The contact conductance of a one-dimensional wire partly embedded in a
superconductor
Raphael Matthews and Oded Agam

arXiv:0709.0371v1 [cond-mat.str-el] 4 Sep 2007

The Racah Institute of Physics, The Hebrew University, Jerusalem, Israel
The conductance through a semi-infinite one-dimensional wire, partly embedded in a superconducting bulk electrode, is studied. When the electron-electron interactions within the wire are
strongly repulsive, the wire effectively decouples from the superconductor. If they are moderately
or weakly repulsive, the proximity of the superconductor induces superconducting order in the segment of the wire embedded in it. In this case it is shown that the conductance exhibits a crossover
from conductive to insulating behavior as the temperature is lowered down. The characteristic
crossover temperature of this transition has a stretched exponential dependence on the length of the
part of the wire embedded in the superconductor. The amount of this stretch is determined by the
nature of the electron interactions within the wire.
I.

INTRODUCTION

In recent years one dimensional interacting electron
systems have attracted a large amount of attention. Part
of the interest in these systems lies in the fact that
electron-electron interactions in 1-D wires, even when
weak, cannot be considered perturbatively. A question of practical importance, dealt with by a number
of authors1 -11 , is that of the contact conductance of a
one dimensional system connected to an external electrode. Most works on the subject picture the junction
as a one dimensional wire connected to the electrode at
a point. Here it has been found that electron-electron
interactions strongly influence the conductance through
the junction. In particular, repulsive interactions in the
wire drive the system to be insulating at low enough temperatures, unless the contact is perfectly clean. Attractive interactions, on the other hand, mask obstructions
at the interface between the two systems. These results
do not qualitatively change whether the electrode is superconducting or metallic.
In this paper we explore the behavior of a junction
between a 1-D wire and a superconducting electrode of
different geometry, specifically, the situation where the
wire is embedded some distance into the electrode, as
illustrated in Fig. 1. In this case one expects that superconducting order is induced in the part of the wire which
is embedded in the superconductor, enhancing the conductance of the junction. On the other hand, for point
contacts with even a small amount of normal reflection
(and repulsive interactions in the wire), it has been shown
that superconducting order suppresses the conductivity
of the junction8,9 . The main goal of this work is to clarify
how these two competing effects determine the behavior
of the junction as a function of the temperature (or the
applied voltage), and length of the embedded wire.
As expected, provided interactions within the wire are
not too strong, superconducting order is induced in the
part of the wire embedded in the bulk, and an effective
gap is formed in the wire whose value is determined by
the tunneling rate between the superconductor and the
wire, as well as the nature of the electron interactions.

Figure 1: A schematic representation of a 1-D wire in contact with a superconductor: (a) A point contact. (b) A wire
embedded a length L into the bulk

Yet, in spite of this proximity effect, the junction conductance exhibits a crossover from a conductive state, governed by Andreev scattering, to an insulating behavior at
low enough temperatures, characterized by a power law
dependence on the temperature. This behavior results
from the finite amount of normal backscattering within
the wire. We found that the characteristic crossover temperature between the two regimes has a stretched exponential dependence on the length of the junction, and the
amount of stretch is determined by the strength of the
electron interactions in the wire.
The article is organized as follows: In the next section
the model whereby a 1-D wire is embedded infinitely deep
into a bulk BCS superconductor is introduced. By integrating out the superconductor degrees of freedom the
effective action of the embedded wire is obtained. In
Sec. II and the Appendix, renormalization group (RG)
techniques are employed in order to characterize the behavior of a 1-D wire embedded in a BCS superconductor. This RG flow allows one to deduce the low energy
properties of the finite part of the wire embedded in the
superconductor. In Sec. III the conductance of the wiresuperconductor junction is evaluated. Finally, the results
are summarized and discussed in Sec. IV.
II.

MODEL

We consider a single 1-D wire with interacting electrons
(including backscattering), embedded inside a standard
BSC superconductor (SC). The action of the system is a

2
sum of three contributions:
¯
¯
¯
S = Ssc (ϕ, ϕ) + SW ψ, ψ + St ψ, ψ; ϕ, ϕ
¯

(1)

where Ssc is the action of the superconductor, SW is the
action of the wire, and St describes the tunneling between
the two systems. ψσ is the electronic field operator in the
wire, ϕσ is the field operator in the SC, and σ denotes
the spin index.
The action of the SC, modeled by the standard BCS
hamiltonian with a constant pairing amplitude ∆sc , has
the form
Ssc =

¯
d4 ξ Φ(ξ)

∂τ + H0
∆sc
∆∗
∂τ − H0
sc

Φ(ξ)

(2)

where the vector ξ = (x, y, z, τ ) contains three space coordinates and an imaginary time, ΦT = (ϕ↑ , ϕ↓ ) is the SC
¯
h2
¯
electron field in Nambu notation, and H0 = − 2m ∇2 − µ
is the free Fermi gas Hamiltonian, with m and µ as the
electron mass and chemical potential, respectively.
The 1-D wire is modeled as a Bosonized Luttinger liquid. Since the literature on this subject is quite extensive

Sc (θc ) =

1
2

β

dτ uc Kc

dx
0

describes the charge sector (after the dependence on the
φc field has been integrated out13 ). A similar expression
describes the spin sector, Ss (φs ), with the subscript c re−1
placed by s and Kc replaced by Ks . Here Kc , Ks , uc ,
and us are model specific parameters describing the interaction strength (K) and the mode velocity (u) of the
charge and the spin fields. For the noninteracting case
Kc = Ks = 1 and uc = us = vf , where vf is the Fermi
velocity of the wire. For an interacting system, values
of Kc > 1 and Ks < 1 correspond to attractive interactions, while Kc < 1, Ks > 1 correspond to repulsive
ones. Generally in an interacting system the velocity of
the two modes differ, uc = us .
The third term of the wire’s action, Sbs , describes
backscattering of two electrons with opposite spins.
Namely a collision which effectively results in a spin flip
between the right and left moving parts of the electronic
†
†
field (∼ ψL,σ ψR,σ ψR,−σ ψL,−σ ). This term has the form

Sbs =

2g
(2πα)2

√
8πφs (x, τ ) ,
dxdτ cos

(6)

where g is the backscattering coupling constant.
Finally, the tunneling between the superconductor and

(ref. [13] and therein), we will only point out the more
relevant details to the topic at hand.
We represent the electronic fields12 ,
√π
1
eirkf x e−i 2 [rφc (x)−θc (x)+σ(rφs (x)−θs (x))] ,
ψr,σ (x) = √
2πα
(3)
in terms of boson charge and spin density fields: ∂x φν (x),
and their conjugates, θν (x), where ν = c/s denotes the
charge and spin sectors, respectively. In the above formula, kf is the Fermi wave number, α is the short distance cutoff, r is the chiral index representing the right
(r = 1) and left (r = −1) moving part of the electronic
field, and σ = 1 for ↑ spin while σ = −1 for ↓ spin.
In terms of the boson fields, the action of the wire
becomes a sum of three contributions:
SW = Sc (θc ) + Ss (φs ) + Sbs (φs ),

(4)

where

1
2
2
(∂τ θc (x, τ )) + (∂x θc (x, τ ))
u2
c

(5)

the wire is described by
St =

d4 ξ ′

ϕσ (ξ ′ )t(ξ ′ ; x, τ )ψσ (x, τ )
¯

dxdτ
σ=↑,↓

+ H.C.,

(7)

where ψσ = r ψr,σ , and t(ξ ′ ; x, τ ) is the tunneling matrix element. In the simplest case this tunneling is instantaneous, homogeneous in space, short distant and SU(2)
spin symmetric. Under these assumptions it takes the
form
˜
t(ξ ′ ; x, τ ) = t(x)δ(x′ − x)δ(y ′ )δ(z ′ )δ(τ ′ − τ )

(8)

˜
where t(x) has the characteristic function form:
˜
t(x) =

t0 for 0 ≤ x ≤ L
0 otherwise

(9)

where L is the length of the part of the wire embedded
in the superconductor (see fig. 1 (b)).
The effective action of a wire embedded in a superconductor is obtained by tracing out the superconductor
degrees of freedom,
¯

e−Sef f (ψ,ψ) =

¯

¯
D[ϕ, ϕ]e−S (ϕ,ϕ;ψ,ψ)
¯

(10)

3
where S is given by (1). Since the action is quadratic in
the superconductor field, this integration is straightforward. The result may be written as a sum of three terms,
Sef f = SW + Sph + Spp , where Sph , and Spp represent
contributions associated with Giever (particle-hole) and
Andreev (particle-particle) tunneling, respectively. At
energies above the superconductor gap ∆sc the particlehole term is dominant and it’s contribution, on integrating it out, will be to renormalize the chemical potential.
On the other hand, in the low temperature regime (well
below the superconductor gap ∆sc ) the particle-hole term
vanishes and the main contribution comes from the Andreev tunneling term, Spp . In this limit, after averaging
over the rapid spatial oscillations, the tunneling becomes
local in space and time. Expressing it in terms of the
bosonic fields we thus have
Sef f = SW + Spp

(11)

where SW is given by (4), and
Spp =

2∆
πα

dxdτ cos

III.

THE RENORMALIZATION GROUP FLOW
EQUATIONS

The effective action (11) describes the physics of the
junction for a temperature up to the order of the superconductor gap, ∆sc (which will subsequently provide the
high energy cut-off of our system).
In order to describe the behavior of the system at much
lower energy scales, and take into account the electronelectron repulsive interactions, we shall employ a real
space RG approach, following Giamarchi & Schulz13,14 .
As usual in these cases the RG procedure manifest itself in a flow of the coupling constants of the problem
as the ultraviolet cut off is reduced from 1/α to 1/α′ .
In our problem these coupling constants are: The interaction strengths, Kν ; The mode velocities, uν ; The
g
dimensionless backscattering constant, y = πus ; And the
α
dimensionless pair tunneling strength ∆ = ∆ us . The
flow equations of these coupling constants (see Appendix
for the details of the derivation) are:

√
√
2πφs (x, τ ) cos 2πθc (x, τ ) ,

2

us
dKc
∆
= Xc
dl
uc
y2
dKs
2
= −Ks Xs ∆2 +
dl
2

(12)
2
t0 N0 πf
p

2

where ∆ ≃
. Here N0 is the normal density of
states of the SC at the Fermi level, and pf is the Fermi
momentum.
In terms of the original fermionic fields, the tunneling term has the form of the regular pairing term in the
standard BCS theory:

d∆
dl
dy
dl
duc
dl
dus
dl

¯
¯
Spp = ∆ dxdτ ψ↑ (x, τ )ψ↓ (x, τ )+ψ↓ (x, τ )ψ↑ (x, τ )
In the absence of electron-electron interactions, this
term, along with the free quadratic kinetic term of the
model can be diagonalized by the standard Bogolubov
transformation. The excitation spectrum of this system
will be gapped with an energy of ∆.

Xc(s) =

Wc(s) =

1
2Ï€
1
2Ï€

2Ï€

dϕ cos2 (ϕ) +

0
2Ï€
0

dϕ cos(2ϕ) ×

(13)
(14)

1
−1
= ∆(2 − (Ks + Kc + y))
2

(15)

= y(2 − 2Ks ) − 2Xs ∆2

(16)

−1
= uc Kc Wc

us
∆
uc

= us Ks Ws ∆2

2

(17)
(18)

where dl = d log α is the dimensionless change in the
ultraviolet cutoff,

us(c)
uc(s)

−gs(c)
2

2

cos2 (Ï•) +

sin2 (Ï•)
us(c)
uc(s)

(19)
−gs(c)
2

2

sin2 (Ï•)

(20)

−1
and gc = Kc while gs = Ks .

We shall restrict our analysis to the spin symmetric
case (i.e. the situation where the interactions between
electrons with parallel and opposite spins are identical).
In this case14 Ks ≃ 1+ y , and equations (14) and (16) for
2
the dimensionless backscattering y and spin interaction

strength Ks reduce to:
dy
= −y 2 − 2Xs ∆2 ,
dl
thus maintaining spin invariance.

(21)

4
A.

Analysis of the Flow Equations
0.33

(0)
(0)
4 − (Ks + Kc

−1

+ y (0) ) > 0,

(22)

since the flow equations can only drive Kc/s to be more
”attractive”. In Fig. 2 we depict the RG flow in the ∆-Kc
plane, in the spin symmetric case, for several initial val0
ues of Kc in the range between 0.32-0.325. Fig. 3 shows
the separatix between relevant and irrelevant tunneling
for initial bare (repulsive) values of Kc and y. One can
observe that Kc must be smaller than 1 for the tunneling
3
to be irrelevant.

relevant tunneling
SC regime
0.325
Kc

The above RG equations imply that the charge and
spin velocities, uc and us , renormalize towards each
other. This is a consequence of the correlations between
spin and charge excitations generated by the proximity
effect. From Eqs. (17), (18), and (20) one can observe
that the signs of Wc and Ws are determined by the ratio
uc /us in such a way that the velocities approach each
other.
Another consequence of the RG equations,(13) and
(14), is that Kc can only grow, while Ks can only be
reduced. This flow stem from the finite value of the tunneling parameter ∆ (and y). Thus the RG behavior of ∆
controls the behavior of the system.
The separatix between the regions where ∆ is relevant
or irrelevant can be obtained numerically for a given set
of bare parameters. It is clear that for initially repulsive
(0)
(0)
interactions (Ks > 1, y (0) > 0, Kc < 1), ∆ will be
relevant if:

irrlevant tunneling
0.32

0.315

−0.2

0

0.2

0.4

0.6

0.8

y

Figure 3: The curve describes the separatix between relevant
tunneling, where the system flows to a superconducting state,
and irrelevant tunneling, where the system flows to a decoupled wire, as a function of the bare values of Kc and y

In the situation where ∆ is relevant, the system flows
to a singlet superconducting state. The interaction parameters become attractive and spin-charge separation is
no longer valid. Yet, before the interaction parameters
obtain their asymptotic values (i.e Kc → ∞, Ks → 0
and y → −∞ which correspond to ”infinitely” attractive
interactions), the RG equations (13- 18) will cease to be
valid since the small parameters of our perturbation theory, ∆ and y, will flow to the strong coupling regime.

B.

Length Dependance of the Effective Gap

0.05
0.04
0.03
0.02
0.01

0.325

0.33

0.335

0.34

Kc

Figure 2: Plots of the flow of ∆ as a function of Kc for initial
(0)
values of Kc between 0.32-0.325. The transition between
irrelevant (∆ → 0) and relevant (∆ → ∞) tunneling occurs
(0)
at Kc ∼ 0.3223. The initial values of ∆(0) and y (0) are 0.05
and 0.1 respectively

In the case where ∆ is irrelevant, the wire effectively decouples from the superconductor. The pair tunneling between the two systems is suppressed and the
superconductor-wire junction becomes insulating. Notice
however that this behavior takes place at very strong re1
pulsive interactions, Kc < 3 , where the system tends to
13
Wigner crystallize .

In what follows we consider a weakly interacting Luttinger liquid, where the bare interaction parameters are
close to unity. Moreover, in order to ensure that the perturbative RG equations (13- 18) remain valid we shall assume that L, the embedding length, is of order or smaller
then the bare superconducting correlation length, vf /∆.
Since L serves as an infrared cutoff for the RG flow, this
condition ensures that the this flow is confined to the
perturbative regime. We shall also assume that the temperature is much smaller than ∆, and approximate the
velocities of the spin and charge sectors by their asymptotic renormalized values: uc = us = vf .
In the limit of weak interactions the RG equations for
Ks/c are of second order in the perturbation parameters
˜
(∆ ≪ 1, y ≪ 1), while the equation for ∆ (Eq. 15) is
of first order (neglecting the y dependance, which is of
second order as well). The equation for the coupling y
(Eq. 16) is also second order in small parameters, at least
if we consider the spin symmetric case . Thus one can
assume Kc , Ks , and y to be approximately constants,
and consider the simplified equation for ∆:
d∆
= γ ∆,
dl

(23)

5
where
1
−1
γ = 2 − (Ks + Kc ).
2

(24)

Integrating the above equations from l = 0 to l =
˜
log(L/α), and using the relation ∆ = ∆α/vf , we obtain
the renormalized value of the gap of the wire embedded
in the superconductor:
∆ef f = ∆

L
α

vf
=ξ
∆ef f

L
α

e
where G0 S = 2G0 N = 4 2π¯ represents the conductance
N
N
h
of an ideal junction where only perfect Andreev reflections take place, and

(26)

THE WIRE-SUPERCONDUCTOR
JUNCTION

At this stage of the RG procedure (pursued up to the
scale L) the junction between the wire and the superconductor may be assumed to be point-like. Thus if the temperature (or the applied voltage) is smaller than ¯ vf /L
h
one may continue to integrate out the high energy degrees of freedom in the part of the wire which is not embedded into the superconductor down to the relevant energy scale. This may be achieved following the procedure
described in the literature9 . The important ingredient,
now, is the magnitude of normal back scattering from
the junction. The latter is of order of rN ≃ r0 e−2L/ξef f
where r0 is system specific reflection amplitude in the
absence of superconductivity, while ξef f is the correlation length (26) within the part of the wire embedded in
the superconductor. This behavior of rN results form the
fact that only the charge that is not converted to the condensate backscatters from the edge of the wire embedded
in the superconductor. According to5 the normal current
reduces exponentially with the distance on a length scale
of ξ , which gives the above estimate for rN . Now, since
ξef f depends on L, the magnitude of the normal reflection has a stretched exponential dependence:
rN ≃ r0 e−aL

γ

2
δG ∝ r0 e−2aL T −2(1−Kc ) .

(25)

1−γ

(27)

where a = 2α1−γ /ξ, and the amount of stretch, γ, is dictated by the nature of the electron-electron interactions
within the wire, as follows from (24).

(28)

2

γ

.

is larger than the bare correlation length ξ = vf /∆.
The above results are valid as long as the RG flow stays
˜
within the perturbative regime, namely ∆ < 1. This con1
dition implies that L should be shorter than ξ(ξ/α) γ −1 .

IV.

G = G0 S − δG,
N

γ−1

In particular, repulsive interactions (Kc < 1 and Ks > 1,
and therefore γ − 1 < 0) reduce the effective gap in the
wire.
This renormalization of the gap implies that the effective correlation length,
ξef f =

The behavior of the backscattering, clearly, manifests
itself in the conductance of the junction. In the limit of
sufficiently high temperatures, one obtains8 :

(29)

The above formula holds for the range of temperature
where δG ≪ G0 S , since δG cannot be larger than G0 S .
N
N
Nevertheless, it has been shown6 -10 , that any scatterer,
at a point contact between a wire with repulsive interactions and a SC, will eventually drive the conductance
to zero as the temperature is lowered. Since any finite
length junction will have some backscattering, the conductance should drop to zero for low enough temperatures, as illustrated in Fig. 4. The crossover temperature, T ∗ from the conductive and the insulating behavior
of the junction depends on the length of the wire embedded in the superconductor and its scaling behavior may
be deduced from (28) and (29):
a

T ∗ ∝ e− 1−Kc L

γ

(30)

This result implies that the temperature scale at which
the effects of backscattering becomes substantial reduce
as a stretched exponent with the length of the junction, and the stretch is determined by the interactions,
through the parameter γ.
.
V.

SUMMARY AND CONCLUSIONS

In this work we studied a junction of a 1-D wire embedded a certain length, L, into a bulk superconductor.
We first characterized the nature of the contact between
the 1-D wire and the superconductor using a real space
RG scheme. We found that repulsive interactions in the
wire compete against the superconducting order being
imposed by the bulk superconductor. The system can
flow to either of two phases, depending on the nature of
the interactions. When the interactions are strongly repulsive the tunneling between the two systems becomes
irrelevant and the wire essentially decouples from the
bulk superconductor. For moderate repulsive interactions, and for attractive interactions, tunneling is relevant, and the bulk superconductor induces superconducting order in the wire. The gap opened in the wire
depend on the tunneling strength, and electron-electron
interactions modify its nominal value.
The finite length of the part of the wire embedded in
the superconductor, L, implies that the RG flow, in general, does not reach its asymptotic (non-perturbative)
limit. Thus L introduces itself in the behavior of the effective gap, and the effective correlation length, in the

6
Dror Orgad. This work has been supported in part by
the Israel Science Foundation (ISF) funded by the Israeli
Academy of Science and Humanities, and by the USAIsrael Binational Science Foundation (BSF).

G

0
2GNN

VI.

T ∗ ~e

−

a
Lγ
1− K c

APPENDIX

The mathematical formulation used in this work follows closely the real space RG procedure used by Giamarchi & Schulz13,14 : In this procedure one evaluates a
correlation function in the wire, of the form (the time
ordering symbol is suppressed):

T

Rϕ (xa , τa ; xb , τb ) = eiγ
Figure 4: A schematic graph of the conductance as a function
of temperature for repulsive interactions in the wire. The decay of the conductance depends on the length of the junction
as a stretch exponent. The longer the junction, the lower the
temperature at which the decay sets in. In the limit of weak
backscattering, represented by the solid curve, we can estimate the temperature dependance of the conductance through
Eq. (29). In the zero temperature limit, represented by the
dashed curve, any initial finite backscattering will eventually
drive the conductance to zero. The crossover temperature,
Eq. (30) between the two behaviors is estimated to scale as a
stretched exponent in the length of the junction.

Here ∆ =

∆α
us

∞
α

dr
α

is the dimensionless tunneling parameter,

,

(31)

where ϕ can symbolize any of the boson fields and γ is
some constant. For the following discussion it will suffice to examine only one of the sectors, for instance the
spin sector. The same considerations can be carried on
straightforwardly for the charge sector.
The fact that the relevant boson fields in the spin sector
is φs leads naturally to the evaluation of the correlation
functions:
Rφs (xa , τa ; xb , τb ) = eiγ

√

2π(φs (xa ,τa )−φs (xb ,τb ))

. (32)

Unfortunately this correlation function cannot be calculated exactly using the complete effective action (11).
Though, if the tunneling and backscattering parameters,
(t and g respectively), are small then it may be computed
perturbatively. To second order in these parameters this
function is found to be:

wire. This, in turn, dictates a stretched exponential behavior of the normal reflection from the junction as function of L. For energy scales (temperature or voltages)
beneath the effective gap, we described the qualitative
picture of the conductance as a function of temperature
and length of the embedded segment of wire, see Fig. 4.
Several simplifications have been used for our analysis:
One is that our model treats a semi-infinite wire with
a single junction, while in practical situations a finite
wire is usually connected to two reservoirs. This idealization holds as long as the segment of the wire outside
the superconductor is long enough compared to ¯ vf /T ,
h
where vf is the Fermi velocity and T is the temperature.
Additional simplification is the assumption that the embedding of the wire into the bulk superconductor does
not introduce inhomogeneities, i.e. the wire can be still
considered to be clean, and that the tunneling to the
superconductor is homogenous along the wire. This approximation holds when the transport mean free path in
the wire is longer than L. In the opposite limit one expects a different behavior of the proximity effect which
will change the conductance of the system.
The authors gratefully acknowledge discussions with

ef
2
Ks f = Ks − Ks ∆2 Xs

√
2π(ϕ(xa ,τa )−ϕ(xb ,τb ))

Rs (ra,b ) = e

ef
ef
−γ 2 Ks f Fs (ra,b )+Ds f sin2 (ϕra,b ,s )

,

(33)

where ra,b = ra − rb , and ϕr,s is the angle between the
vector r = (x, us Ï„ ) and the x axis. The function Fs is
(at zero temperature)13 :
Fs (x, Ï„ ) =

1
ln
2

x2 + (us |τ | + α)2
α2

.

(34)

Apart from the term proportional to sin2 (Ï•ra,b ,s ), this
functional form is identical to the free correlation function:
(0)
Rs (ra − rb ) = e−γ

2

Ks Fs (ra −rb )

,

(35)

ef
but with an effective Luttinger interaction constant Ks f
modified by the perturbations:

r
α

−1
3−Ks −Kc

y=

g
Ï€us

+

y2
2

dr
α

r
α

3−4Ks

.

(36)

is the dimensionless spin backscattering param-

7
Applying the same renormalization scheme to Def f ,
will generate the flow of the parameter Ds :

eter, and Xs is a geometrical term given by:

Xs =

2Ï€

1
2Ï€

dϕ cos2 (ϕ)+

0

uc
us

−1
−Kc
2

2

sin2 (Ï•)

Integrating out high energy degrees of freedom, near
the ultraviolet cutoff, corresponds to integrating out a
”small ring” between α → α′ = α + dα, where α is the
small distances parameter of the model. After integration
and rescaling, an infinitesimal change is generated in the
ef
ef
expressions for Ks f . In order to keep Ks f constant
with the reduction of the cutoff, it is required that the
bare parameters change. For instance, we find that:
2
Ks (α′ ) = Ks (α)−Ks (α) ∆2 (α)Xs (α)+

dDs
2
= Ks (l)Ws (l)∆2 (l).
dl

(37)

y 2 (α) dα
(38)
2
α

which generates an (exact) differential equation for Ks :
dKs
y 2 (l)
2
= −Ks (l) Xs ∆2 (l) +
dl
2

(39)

It should be noted that Ds is initially zero but is generated under renormalization.
The parameter Ds controls the renormalization of the
velocity parameter. As long as the space and time directions are isotropic the velocity parameter does not
flow under renormalization, but they should flow in the
anisotropic case. Indeed, assuming that initially the correlation function is described by the function Ks Fs (ra,b ),
then a small change of dus will generate the term:
2
Ks
us sin (ϕra,b ,s ) · dus . Up to a factor, this is exactly the
anisotropy term. Therefore, the renormalization of Ds
is equivalent to that of the velocity us by the following
u
relation (e.g. Eq. 33): dus = Ks dDs .
dl
s dl
The flow equations for the charge sector can be obtained by an identical procedure. The equations obtained
are:

(here dl = dα ).
α
In a similar fashion one obtains differential equations
for the parameters ∆ and y:
d∆2
−1
= ∆2 (4 − Ks − Kc ),
dl
dy 2
= y 2 (4 − 4Ks ).
dl

The sin2 (Ï•ra,b ,s ) contribution to the correlation function arises from the fact that the tunneling perturbation
couples the spin and charge sectors, which were uncoupled without this term. Mathematically, this term characterizes the anisotropy between the space (x) and time
(us τ ) directions. It’s pre-factor Def f is given by:
ef
2
Ds f = Ds + Ks ∆2 Ws

∞
α

dr
α

r
α

−1
3−Ks −Kc

dKc
= = Xc (l)
dl

,

(42)

1
Ws =
2Ï€

1
2
3

4

0

uc
dϕ cos(2ϕ) cos2 (ϕ)+
us

2

sin2 (Ï•)

2

,

us
∆(l)
uc

(44)
2

.

(45)

Finally these equations cannot be exactly correct, since
they do not maintain the spin invariance SU(2) of a model
that was spin invariant to begin with. (A spin symmetric
model is one where the interactions between electrons
of opposite and parallel spin are identical). Since the
perturbations do not break this symmetry, something in
the above result is insufficient. Indeed, it turns out that
the remedy for this problem lies in the inclusion of the
third order terms of perturbation theory. This correction
is presented in reference14 . It affects only the equations
for ∆ (Eq. (40))and y (Eq. (41)) which become:
d∆
1
−1
= ∆(2 − (Ks + Kc + y)),
dl
2
dy
= y(2 − 2Ks ) − 2Xs ∆2 .
dl

where Ws is another geometric term factor:
2Ï€

us
∆(l)
uc

dDc
−2
= Kc (l)Wc (l)
dl

(40)
(41)

(43)

−1
−Kc
2

(46)
(47)

.

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5
6

7
8
9

G. E. Blonder, M. Tinkham and T.M. Klapwijk, Phys.
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8

10

11
12

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There are actually quite a few different conventions used in
√
the bosonization literature. Apart from a factor of π in

13
14

the normalization of the bosonic fields, we follow the one
used by13 .
T. Giamarchi, Quantum Physics in One Dimension.
Clarendon Press, Oxford (2004).
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(1988).