spin-1/2 and spinless
electrons. This will give us another way of understanding
the RG equations discussed in Sec. II.

i
vF

Vn ei2kF n .

(30)

n

If the region in which the electrons interact with each
other only extends from n = −L/2 to L/2, then for L >>
π/kF , we get
rB (kF ) = −

A. Spin-1/2 electrons

iV
U
L
[1 −
ln ] .
vF
2πvF
2

(31)

If U > 0, we see that the conductance G = (2e2 /h)[1 −
|rB (kF )|2 ] increases with L till it reaches a maximum
when ln(L/2) ≃ 2πvF /U . However, the Born approximation that we are using here cannot be trusted up to such
large length scales because we are not self-consistently
modifying the densities at the different sites in response
to the renormalization of the reflection amplitudes (produced by the interaction U ). Because of this lack of selfconsistency, we can only trust the Born approximation

For the interaction given in Eq. (19), the HF approximation takes the form
[ ρn (c† cn,↑ + c† cn,↓ ) − ρ2 ] ,
n
n,↑
n,↓

Hint,HF = U

(28)

For small values of |V |/vF (where vF = 2γ sin kF is the
Fermi velocity), r(kF ) = −iV /vF from Eq. (22), and we
have

where ρ = kF /π is the value of the density very far from
¯
the impurity. In the limit |n| → ∞, we find that the
density has oscillations given by

≃ −

(ρn − ρ) (c† cn,↑ + c† cn,↓ ) ,
¯ n,↑
n,↓

Hint,HF = U

n

(26)
where we have set < c† cn,↑ >=< c† cn,↓ >= ρn .
n,↑
n,↓
[This is called the restricted HF. One can also make
5

Hence,

results up to length scales where the reflection amplitude
has not changed very much from the value of −iV /vF
that it has in the non-interacting theory.
We note that Eq. (31) is consistent with Eq. (16)
for small values of V and (U/2πvF ) ln l, since (g2 (l) −
2g1 (l))/2πvF = −U/2πvF for such values of l.

Vn = V δn,0 −

− r∗ (kF )e−i2kF |n| ] ,
(37)
where the second term is only valid for |n| → ∞, and

B. Spinless electrons

Vn+1/2 =

For the interaction given in Eq. (20), the HF approximation takes the form
U
2

Hint,HF =

− r∗ (kF ) e−i2kF |n|−isgn(n)kF ] ,
(38)
for |n| → ∞; here sgn(n) ≡ n/|n|. For small values of
|V |/vF , r(kF ) = −iV /vF . Substituting everything in the
Born approximation (35), and assuming that the region
in which the electrons interact with each other extends
from n = −L/2 to L/2, where L >> π/kF , we find that

n

− < c† cn+1 > c† cn
n
n+1
(32)

where ρn ≡< c† cn >, and we have dropped some conn
stants. In the absence of any impurities, ρn = kF /π and
< c† cn >= (sin kF )/π for all values of n.
n+1
Once again, we impose the condition that the interaction should have no effect on the conductance if there is
no impurity. We therefore modify Eq. (32) to the form
U
2

Hint,HF =

rB (kF ) = −

Eq. (39) is consistent with Eq. (11) for small values of
V and (U/2πvF ) ln l, since α = (U/2πvF )[1 − cos(2kF )].

[(ρn+1 + ρn−1 − 2ρ) c† cn
¯ n
n

sin kF †
) cn+1 cn
π
sin kF †
) cn cn+1 ] ,
− (< c† cn > −
n+1
π
(33)

IV. NON-EQUILIBRIUM GREEN’S FUNCTION
FORMALISM

In this section, we will introduce the NEGF formalism
which will allow us to study the conductance of a wire
with any length, both short (of the order of π/kF ) and
long (where a continuum description and RG analysis
may be expected to be reliable).

where ρ = kF /π as before.
¯
Let us again use the Born approximation to understand
the effect of Eq. (33), assuming that both V and U are
small. If one adds a perturbation of the form

A. Self-energy, density and conductance

[ V n c† cn
n
n

An important concept in the NEGF formalism is a
“self-energy” which describes the amplitude for an electron to leave or enter the wire from the leads (reservoirs) which are maintained at some chemical potentials
and temperature [10,20–23]. The self-energy is a nonHermitian term in the single-particle Hamiltonian of the
wire. To see how this arises, let us begin by modeling
one of the reservoirs by a tight-binding Hamiltonian

∗
+ Vn+1/2 c† cn + Vn+1/2 c† cn+1 ]
n
n+1

(34)
to the Hamiltonian in Eq. (17), the Born approximation
for the reflection amplitude is given by
rB (kF ) = −

i
vF

[Vn ei2kF n
n
∗
+(Vn+1/2 + Vn+1/2 ) eikF (2n+1) ] .

( c† cn+1,σ + c†
n,σ
n+1,σ cn,σ ) .

H = −γ

(40)

n,σ

(35)

The energy of an electron in the reservoir is related to
its wavenumber as E = −2γ cos k. The reservoir has
a chemical potential µ and an inverse temperature β =
1/(kB T ). The reservoir is semi-infinite; the last site at
one end of the reservoir is coupled to the first site of the
wire by a hopping amplitude γ ′ (see Fig. 1).

In our case,
U
[ ρn+1 + ρn−1 − 2ρ ] ,
¯
2
U
sin kF
=−
[ < c† cn+1 > −
].
(36)
n
2
π

Vn = V δn,0 +
Vn+1/2

iV
U
L
[1 +
(1 − cos(2kF )) ln ] .
vF
2πvF
2
(39)

− (< c† cn+1 > −
n

Hpert =

iU
[ r(kF ) ei2kF |n|+isgn(n)kF
8π|n|

[ ρn+1 c† cn + ρn c† cn+1
n
n+1

− < c† cn > c† cn+1 ] ,
n
n+1

iU cos(2kF )
[ r(kF )ei2kF |n|
4π|n|

6

H + ΣL (E) + ΣR (E) =

γ
−4

γ
−3

γ

γ′

γ

−2

−1

0



 σ(E)

 γ1
− 
 0

 ···
···

1

Reservoir from −∞ to 0

Wire

The reservoir gives rise to a self-energy at the first site
of the wire of the form [10,22]
c† c1,σ ,
1,σ

(41)

σ

dE
GΓL G† fL
2π

where
′2

σ(E) = −

γ
eik
γ

(42)

γ ′2 −k
e
− iη
γ

γ ′2 −k
e
− iη
γ

···
···
0
···
σ(E)




 .




and

dE
GΓR G† fR
2π

fa (E) = [eβ(E−µa ) + 1]−1

(45)

(46)

(47)

is the Fermi function, and µa is the chemical potential
in reservoir a. We note that if E lies outside the range
[−2γ, 2γ], the matrix Γa (E) ∼ 2η is infinitesimal. In
that case, the density matrix can still receive a contribution from certain states; these are typically bound states
which have a discrete set of energies Eb . The reason that
such states can make a finite contribution in Eq. (46)
even though Γa is infinitesimal is that G takes the form
1/(E − Eb + iη), and

(43)

for E = −2γ cosh k ≤ −2γ (with k ≥ 0), and
σ(E) =

···
0
γ1
···
γ1

respectively, where Γa (E) = i(Σa − Σ† ),
a

for −2γ ≤ E = −2γ cos k ≤ 2γ (with 0 ≤ k ≤ π),
σ(E) = −

0
γ1
0
···
0

(The effects of the reservoirs on the wire is completely
taken into account by the self-energy terms). We would
like to emphasize that the relation between E and k is
given entirely by the dispersion in the reservoirs (we assume that the dispersion is the same in both the reservoirs), and not on the form of the Hamiltonian H inside
the wire.
The density matrices due to electrons coming in from
the left and right reservoirs are given by [10,20–22]

FIG. 1. (Color online) Picture of a semi-infinite reservoir
going from site number 0 to −∞ and a wire beginning from
site number 1.

Σ(E) = σ(E)

γ1
0
γ1
···
···



(44)

for E = 2γ cosh k ≥ 2γ (with k ≥ 0). Here η is an
infinitesimal positive number which appears only if E
lies outside the range [−2γ, 2γ]. For −2γ < E < 2γ, the
self-energy already has an imaginary piece, so it is not
necessary to add an iη term.

Limη→0+

2η
= 2 π δ(E − Eb ) .
(E − Eb )2 + η 2

(48)

Finally, the current is given by the expression

γ′
Left
Reservoir
µL

γ1
1

γ1

γ1

2 ······ N

γ′
Right
Reservoir
µR

I = −

e
h

dE trace(GΓL G† ΓR ) [fL (E) − fR (E)] ,
(49)

and the conductance is G = e I/(µL − µR ) . If E
lies outside the range [−2γ, 2γ], ΓL (E) and ΓR (E) are
both infinitesimal, and the current does not get any contribution from states lying in that energy range. (The
difference between the density matrix and the current
for such states is that the density matrix has only one
factor of η in the numerator while the current has two
factors of η).
We will be interested below in the case of linear response, i.e, the limit µL → µR = µ. In that limit, the
Fermi wavenumber kF is given by µ = −2γ cos kF . Further, the conductance takes the form

FIG. 2. (Color online) A quantum wire system with a
N -site wire in the middle and semi-infinite reservoirs on its
left and right.

Now let us consider the complete system which consists
of a wire with N sites in the middle, and reservoirs on
its left and right as shown in Fig. 2. If the wire is also
modeled by a tight-binding Hamiltonian (with a hopping
amplitude γ1 ), the Green’s function of the wire at energy
E is given by a N × N matrix
G(E) = [ E I − H − ΣL (E) − ΣR (E) ]−1 ,

7

G=

e2
h

2γ

dE trace(GΓL G† ΓR )
−2γ

β
.
[2 cosh β (E − µ)]2
2
(50)

• Repeat the previous step till the density changes
no further, i.e., between two successive iterations,
the maximum change in density on any lattice site
is less than about 10−4 .

(We have to multiply the above expression by 2 for spinhalf electrons).

• Use the converged density to compute the conductance.

B. Interactions

V. NUMERICAL RESULTS

Let us now consider how interactions can be studied
within the NEGF formalism. Note that this formalism
works with a one-particle Hamiltonian, e.g., the selfenergy in Eq. (41) is given in terms of the energy of
a single electron which is entering or leaving the wire.
Hence, the only way to deal with interactions is to do
a HF decomposition as shown in Eqs. (26) and (32);
these take the form of corrections to the on-site chemical
potential or the hopping amplitude.
However, there is a difficulty in using the expressions
in (26) or (32). We want to have interactions between the
electrons only in the wire, not in the reservoirs. The HF
decompositions shown in (26) and (32) will then lead to
one-body terms only in the wire; these terms will backscatter electrons coming from the reservoirs, and hence
reduce the conductance from its ideal value of 2e2 /h or
e2 /h. We would like to ensure that there is no such scattering if there are no impurities inside the wire and if the
leads connect adiabatically to the wires, i.e., if the entire
system with wire and reservoirs is translation invariant.
We know that two-body scatterings between electrons
conserve momentum and therefore do not affect the conductance in the absence of impurities and scattering from
the lead-wire junctions. We therefore modify the form of
the HF decomposition from Eqs. (26) and (32) to Eqs.
(27) and (33) respectively; the extra terms in those equations ensure that the HF terms vanish identically if there
are no impurities. [An alternative way of ensuring adiabaticity between the leads and the wires is to turn on the
interaction strength U smoothly from zero in the leads
to a finite value inside the wire. This, however, is harder
to implement numerically. Subtracting the mean values
as in (27) and (33) is easier to implement, and it serves
the same purpose because the subtracted quantities approach zero near the ends of the wires (this will become
clear from the numerical results presented below).]
In the presence of interactions, we will implement a
self-consistent NEGF calculation as follows:

In this section, we will present the results obtained by
the NEGF formalism, first for non-interacting electrons,
and then for interacting spin-1/2 and spinless electrons,
using the procedure described in Sec. IV B. We will study
the dependence of the conductance G on the wire length
L, the inverse temperature β, the impurity strength V ,
and the interaction parameter U . Finally, we will make
a quantitative comparison between our numerical results
and the RG equations for the case of spinless electrons.
We present some details about our choice of parameters
for the calculations. We always take the wire to have an
odd number of sites, and the impurity to be on the middle
site. The hopping amplitudes in the reservoirs and in
the wire is chosen to be the same (γ = γ1 = γ ′ = 1);
the values of the inverse temperature β = 1/(kB T ) is
quoted in units of 1/γ. (We set both ¯ and the lattice
h
spacing equal to 1). We choose kF = π/10; hence, vF =
2 sin kF = 0.618, µ = −2 cos kF = −1.902, and EF =
2(1 − cos kF ) = 0.098. The thermal coherence length is
given by LT = 0.618β. In the energy integrations for the
current and conductance, we take the energy step size to
be dE = 2 × 10−4 . In the integration outside the energy
range [−2γ, 2γ], we take the quantity η to be 5 times dE.
Finally, in all the figures, the conductance G is expressed
in units of 2e2 /h and e2 /h for the spin-1/2 and spinless
cases respectively.
Our choice of parameters was dictated partly by experiments on quantum wires in semiconductor heterojunctions [1–6], and partly by our numerical limitations
(which prevent us from going to very large wire lengths).
Experimentally, the ratio of the wire length to the de
Broglie wavelength of the electrons (LkF /π) ranges from
about 20 to 200, the ratio of the wire length to the inverse
temperature L/β ranges from about 1/2 to 10, and the
parameter U/(2πvF ) appearing in the Eq. (31) is about
0.2 - 0.3 [9]. In our calculations, LkF /π goes from about
1 to 30, L/β goes from about 1/40 to 6, and the interaction parameters take the values U/(2πvF ) = 0.0773 in the
spin-1/2 case (Eq. (31)) and (U/2πvF )[1 − cos(2kF )] =
0.0148 in the spinless case (Eq. (39)) for U = 0.3. The
much smaller value of the interaction parameter in the
spinless case leads to smaller changes in the conductance
in that case as we will see.

• Start with the Hamiltonian with no interactions,
and calculate the density matrix. The diagonal elements of the density matrix give the densities at
different sites.
• Use the HF approximation to compute the Hamiltonian with interactions, and use that to calculate
the density matrix again.

8

A. Non-interacting electrons
0.22

Let us first discuss some properties of a system in which
there is a δ-function impurity of strength V in the middle of the wire, and there are no interactions between
the electrons. Fig. 3 shows the Friedel oscillations in the
density < c† cn > for V = 0.3 for two different temperan
tures. Note that the density at the middle site is much
lower than the mean value of ρ = kF /π = 1/10 due to
¯
the repulsive nature of the impurity. We also observe
that the oscillations die out beyond a length scale of order LT = 0.618β in the lower figure. This is a simple
illustration of the idea of a thermal coherence length.

β = 300

0.2

0.18

ρ

0.16

0.14

0.12

0.1

0.08
0.12
0

50

β = 300

0.11

n

100

150

FIG. 4. (Color online) Friedel oscillations in the density of non-interacting electrons caused by an impurity with
V = −0.3 placed in the middle of a wire with 151 sites and
β = 300.

0.1

ρ

0.09

0.08

because of the presence of a bound state whose wave
function has a peak at that site. Indeed, in the absence
of interactions, the difference in the site densities for V =
0.3 and −0.3 is given entirely by the bound state. We
see from Figs. 3 and 4 that the density difference at the
site of the impurity is about 0.15; this agrees well with
the value of tanh kb = 0.148 given by Eq. (23). (For
V = −0.3, we have kb = 0.149).
Finally, Fig. 5 shows the quantity ρ1 ≡< c† cn+1 > for
n
V = 0.3 and −0.3 for β = 300. There is a strong similarity between these and the density plots shown in Figs. 3
(upper figure) and 4. This is because the electrons at the
Fermi energy (which dominate the long distance properties of the system) have a wavelength which is π/kF = 10
times longer than the lattice spacing; hence there is very
little difference between < c† cn > and < c† cn+1 >.
n
n
For non-interacting electrons, the conductance depends on the temperature, but not on the length of the
wire. For a δ-function impurity of strength V , the conductance for spinless electrons is given by

0.07

0.06

0.05

0.04

0

50

n

100

150

0.12

β = 50

0.11

0.1

ρ

0.09

0.08

0.07

0.06

G=

0.05

0.04

0

50

n

100

T (E) =

150

e2
h

2

dE T (E)
−2

4 sin2 k
.
4 sin2 k + V 2

β
[2 cosh

β
2 (E

− µ)]2

,
(51)

FIG. 3. (Color online) Friedel oscillations in the density of
non-interacting electrons caused by an impurity with V = 0.3
placed in the middle of a wire with 151 sites. The upper and
lower figures have β = 300 and 50 respectively.

where E = −2 cos k. For kB T << EF = µ + 2, Eq. (51)
has a Sommerfeld expansion of the form [27]

Fig. 4 shows the Friedel oscillations in the density
for V = −0.3 for β = 300. The density in the middle
site is now much higher than the mean value; this is

For kF = π/10 and V = 0.3, T (µ) = 0.809 and
(π 2 /6)T ′′(µ) = 42.1; this implies a significant temperature dependence of the conductance. Fig. 6 shows that

G =

9

π2
e2
[ T (µ) +
(kB T )2 T ′′ (µ) + · · · ] . (52)
h
6

0.11

0.85

0.1
0.8

0.09

G
0.08

V = 0.3

ρ1

V = 0.3

0.75

0.07

0.7

0.06

0.05

0

25

50

75

n

100

125

20

150

0.18

60

100

140

β

180

220

260

300

FIG. 6. (Color online) Conductance versus β for
non-interacting electrons in the presence of an impurity with
V = 0.3 placed in the middle of a wire with 201 sites.

V = − 0.3
0.16

In Fig. 7, we show the conductance as a function of
the wire length L for three different value of β, with
U = 0.3 in the upper figure and U = −0.3 in the lower
figure. The trends in Fig. 7 are in accordance with Eq.
(31). Firstly, the conductance increases with the length
scale if U > 0 and decreases if U < 0. For U = 0.3,
we have U/(2πvF ) = 0.0773; we expect the HF approximation to be reasonable for such a small value. [Due to
the RG flows of g1 and g2 in Eq. (16), the conductance
should start decreasing for positive U and increasing for
negative U at a very large length scale of the order of
L ∼ exp(1/0.0773) ∼ 420000, provided that LT is also
larger than 420000. Such a wire length is beyond our
numerical capability.]
Secondly, for fixed β, G increases till L reaches a value
of the order of LT = 0.618β beyond which G stops changing. This happens because there is complete thermal decoherence once L exceeds LT ; hence G does not vary
any more with L in that regime. Thirdly, the conductance for two different values of β, say, β1 and β2 (where
β1 < β2 ), start separating from each other at a value
of the wire length Lc such that Lc /β1 is much less than
0.618. This happens because if the inverse temperature
is β1 or higher, there is complete thermal coherence, and
G does not depend on β any more. Thus, for any temperature β, there is a range of wire lengths going from Lc up
to LT where there is partial thermal coherence, and the
conductance evolves slowly in this range of lengths. We
therefore conclude that the RG flow does not suddenly
stop at a length scale which is the smaller of the wire
length L and the thermal coherence length LT ; rather,
the flow continues to occur (but slows down) in an intermediate range of length scales.
Finally, Fig. 7 shows that even for the smallest values

0.14

ρ1
0.12

0.1

0.08

0

25

50

75

n

100

125

150

FIG. 5. (Color online) Plots of ρ1 ≡< c† cn+1 > for
n
non-interacting electrons in the presence of an impurity with
V = 0.3 (upper figure) and −0.3 (lower figure) placed in the
middle of a wire with 151 sites, with β = 300.

the conductance changes appreciably with β till β reaches
about 100. We have to keep this in mind when studying the temperature dependence of the conductance of a
system of interacting electrons.
B. Spin-1/2 electrons

We now consider the spin-1/2 model with an on-site
interaction between the electrons. In the absence of interactions, the conductance at zero temperature is independent of the wire length and is given by Eq. (52), with
a factor of 2 for spin; namely,
G0 =

2e2
4 sin2 kF
.
h 4 sin2 kF + V 2

(53)

For kF = π/10 and |V | = 0.03, G0 = (2e2 /h) 0.809.
10

0.9

0.9

U = 0.3

U = 0.3
0.89

β = 400

0.89

L = 101
0.88

β = 200

L = 71

G

0.88

L = 41

0.87

G
β = 100

0.87

0.86

0.85

0.86

0.84
50

0.85

0

40

80

120

0.74

L

160

200

240

280

100

150

200

β

250

300

350

400

FIG. 8. (Color online) Conductance versus β for interacting spin-1/2 electrons for three different wire lengths, for
V = 0.3 and U = 0.3.

U = − 0.3

0.72

G varies relatively slowly in that region.
In Fig. 9, we show the conductance as a function of
V for three different values of U , with L = 101 and β =
100. Note that for U = 0, the conductance is an even
function of V given by Eq. (53). But in the presence
of interactions, G is no longer precisely an even function
of V , particularly for large values of V . This is more
clearly visible in Fig. 10 which shows the conductance as
a function of U for V = 1 and -1, with L = 101 and β =
100. For any value of U , we find that the conductance
deviates less from the non-interacting value for V > 0
compared to V < 0, for the same value of |V |. For small
values of U , this can be explained as follows. We saw in
Figs. 3 and 4 that the density at the site of the impurity
deviates from the mean density of ρ = 1/10 by a smaller
¯
amount for V > 0 compared to V < 0; the numerical
values for ρ0 − 1/10 are given by -0.046 and 0.095 for
V = 0.3 and -0.3 respectively. The effective impurity
strength is given by

β = 100

0.7

G
0.68

β = 200
0.66

0.64

β = 400

0.62

0.6

0

30

60

90

120

L

150

180

210

240

270

300

FIG. 7. (Color online) Conductance versus wire length for
interacting spin-1/2 electrons for three different values of β,
for V = 0.3. U = 0.3 and -0.3 in the upper and lower figures
respectively.

of L, the conductance differs from the non-interacting
value of 0.809 by an appreciable amount. This is due to
the large deviations of the density from the mean value at
the sites close to the impurity. In that region, the density
does not satisfy the 1/|n| form given in Eq. (25). Since
that form is intimately tied to the RG results (the sum
over 1/|n| gives ln(L)), we do not expect the numerically
obtained value of the conductance to agree well with the
RG analysis for short wire lengths.
In Fig. 8, we show the conductance as a function of β
for three different value of L, with U = 0.3. Once again,
we see the same trends as in Fig. 7. Namely, for any
two values of L, there is a β where the values of G start
separating from each other. Then there is a higher value
of β where G stops changing. The region between the
two is where there is only partial thermal coherence, and

Veff = V + U (ρ0 − ρ) ,
¯

(54)

which equals 0.3 − 0.046U and −0.3 + 0.095U for V =
0.3 and −0.3 respectively. For U small and positive,
this means that the magnitude of the effective impurity strength and therefore the reflection probability is
larger for V = 0.3; hence the conductance is smaller for
V = 0.3. The opposite statement is true if U is small and
negative. To conclude, the different values of the density
deviation at the impurity site for positive and negative
values of V are responsible for the asymmetry in G as a
function of V . (This asymmetry is more prominent for
the case of spinless electrons as we will see below).
The fact that the deviation of the conductance from
the non-interacting value is smaller for V > 0 than for
V < 0 is clearly a result of the bound state which raises
11

1

0.5

0.9

0.45

0.8

0.4

0.7

0.35

V = −1

V=1

U = 0.3
0.3

0.6

G

G

0.25

0.5

U=0

0.4

0.2

0.15

0.3

0.1

0.2

0.1
−1

U = −0.3
−0.8

−0.6

−0.4

−0.2

0

V

0.2

0.4

0.6

0.8

0.05
−0.5

1

−0.4

−0.3

−0.2

−0.1

0

U

0.1

0.2

0.3

0.4

0.5

FIG. 9. (Color online) Conductance G versus V for interacting spin-1/2 electrons for U = 0.3, 0 and -0.3, with L = 101
and β = 100.

FIG. 10. (Color online) Conductance G versus U for interacting spin-1/2 electrons for V = 1 and -1, with L = 101 and
β = 100.

the density at the impurity site for V < 0; hence this is a
short distance effect. [The contribution of the scattering
states to the reflection probability |r|2 is an even function
of V as one can see from Eq. (31). The RG equations
which are derived from continuum theories take into account only long distance effects; these come from the
scattering states only.]
In Fig. 11, we show the Friedel oscillations for three
different values of U with V = 0.3, L = 101 and β = 100.
We see that a repulsive interaction (U > 0) suppresses
the Friedel oscillations, while an attractive interaction
(U < 0) with the same magnitude enhances the oscillations by a much larger amount. This is because attractive interactions tend to lead to the formation of a charge
density wave; if a charge density wave is already present
(due to the impurity), attractive interactions enhance it.
Since the scattering from the Friedel oscillations renormalize the scattering from the impurity, this difference in
the magnitude of the oscillations helps to explain why the
deviation of the conductance G from the non-interacting
value is larger for U < 0 compared to U > 0, as can be
seen in Fig. 7.
For spin-1/2 electrons, we can do a self-consistent HF
calculation in two ways, restricted and unrestricted. In a
restricted HF calculation, the site densities of spin-up and
spin-down electrons are taken to be equal at all stages.
In a unrestricted HF calculation, spin-up and spin-down
electrons are allowed to have different densities. We have
done both kinds of calculations. For the range of the interaction −0.5 ≤ U ≤ 0.5, we find that they give the
same results; the spin-up and spin-down densities converge to the same values even if one begins with different
initial values for them. Thus we do not find any spin

density wave within our range of parameters.
C. Spinless electrons

We now consider a model of spinless electrons with
nearest neighbor interactions between the electrons.
Comparing the forms of Eqs. (31) and (39), we see
that the behaviors of the spin-1/2 model for U > 0
and U < 0 should be similar to that of the spinless
model for U < 0 and U > 0 respectively. In Fig.
12, we show the conductance as a function of the wire
length L for three different value of β, with V = 0.3
and U = 0.3. The trends are in agreement with Eq.
(31), with α = U (1 − cos(2kF ))/(2πvF ) = 0.0148. For
fixed β, G decreases till L reaches a value of the order of
LT = 0.618β.
In Fig. 13, we show the conductance as a function of β
for three different value of U , with V = 0.3 and L = 201.
As we saw earlier in Fig. 6, G has a significant dependence on β for small values of β even for non-interacting
electrons. It is only when we go to large values of β
that we see the trend expected from the RG equations
for interacting electrons; namely, as β increases, the conductance decreases if U > 0 and increases if U < 0.
Since G has an appreciable dependence on β even for
non-interacting electrons, it is easier to consider the dependence of G on the wire length L at zero temperature
in order to see how well the numerical results compare
with the RG expression given in Eq. (12). That equation
has two parameters, namely, the interaction parameter
α and the short distance scale d (with its corresponding
transmission probability |t(d)|2 ). To begin, let us choose
α = 0.0148 as given by the analytical expression in Eq.
12

0.81

0.12

U = 0.3

U = 0.5

0.11

0.1
0.8

β = 200

0.09

G
0.08

ρ

β = 100

0.79

0.07

0.06
0.78

β = 50

0.05

0.04

0.03

0.77

0

10

20

30

40

50

n

60

70

80

90

100

0.12

100

150

L

200

250

300

FIG. 12. (Color online) Conductance versus wire length for
interacting spinless electrons for three different values of β, for
V = 0.3 and U = 0.3.

U=0

0.11

50

0.1

0.09
0.85

0.08

ρ

U = − 0.5

0.07

U=0
0.06
0.8

U = 0.5

0.05

G
0.04

0.03

0.75

0

10

20

30

40

50

n

60

70

80

90

100

0.12
0.7
20

U = − 0.5

0.11

60

100

140

β

180

220

260

300

0.1

FIG. 13. (Color online) Conductance versus β for interacting spinless electrons for three different values of U , for a wire
with 201 sites and V = 0.3.

0.09

0.08

ρ

0.07

(39) in terms of U and vF . In Fig. 14, we show a comparison between the expression in (12) for two values of d,
namely, d = 11 with |t(d)|2 = 0.805 (the dash dot line A),
and d = 51 with |t(d)|2 = 0.795 (the dashed line B), with
our numerical results shown by astersisks. (The values of
|t(d)|2 for d = 11 and 51 have themselves been obtained
by our numerical calculations). We see that the RG expression with d = 51 fits the numerical results better than
the expression with d = 11. This difference between the
lines A and B shows that the interactions change the conductance by an appreciable amount between d = 11 and
d = 51, and this change is not captured accurately by the
expression in Eq. (12). [The conductance at short dis-

0.06

0.05

0.04

0.03

0

10

20

30

40

50

n

60

70

80

90

100

FIG. 11. (Color online) Friedel oscillations for spin-1/2
electrons for U = 0.5, 0 and -0.5 (upper, middle and lower
figures respectively) for V = 0.3, L = 101 and β = 100.

13

0.81

0.5

0.45

0.805

0.4
0.8

G

0.35

0.795

0.3

V=1

G
A

0.79

0.25

B

V = −1

0.2

C

0.785

0.15

0.78

50

100

150

L

200

250

0.1

300

0.05
−0.5

FIG. 14. (Color online) Conductance G versus L for interacting spinless electrons compared with the RG expression
in Eq. (12), for V = 0.3, U = 0.3, and β = ∞. Line
A (dash dot line) corresponds to d = 11, |t(d)|2 = 0.805
and α = 0.0148. Line B (dashed line) corresponds to
d = 51, |t(d)|2 = 0.795 and α = 0.0148. Line C (solid line)
corresponds to d = 51, |t(d)|2 = 0.795 and α = 0.016. The
asterisks show the numerical results.

0.8

0.7

G
0.6

U = −0.3

0.4

U=0
0.3

0.1
−1

U = 0.3
−0.8

−0.6

−0.4

−0.2

0

V

0.2

0.4

0.6

0.8

−0.2

−0.1

0

U

0.1

0.2

0.3

0.4

0.5

that even line B corresponding to d = 51 starts deviating from the numerical results at large length scales. We
have therefore tried varying the parameter α also, keeping d and |t(d)|2 fixed at 51 and 0.795 respectively. We
find that α = 0.016 (the solid line C in Fig. 14) gives an
excellent fit to the numerical results.
In Fig. 15, we show the conductance as a function of V
for three different values of U , with L = 101 and β = 100.
We see that G is not an even function of V in the presence
of interactions. In Fig. 16, we show the conductance as
a function of U for V = 1 and -1, with L = 101 and
β = 100. For any value of U , we again find that the
conductance deviates less from the non-interacting value
for V = 1 compared to V = −1. The explanation for
this is similar to that for the similar phenomenon in the
spin-1/2 model, except that the roles of U and −U are
interchanged in the two models. The larger change in the
conductance for V = −1 is again due to the presence of
a bound state.

0.9

0.2

−0.3

FIG. 16. (Color online) Conductance G versus U for interacting spinless electrons for V = 1 and -1, with L = 101 and
β = 100.

1

0.5

−0.4

1

VI. DISCUSSION

FIG. 15. (Color online) Conductance G versus V for interacting spinless electrons for U = 0.3, 0 and -0.3, with L = 101
and β = 100.

We have used the NEGF formalism to study the dependence of the conductance of a quantum wire with
both spin-1/2 and spinless electrons on various parameters such as the wire length, temperature, impurity potential, and the strength of the interactions between the
electrons. The advantage of the NEGF formalism is that
it can be used to compute the density and conductance
at any length scale. At large length scales, our numerical
results agree with those obtained by an RG analysis of
continuum theories. We find that the trends of the RG
results can be understood using the Born approximation

tances is affected substantially by quantities such as the
large value of ρ − ρ at the site of the impurity. The RG
¯
analysis, on which (12) is based, does not take such effects
into account.] Since the wavelength of the Friedel oscillations is given by λ = 10, we conclude that the RG results
agree reasonably well with the numerical results only if
we begin the integration of the RG flows from a distance
which is significantly larger than λ. However, we observe

14

for scattering from a weak impurity and from the density
oscillations produced by the impurity.
Our numerical results differ in detail in two ways
from those obtained analytically from the RG equations.
Firstly, the dependence of G on the wire length and temperature fits the expression in Eq. (12) only if we take
the starting point of the RG equation to be significantly
larger than the short distance scale λ, and we allow α to
be a little different from its analytically obtained value.
Secondly, the conductance is not an even function of the
impurity potential V (as one expects from the RG analysis); this is due to the existence of a bound state for an
attractive impurity. These differences between the numerical results and the results based on the RG analysis
seem to be due to effects at short distances, where the
density deviates significantly from the mean density.
Before ending, we would like to briefly compare our
work with that of Refs. [16–18]. Using the functional RG
technique, the authors of those papers have shown that
there is excellent agreement at very large length scales
between the numerical results and the asymptotic scaling
forms given by the RG equations derived from continuum
theories. We have not been able to go up to such large
length scales using the NEGF formalism. However, even
at the length scales studied by us, our numerical results
agree quite well with the RG equations if we start at a
short distance scale which is about 5 times larger than λ.
As mentioned towards the beginning of Sec. V, the
length scales L and LT that we have considered are comparable to those studied experimentally. Our observations about the short distance effects may therefore have
implications for fitting experimental data to expressions
obtained by an RG analysis.

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Acknowledgments
We thank Supriyo Datta, Avik Ghosh and V. Ravi
Chandra for many stimulating discussions. We thank the
Department of Science and Technology, India for financial support under projects SR/FST/PSI-022/2000 and
SP/S2/M-11/2000.

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15