Critical Behavior of a Point Contact in a Quantum Spin Hall Insulator
Jeﬀrey C.Y. Teo and C.L. Kane

arXiv:0904.3109v1 [cond-mat.mes-hall] 21 Apr 2009

Dept. of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104
We study a quantum point contact in a quantum spin Hall insulator. It has recently been shown
that the Luttinger liquid theory of such a structure maps to the theory of a weak link in a Luttinger
liquid with spin with Luttinger liquid parameters gρ = 1/gσ = g < 1. We show that for weak
interactions, 1/2 < g < 1, the pinch-oﬀ of the point contact as a function of gate voltage is
controlled by a novel quantum critical point, which is a realization of a nontrivial intermediate ﬁxed
point found previously in the Luttinger liquid model with spin. We predict that the dependence of
the conductance on gate voltage near the pinch-oﬀ transition for diﬀerent temperatures collapses
onto a universal curve described by a crossover scaling function associated with that ﬁxed point.
We compute the conductance and critical exponents of the critical point as well as the universal
√
scaling function in solvable limits, which include g = 1 − ǫ, g = 1/2 + ǫ and g = 1/ 3. These
results, along with a general scaling analysis provide an overall picture of the critical behavior as a
function of g. In addition, we analyze the structure of the four terminal conductance of the point
contact in the weak tunneling and weak backscattering limits. We ﬁnd that diﬀerent components
of the conductance can have diﬀerent temperature dependence. In particular, we identify a skew
conductance GXY , which we predict vanishes as T γ with γ ≥ 2. This behavior is a direct consequence
of the unique edge state structure of the quantum spin Hall insulator. Finally, we show that for
strong interactions g < 1/2 the presence of spin non conserving spin orbit interactions leads to a
novel time reversal symmetry breaking insulating phase. In this phase, the transport is carried by
spinless chargons and chargeless spinons. These lead to nontrivial correlations in the low frequency
shot noise. Implications for experiments on HgCdTe quantum well structures will be discussed.
PACS numbers: 71.10.Pm, 72.15.Nj, 85.75.-d

I.

INTRODUCTION

A quantum spin Hall insulator (QSHI) is a time reversal invariant two dimensional electronic phase which
has a bulk energy gap generated by the spin orbit
interaction1,2 . It has a topological order3 which requires
the presence of gapless edge states similar to those that
occur in the integer quantum Hall eﬀect. In the simplest version, the QSHI can be understood as two time
reversed copies of the integer quantum Hall state4 for up
and down spins. The edge states, which propagate in
opposite directions for the two spins, form a unique one
dimensional system in which elastic backscattering is forbidden by time reversal symmetry1 . This state occurs in
HgCdTe quantum well structures5 , and experiments have
veriﬁed the basic features of the edge states, including the
Landauer conductance6 2e2 /h, as well as the non locality
of the edge state transport7 .
In the presence of electron interactions, the edge states
form a Luttinger liquid8,9,10,11,12,13,14 . For strong interactions (when the Luttinger liquid parameter g < 3/8)
random two particle backscattering processes destabilize
the edge states, leading to an Anderson localized phase.
For g > 3/8 (or a suﬃciently clean system), however, one
expects the characteristic power law behavior for tunneling of a Luttinger liquid.
A powerful tool for probing edge state transport experimentally is to make a quantum point contact. As
depicted in Fig. 1(a,b), a gate voltage controls the coupling between edge states on either side of a Hall bar as
the point contact is pinched oﬀ. Recently, the point con-

tact problem for a QSHI has been studied10,11 . Hou, Kim
and Chamon10 made the interesting observation that the
QSHI problem maps to an earlier studied model15,16 of
a weak link in a spinful Luttinger liquid (SLL), in which
the charge and spin Luttinger parameters are given by
gρ = g and gσ = 1/g 17 . For suﬃciently strong interactions (g < 1/2) they found that the simple perfectly
transmitting and perfectly reﬂecting phases are both unstable. They showed that as long as spin is conserved
at the junction the low energy behavior is dominated by
a non trivial “mixed” ﬁxed point of the SLL, in which
charge is reﬂected but spin is perfectly transmitted. This
charge insulator/spin conductor (IC) phase leads to a
novel structure in the four terminal conductance of the
point contact.
In this paper, we will focus on the QSHI point contact for weaker interactions, when 1/2 < g < 1. In this
regime the open limit (or weak backscattering, “small v”)
and the pinched oﬀ limit (or weak tunneling, “small t”)
are both stable perturbatively. This is diﬀerent from the
behavior in an ordinary Luttinger liquid15,16,18 or a fractional quantum Hall point contact19,20 . In those cases
the perfectly transmitting limit is unstable for g < 1.
Weak backscattering is relevant and grows at low energy,
leading to a crossover to the stable perfectly reﬂecting
ﬁxed point. The fact that both the small v and the small
t limits are stable for the QSHI point contact means that
there must be an intermediate unstable ﬁxed point which
separates the ﬂows to the two limits. This unstable ﬁxed
point describes a quantum critical point where the point
contact switches on as a function of the pinch-oﬀ gate
voltage. We will argue that in the limit of zero tempera-

2

VG

1

QSHI
4

2

VG

1

QSHI
VG

QSHI
3

2

QSHI
VG

4

(a)

3

(b)
G
2e2/h

G*
α

T
(c)

VG
*
VG

FIG. 1: A quantum point contact in a QSHI, controlled by
∗
a gate voltage VG . In (a) VG < VG , and the point contact
is pinched oﬀ. The spin ﬁltered edge states are perfectly
∗
reﬂected. In (b) VG > VG , and the point contact is open.
The edge states are perfectly transmitted. In (c) we plot the
conductance (later deﬁned as GXX ) as a function of VG for
diﬀerent temperatures. As the temperature is lowered, the
pinch-oﬀ curve sharpens up with a width T α . The curves
cross at a critical conductance G∗ , and the shape of the curve
has the universal scaling form (1.1). The plotted curves are
based on Eq. 3.9, valid for g = 1 − ǫ, which is computed in
Section III.C.

ture the point contact switches on abruptly as a function
∗
of gate voltage VG , with conductance G = 0 for VG < VG
∗
and G = 2e2 /h for VG > VG . At ﬁnite but low temperature T , the shape of the pinch-oﬀ curve G(VG , T ) is
controlled by the crossover between the unstable and stable ﬁxed points, and is described by a universal crossover
scaling function,
lim

∆VG ,T →0

G(VG , T ) =

2e2
∆VG
Gg (c αg ).
h
T

(1.1)

∗
Here ∆VG = VG − VG and c is a non universal constant.
αg is a critical exponent describing the unstable intermediate ﬁxed point. Gg (X) is a universal function which
crosses over between 0 and 1 as a function of X. αg and
Gg (X) are completely determined by the Luttinger liquid
parameter g. This behavior means that as temperature is
lowered, the pinch-oﬀ curve as a function of VG sharpens
up with a characteristic width which vanishes as T αg , as
shown schematically in Fig. 1(c). The curves at diﬀerent
low temperatures cross at G∗ = Gg (0), the conductance
g
of the critical point. Eq. 1.1 predicts that data from
diﬀerent temperatures can be rescaled to lie on the same
universal curve.
The crossover scaling function Gg (X) is similar to the
scaling function that controls the lineshape of resonances

in a Luttinger liquid16,21 and in a fractional quantum Hall
point contact19 . That scaling function was computed exactly for all g by Fendley, Ludwig and Saleur22 using the
thermodynamic Bethe ansatz. That problem, however,
was simpler than ours because the critical point occurs
at the weak backscattering limit, which is described by
a boundary conformal ﬁeld theory with a trivial boundary condition23 . The intermediate ﬁxed point relevant to
our problem has no such simple description. Thus, even
the critical point properties αg and G∗ (which were simg
ple for the resonance problem) are highly nontrivial to
determine.
Intermediate ﬁxed points in Luttinger liquid problems
were ﬁrst discussed in Refs. 15 and 16 in the context of
SLLs. However, for that problem they occur in a rather
unphysical region of parameter space gσ > 2, because
spin rotational invariance requires gσ = 1. To our knowledge, the QSHI point contact provides the ﬁrst physically
viable system to directly probe these non trivial ﬁxed
points.
The existence of the intermediate ﬁxed points can be
inferred from the stability of the simple perfectly transmitting or reﬂecting ﬁxed points15,16 . However their
properties are diﬃcult to compute, and a general characterization of these critical points remains an unsolved
problem in conformal ﬁeld theory24 . Two approaches
have been used to study their properties. In Ref. 16,
a perturbative approach was introduced which applies
when the Luttinger parameters are close to their criti∗
∗
cal values gσ and gρ , where the simple ﬁxed points become unstable. (For instance, for the weak backscatter∗
∗
∗
ing limit, gρ = 1/2, gσ = 3/2). For gρ,σ = gρ,σ − ǫ, the
ﬁxed point is accessible in perturbation theory about the
simple ﬁxed point, and it’s properties can be computed
in a manner analogous to the ǫ expansion in statistical
mechanics.
An alternative approach is to map the theory for speciﬁc values of gρ and gσ onto solvable models. In Ref.
25, Yi and Kane recast the Luttinger liquid barrier problem as a problem of quantum Brownian motion (QBM)
in a two dimensional periodic potential. When gρ = 1/3,
gσ = 1 and the potential has minima with a honeycomb
lattice symmetry, a stable intermediate ﬁxed point which
occurs in that problem was identiﬁed with that of the
3 channel Kondo problem. This, in turn is related to
the solvable SU (2)3 Wess Zumino Witten model26 , allowing for a complete characterization of the ﬁxed point.
This idea was further developed by Aﬄeck, Oshikawa and
Saleur24 , who provided a more general characterization of
the ﬁxed point in terms of the boundary conformal ﬁeld
√
theory of √ three state Pott’s model. For gρ = 1/ 3
the
and gσ = 3 the QBM model with triangular lattice symmetry has an unstable intermediate ﬁxed point, which we
will see is related to the ﬁxed point of the QSHI problem. In Ref. 25 symmetry arguments were exploited to
determine the critical conductance G∗ in that case.
In this paper we will compute αg and Gg (X) (along
with a multiterminal generalization of the conductance)

3
in three solvable limits:
(i) For g = 1 − ǫ, we will perform an expansion for
weak electron interactions. For non interacting electrons
the point contact can be characterized in terms of a scattering matrix Sij , for arbitrary transmission. Weak interactions lead to a logarithmic renormalization of Sij .
Following the method developed by Matveev,Yue and
Glazman27 , this allows Gg (X) and αg to be calculated
exactly in the limit g → 1.
(ii) For g = 1/2 + ǫ we ﬁnd that the intermediate
ﬁxed point approaches the charge insulator/spin conductor ﬁxed point, allowing for a perturbative calculation of
the ﬁxed point properties G∗ and αg to leading order in
g
ǫ. Moreover, for g = 1/2 the Luttinger liquid theory can
be fermionized, which allows the full crossover function
Gg (X) to be determined in that limit.
√
(iii) For g = 1/ 3 the self duality argument developed
in Ref. 25 allows us to compute the ﬁxed point conductance G∗ exactly.
These three results, along with the general scaling
analysis provide an overall picture of the critical behavior
of the QSHI point contact as a function of g.
In addition to the analysis of the pinch-oﬀ transition
discussed above, we will touch on two other issues in this
paper. First, we will introduce a convenient parameterization of the four terminal conductance as a 3×3 conductance matrix. In this form symmetry constraints on the
conductance are reﬂected in a natural way. Moreover, we
will predict that diﬀerent components of the conductance
matrix have diﬀerent temperature dependence at the low
temperature ﬁxed points. In particular, we will introduce
a “skew” conductance GXY , which is predicted to vanish
as T γ with γ ≥ 2. For non interacting electrons we will
show that GXY = 0, and for weak interactions γ = 2.
This behavior is a direct consequence of the spin ﬁltered
nature of the edge states, and does not occur in a generic
four terminal conductance device. It is thus a powerful
diagnostic for the edge states.
Secondly, we will examine the role of spin orbit terms
at the point contact which respect time reversal symmetry but violate spin conservation. For g > 1/2 we will
provide evidence that such terms are irrelevant at the intermediate critical ﬁxed point, so that they are unimportant for the critical behavior of the point contact. However, for g < 1/2, such terms are relevant. Hou, Kim and
Chamon10 pointed out that these terms are relevant perturbations at the charge insulator/spin conductor ﬁxed
point for g < 1/2, but they did not identify the stable
phase to which the system ﬂows at low energy. We will
argue that the system ﬂows to a time reversal symmetry breaking insulating state in which the four terminal
conductance Gij = 0. Since spin orbit interaction terms
will generically be present in a point contact, the true
low energy behavior of a point contact will be described
by this phase. An interesting consequence of the broken
time reversal symmetry of this phase is that the weak
tunneling processes which dominate the conductance at
low, but ﬁnite temperature are not electron tunneling

processes. Rather, they involve the tunneling of neutral
spinons and spinless chargons. This has nontrivial implications for four terminal noise correlation measurements.
A related eﬀect has been predicted by Maciejko et al.12
for the insulating state of a single impurity on a single
edge of a QSHI. This insulating state, however, requires
stronger electron electron interactions. It occurs in the
regime g < 1/4, where weak disorder already leads to
Anderson localization.
This paper is organized as follows. In section II we
discuss our model and analyze ﬁve stable phases. In addition to the simple ﬁxed points, where charge and spin
are either perfectly reﬂected or perfectly transmitted, we
discuss the time reversal symmetry breaking insulating
phase which occurs for strong interactions with spin orbit. In section III we discuss the critical behavior of the
conductance at the pinch-oﬀ transition. We will begin
in section III.A with a general discussion of the scaling theory and phase diagram, along with a summary
of our results. Readers who are not interested in the
detailed calculations can go directly to this subsection.
In the following subsections we describe the calculations
√
for g = 1/ 3, g = 1 − ǫ and g = 1/2 + ǫ in detail. In
section IV we conclude with a discussion of experimental
and theoretical issues raised by this work. In appendix
A we show describe our parameterization of the four terminal conductance and show that in this representation
symmetry constraints have a simple form.

II.

MODEL AND STABLE PHASES

In this section we will describe the Luttinger liquid theory of the QSHI point contact. We will begin in section
II.A by describing the Luttinger liquid model ﬁrst for a
single edge and then relating the four edges to the theory of the SLL. We then discuss the four terminal conductance. In section II.B we describe the simple limits of our
model which correspond to stable phases. The simplest
limits are the perfect transmission limit, or charge conductor/spin conductor (CC), the perfect reﬂection limit,
or charge insulator/spin insulator (II). In addition we will
discuss the “mixed” phases, including the charge insulator/spin conductor (IC) and the charge conductor/spin
insulator (CI).
For most of this section we will assume that spin is conserved. While spin nonconserving spin orbit interactions
are allowed and will generically be present we will argue
that they are irrelevant for the ﬁxed points and crossovers
of physical interest. An exception to this, however, occurs for strong interactions when g < 1/2. This will be
discussed in section II.B.5, where we will show that there
are relevant spin orbit terms which destabilize the CC,
II and IC phases. We will argue that these perturbations ﬂow to a diﬀerent low temperature phase, which
we identify as a time reversal symmetry breaking insulator (TBI). In that section we will explore the transport
properties of that state.

4
Much of the theory presented in this section is contained either explicitly or implicitly in the work Hou, Kim
and Chamon10 , as well as in Refs. 11,15,16. We include it
here to establish our notation and to make our discussion
self contained. We will highlight, however, three results
of this section which are original to this work. They include (1) our analysis of the four terminal conductance,
which predicts that diﬀerent components of the conductance matrix have diﬀerent temperature dependence. In
particular, we ﬁnd that the skew conductance GXY vanishes at low temperature as T γ with γ ≥ 2. (2) In section II.B.5 we introduce the TBI phase discussed above.
(3) We introduce a perturbative analysis of the IC and
CI phases in section II.B.3 and II.B.4. While this was
partially discussed in Ref. 16, we will show that a full
analysis requires the introduction of a pseudo-spin degree
of freedom in the perturbation theory. This new pseudospin does not aﬀect the lowest order stability analysis of
the IC phase, but it will prove crucial for the second order renormalization group ﬂows, which will be used in
the ǫ expansion in Section III.D.

Then,
i
H0 =

v0
(1 + λ4 ) (∂x φi,in )2 + (∂x φi,out )2
4π
− 2λ2 ∂x φi,in ∂x φi,out ] ,

where λi = ui /(2πv0 ). Changing variables
φi,in
φi,out

=

1
2g

1+g 1−g
1−g 1+g

H0 =

v
˜
˜
(∂x φi,in )2 + (∂x φi,out )2 ,
4πg

4

∞

H0 =
i=1

0

i
dxi H0 ,

z
˜
˜
[φi,a (x), φj,b (y)] = iπgδij τab sgn(x − y).

†
†
i
H0 = iv0 (ψi,in ∂x ψi,in − ψi,out ∂x ψi,out )

†
†
+ u2 ψi,in ψi,in ψi,out ψi,out
(2.2)
1
†
†
+ u4 (ψi,in ψi,in )2 + (ψi,out ψi,out )2 .
2
Here ψi,in and ψi,out are a time reversed pair of fermion
operators with opposite spin which propagate toward and
away from the junction. v0 is the bare Fermi velocity,
and u is electron interaction strength. u2 and u4 are forward scattering interaction parameters. The boundary
condition on the fermions at x = 0 is determined by the
transmission of the point contact, and will be discussed
in various limits below.

1.

z
[φi,a (x), φj,b (y)] = iπδij τab sgn(x − y).

1 + λ4 − λ2
.
1 + λ4 + λ2

2.

(2.4)

(2.9)

Mapping to Spinful Luttinger liquid

Consider an open point contact in a Hall bar geometry
with edge states on the top and bottom edges which continuously connect leads 1 and 2 and leads 3 and 4. We
then deﬁne left and right moving ﬁelds with spin ↑, ↓ as
φR↑
φL↓
φL↑
φR↓

= φ1,in (−x)θ(−x) + φ2,out (x)θ(x)
= φ2,in (x)θ(x) + φ1,out (−x)θ(−x)
= φ3,in (x)θ(x) + φ4,out (−x)θ(−x)
= φ4,in (−x)θ(−x) + φ3,out (x)θ(x).

(2.10)

It is then useful to deﬁne sum and diﬀerence ﬁelds as
φaσ =

1
(ϕρ + σϕσ + aθρ + aσθσ ),
2

(2.11)

where a = R, L = +, − and σ =↑, ↓= +, −. Then, θα
and ϕα obey,
[θα (x), ϕβ (y)] = 2πiδαβ θ(x − y),

where a = in, out, and xc is a short distance cutoﬀ. ψi,a
obey the Kac Moody commutation algebra,

(2.8)

The Luttinger liquid parameter g determines the power
law exponents for various quantities. For instance the
tunneling density of states scales as ρ(E) ∝ E (g+1/g)/2−1 .

Bosonization of a single edge

We ﬁrst consider the Luttinger liquid theory for a single edge. We thus bosonize according to
1
(2.3)
eiφi,a ,
ψi,a = √
2πxc

(2.7)

(1 + λ4 )2 − λ2 and
2

Here v = v0

(2.1)

with

(2.6)

˜
where φi,a obey

Model

The edge states on the four edges in Fig. 1(a,b) emanating from the point contact may be described by the
Hamiltonian

˜
φi,in
˜
φi,out

transforms (2.5) into a theory of decoupled chiral bosons

g=
A.

(2.5)

(2.12)

and (2.3, 2.5) become10
H0 =

∞
−∞

dx

v
1
ga (∂x ϕa )2 + (∂x θa )2 .
4π
ga
a=σ,ρ
(2.13)

where
gρ = g,

gσ = 1/g,

(2.14)

5
O ΘOΘ−1 MX OM−1 MY OM−1
X
Y
θρ
θρ
−θρ
θρ
ϕρ −ϕρ
ϕρ
ϕρ
θσ −θσ
θσ
−θσ
ϕσ ϕσ + π
−ϕσ
−ϕσ

a third voltage VZ which biases leads (13) relative to (24).
Spin conservation then requires
IZ = GZZ VZ ,

and g and v are given in the previous section.
It is useful to list the eﬀect of symmetry operations
on the charge-spin variables, because symmetries constrain the allowed tunneling operators. Charge conservation leads to gauge invariance under the transformation ϕρ → ϕρ + δρ . The conservation of spin Sz leads
to invariance under ϕσ → ϕσ + δσ . The eﬀects of
time reversal and mirror symmetries is shown in Table
I. Time reversal symmetry is speciﬁed by the operation
Θψaσ Θ−1 = iσψaσ . The mirror MX interchanges leads
¯¯
14 ↔ 23 while MY interchanges leads (12 ↔ 34).
Four Terminal Conductance

The central measurable quantity is the four terminal
conductance, deﬁned by
Gij Vj ,

Ii =

(2.15)

j

where Ii is the current ﬂowing into lead i. Gij is in
general characterize by 9 independent parameters. In
Appendix A we introduce a convenient representation for
these parameters, which simpliﬁes the representation of
symmetry constraints. Here we will summarize the key
points of that analysis.
The presence of both time reversal symmetry and spin
conservation considerably simpliﬁes the conductance. It
is characterized by three independent conductances
IX
IY

=

GXX GXY
GY X GY Y

VX
VY

.

e2
.
h

(2.18)

with
GZZ = 2

TABLE I: The eﬀect of discrete symmetry operations on the
boson ﬁelds θρ and θσ .

3.

(2.17)

(2.16)

Here IX = I1 + I4 is the current ﬂowing from left to right
in Fig. 1, while IY = I1 + I2 is the current ﬂowing from
top to bottom. Similarly, VX is a voltage biasing leads
(14) relative to (23) and VY biases leads (12) relative to
(34). GXX is thus the two terminal conductance measured horizontally, while GY Y is the two terminal conductance measured vertically. GXY = GY X is a “skew
conductance”, which vanishes in the presence of mirror
symmetry. Given these three parameters, the full four
terminal conductance matrix Gij can be constructed using Eq. (A6).
A second consequence of spin conservation is the quantization of a particular combination of Gij . In particular,
in appendix A we deﬁne a third current IZ = I1 + I3 and

Since spin nonconserving spin orbit terms are allowed,
spin conservation will not be generically present in the
microscopic Hamiltonian of the junction. Nonetheless,
we will argue that the low temperature ﬁxed points possess a emergent spin conservation, as well as mirror symmetry, so that (2.18) should hold, albeit with corrections
which vanish as a power of temperature.
B.

Stable Phases

In this section we describe various stable ﬁxed points
which admit simple descriptions using bosonization. We
will ﬁrst focus on the limit in which spin is conserved at
the junction. There are then four simple ﬁxed points15,16 .
These include the perfectly transmitting (CC) limit, in
which both charge and spin conduct, and the perfectly
reﬂecting limit (II) in which both charge and spin are
insulating. The “mixed” ﬁxed points, denoted IC (CI)
are perfectly reﬂecting for charge (spin) and perfectly
transmitting for spin (charge).
In the presence of spin non conserving spin orbit terms
(which preserve time reversal symmetry) an additional
ﬁxed point is possible in which time reversal symmetry
is spontaneously broken. We will see that in the presence
of spin orbit terms this time reversal breaking insulator
(TBI) phase is the stable phase when g < 1/2.
1.

Weak backscattering (CC) limit

We ﬁrst consider the limit where the point contact is
nearly open and assume spin is conserved. It will prove
useful to follow Ref. 16 and write (2.13) as a 0+1 dimensional Euclidean path integral for θρ,σ (τ ) ≡ θρ,σ (x =
0, τ ). This formulation is not essential for carrying out
the perturbative analysis of this ﬁxed point. However,
it is of conceptual value for discussing the duality between diﬀerent phases, which can be understood in terms
of instanton processes in which θρ,σ (τ ) tunnels between
degenerate minima at strong coupling. This is accomplished by setting up the path integral for θσρ (x, τ ) and
then integrating out θσ,ρ (x, τ ) for x = 0. The resulting
theory for θσ,ρ (τ ) has the form of a quantum Brownian
motion model24,25,28,29,30 , described by the Euclidean action
SCC =

1
β

α,ωn

1
|ωn ||θa (ωn )|2 −
2πgα

β
0

dτ
VCC (θσ , θρ ),
τc
(2.19)

6
(a) ve

or

(b)

vρ

(c)

At the ﬁxed point the conductance matrix elements are

vσ

GXX = 2e2 /h
GY Y = GXY = 0.
te
(d)

tσ
(e)

or

tρ

At ﬁnite temperature, there will be corrections to these
values. The leading corrections will depend on the least
irrelevant operators. We ﬁnd

(f)

δGXX =

2
−c1 ve T g+g
2
−c2 vρ T 4g−2

δGY Y =

2
c3 ve T g+g −2
2 4/g−2
c4 vσ T

−1

FIG. 2: Schematic representation of tunneling processes in
(a,b,c) the CC phase (“small v”) and (c,d,e) the II phase
(“small t”). (a) and (d) describe single electron processes,
while the others are two particle processes. The duality relating ve ↔ te , vρ ↔ tσ and vσ ↔ tρ can clearly be seen.

where ωn = 2πn/β are Matsubara frequencies, and
β = 1/kB T . We have included the short time cutoﬀ
τc = xc /v in the second term to make the potential
V (θρ , θσ ) dimensionless. The theory can be regularized
by evaluating frequency sums with a exp(−|ωn |τc ) convergence factor.
The potential V (θρ , θσ ) is given by an expansion in
terms of tunneling operators, which represent the processes depicted in Fig. 2(a,b,c),
VCC = ve cos(θρ + ηρ ) cos θσ + vρ cos 2θρ + vσ cos 2θσ .
(2.20)
ve represents the elementary backscattering of a single
electron across the point contact. The phase of cos θσ
in that term is ﬁxed by time reversal symmetry. The
phase ηρ of cos θρ is arbitrary, though mirror symmetry, if present, requires ηρ = nπ. In addition we include compound tunneling processes. vρ represents the
backscattering of a pair of electrons with opposite spins.
We have chosen to deﬁne θρ such that the phase of this
term is zero. Note that this process involves the tunneling of spin (not charge) between the top and bottom
edges. Similarly, vσ represents the transfer of a unit of
spin from the right to the left moving channels, and involves the tunneling of charge 2e between the top and
bottom edges. In general higher order terms could also
be included. However, those terms are less relevant.
The low energy stability of this ﬁxed point is determined by the scaling dimensions ∆(vα ) of the perturbations, which determine the leading order renormalization
group ﬂows,
dvα /dℓ = (1 − ∆(vα ))vα .

(2.21)

These are given by

∆(vσ ) =

2gσ

(2.22)

= 2g −1 .

It is therefore clear that all operators are irrelevant for
1/2 < g < 2, so that the CC phase is stable. For g < 1/2
vρ becomes relevant, and for g > 2 vσ becomes relevant.

√
g > 1/ 3
√
g < 1/ 3
√
g< 3
√ ,
g> 3

−2

−1

(2.24)

where c√ are nonuniversal constants. Note that for
i
g < 1/ 3 the exponents for GXX and GY Y are different. In addition, there will be power law corrections
to GXY when the mirror symmetries Mx , My are violated. However, this correction is zero when computed
from (2.19,2.20), even when ηρ = 0, due to the symmetry of (2.20) under θσ → −θσ . Computing GXY requires a higher order irrelevant operator. For instance
λ1 ∂x ϕσ sin θρ cos θσ and λ2 ∂x ϕρ cos θρ sin θσ break both
MX and MY , while preserving time reversal. This leads
to
δGXY = c5 λ1 λ2 T g+g

−1

.

(2.25)

Note that the temperature exponent of GXY is at least 2 even for weak interactions g ∼ 1. This is because the tunneling terms λ1 and λ2 include an extra derivative term.
This is related to the fact (which we will show in Section
III.C) that for non interacting electrons GXY = 0. Weak
interactions then introduce inelastic processes which give
GXY ∝ T 2 . The vanishing of GXY is a unique property
of the spin ﬁltered edge states of the QSHI, which does
not occur for a generic four terminal conductance.
2.

Weak Tunneling (II) limit

When the point contact is pinched oﬀ, θρ,σ are eﬀectively pinned, and a theory can be developed in terms of
electron tunneling process across the point contact. This
theory is most conveniently expressed in terms of the dis˜
continuity θσ,ρ ≡ ϕright − ϕleft across the junction31 . The
σ,ρ
σ,ρ
theory takes the form
SII =

∆(ve ) = (gρ + gσ )/2 = (g + g −1 )/2
∆(vρ ) =
2gρ
= 2g

(2.23)

1
β

α,ωn

gα
˜
|ωn ||θa (ωn )|2 −
2π

β
0

dτ
˜ ˜
VII (θσ , θρ ),
τc
(2.26)

with
˜
˜
˜
˜
VII = te cos(θρ + ηρ ) cos θσ + tρ cos 2θρ + tσ cos 2θσ .
(2.27)
As depicted in Fig. 2(d,e,f) te represents the tunneling
of a single electron from left to right across the junction.

7
tσ describes the transfer of a unit of spin across the junction. tρ describes the tunneling of a pair of electrons with
opposite spins.
The relationship between SII and SCC can be understood in two ways. First, since both SII and SCC describe tunneling between the middles of two disconnected
Luttinger liquids (either on the top and bottom of the
junction or the left and right) the two theories are identical. It is straightforward to see that if we make the
identiﬁcation
˜
θρ ↔ θσ
˜
θσ ↔ θρ

2π

tσ

Using this identiﬁcation, the scaling dimensions ∆(tα )
can be read oﬀ from Eq. 2.22. Thus, like the CC phase,
the II phase is stable when 1/2 < g < 2. The low temperature conductance can also be read from (2.23), (2.24)
and (2.25) using the identiﬁcation
GXX ↔ GY Y .

(2.31)

Another way to understand this duality, which will
prove useful below, is to consider an instanton expansion
for strong coupling. For large ve (θρ , θσ ) will be tightly
bound at the minima of V (θρ , θσ ), shown in Fig. 3(a).
(Here we assume for simplicity ηρ = 0.) The partition
function describing the path integral of (2.19) can then
be expanded in instanton processes, in which (θρ , θσ )
switches between nearby minima at discrete times. Evaluating the ﬁrst term in (2.19) for a conﬁguration of instantons leads to an interaction between the instantons
which depends logarithmically on time. The expansion
describes the partition function for a one dimensional
“Coulomb gas”, where the “charges” correspond to the
tunneling events. This Coulomb gas has exactly the same
form as the expansion of (2.26) in powers of te , tρ and
tσ . Thus, we can identify te , tρ and tσ as the fugacity of
the instantons.
This duality argument also works in reverse. Starting from (2.26) we can derive (2.19) by considering large
˜
˜
te and expanding in instantons in θρ and θσ connecting
minima in Fig. 3(b), which have fugacities ve , vρ and vσ .
3.

Charge Insulator/Spin Conductor (IC)

We next study the mixed charge insulator spin conductor phase. To generate the eﬀective action for this

vρ
(b)

θσ

2π

∼
tρ

Thus, the “small v” and “small t” theories are dual to
each other, with the identiﬁcation

∼
θσ
∼
θρ

θρ
2π

(c)

∼
θρ

ve
2π

(a)

(2.29)

(2.30)

vσ

tρ

(2.28)

g ↔ g −1 .

θρ

te

2π

ve ↔ te
vρ ↔ tσ
vσ ↔ tρ

2π

∼
θσ

2π

it follows that
SII (gρ , gσ , te , tρ , tσ ) = SCC (gσ , gρ , ve , vσ , vρ ).

θσ

τz=-1 τz=1

∼
τx=-1

vσ

2π

τx=1

(d)

FIG. 3: (a) Positions of the minima of V (θρ , θσ ) in Eq. 2.20.
When the minima are deep instanton tunneling events between the minima, denoted by te , tρ and tσ correspond to
the transfer of charge and spin across the junction, and deﬁne the dual theory (2.26,2.27). (b) Positions of the minima
˜ ˜
of V (θρ , θσ ) in the dual theory (2.26,2.27). Instanton process
ve , vρ and vσ correspond to backscattering of charge and spin
in the original theory. (c) The IC phase viewed from the CC
limit. When vρ is large and vσ = 0, the minima of V (θρ , θσ )
in Eq. 2.20 are on one dimensional valleys, and deﬁne the
IC phase. When vσ is small but ﬁnite the valleys have a periodic potential vσ τ z cos θσ , with opposite signs τ z = ±1 in
˜
neighboring valleys. Instanton tunneling processes between
˜
the valleys, denoted tρ , switch the sign of τ z . (d) The IC
phase viewed from the II limit, in which tρ = 0 and tσ is
˜
˜
large. The valleys have periodic potential tρ τ x cos θρ with
τ x = ±1, whose sign is switched by instanton processes vσ .
˜

phase, including the leading relevant operators it is useful to use the instanton analysis discussed at the end of
the previous section. Consider (2.26,2.27) for large vρ ,
keeping ve and vσ small. θρ will be pinned in the minima
of − cos 2θρ , θρ = nπ, while θσ remains free to ﬂuctuate.
(θρ , θσ ) are thus conﬁned to “valleys” along the vertical
lines in Fig. 3(c).
There are two types of perturbations to be considered.
First, ve will lead to a periodic potential along the vertical lines, with minima at the dots. Note, however, that on
alternate lines the sign of the periodic potential changes,
since cos θρ cos θσ ∼ (−1)n cos θσ for θρ = nπ.
Next consider an instanton process where θρ tunnels
between neighboring valleys. In this process, θρ → θρ ±π,
but θσ is unchanged. It follows that the ve perturbation
discussed above changes sign. Thus, the instanton pro-

8
cess does not commute with the ve term.
The expansion of the partition function in both instantons and ve can be generated by the action for the IC
1
0
phase give by SIC = SIC + SIC with
0
SIC

1
=
β

ωn

1
gρ
˜
|ωn ||θσ (ωn )|2 ,
|ωn ||θρ (ωn )|2 +
2π
2πgσ
(2.32)

and
1
SIC =

β
0

dτ ˜ x
˜
tρ τ cos θρ + vσ τ z cos θσ .
˜
τc

As in section the corrections to GXY will depend
on a higher order irrelevant operator. For instance,
˜
˜
λ1 τ y sin θρ sin θσ and λ2 τ y cos θρ cos θσ lead to
δGXY = d3 λ1 λ2 T 2g

4.

0
SCI =

1
β

(2.38)
and
β

dτ ˜ z
˜
tσ τ cos(θσ + ησ ) + vρ τ x cos(θρ + ηρ ) .
˜
0 τc
(2.39)
The leading relevant operators have dimensions
1
SCI =

g
1
=
2gσ
2
g
gρ
= .
∆(˜ρ ) =
v
2
2
˜
∆(tσ ) =

(2.35)

This, however, does not mean that the full four terminal conductance is zero because spin conservation still
requires GZZ = 2e2 /h. This leads to the non trivial
structure in the four terminal conductance predicted in
Ref. 10.
At ﬁnite temperature, there will be corrections to the
conductance. We ﬁnd
−2

−1

−2

(2.40)

This phase is thus stable when g > 2 and has conductance

5.

GXX = GY Y = GXY = 0.

δGY Y = d2 vσ T g
˜2

ωn

1
gσ
˜
|ωn ||θρ (ωn )|2
|ωn ||θσ (ωn )|2 +
2π
2πgρ

(2.41)

(2.34)

Thus, the IC phase is stable when g < 1/2.
In section III.D we will require the renormalization
˜
group ﬂow to third order in tρ and vσ . There, the non
˜
trivial interaction between them introduced by the pseudospin will play a crucial role.
The conductivity at the IC ﬁxed point is given by

−1

(2.37)

For g > 2 the perturbation vσ cos 2θσ in (2.20) becomes
relevant and drives the system to the CI phase. This may
be described in a manner similar to the IC phase. It is
0
1
described by the action SCI = SCI + SCI with

GXX = GY Y = 2e2 /h
GXY = 0.

1
1
=
2gρ
2g
gσ
1
∆(˜σ ) =
v
=
.
2
2g

.

Charge conductor/Spin insulator (CI)

˜
∆(tρ ) =

˜ρ
δGXX = d1 t2 T g

−2

As in (2.25), GXY is suppressed more strongly at low
temperature than GXX and GY Y , and the exponent is
larger than 2 for g < 1/2.

(2.33)

˜
Here tρ describes the instanton tunneling process. The
tilde distinguishes it from the ordinary charge tunneling
˜
process, which involves charge 2e. tρ describes a tunneling of charge e without spin. vσ describes the periodic
˜
potential as a function of θσ generated by ve . We have
introduced a pseudo spin degree of freedom τ z = ±1
to account for the sign of cos θρ in the diﬀerent valleys.
Since the instanton process switches the sign, it is associated with τ x . Expanding the partition function deﬁned
˜
by (2.32,2.33) in powers of tρ and vσ precisely generates
˜
the expansion of (2.19,2.20) in instantons.
It is also instructive to derive (2.32,2.33) starting from
the opposite limit of the II phase described by (2.26,2.27).
In this case, consider large tσ , which leads to the horizon˜
˜
tal valleys as a function of θρ and θσ in Fig. 3(d). The
˜
roles of the two terms in (2.33) are thus reversed. tρ describes the periodic potential along the valleys, which has
a sign speciﬁed by τ x = ±1. vσ describes the instanton
˜
processes which switch the sign of τ x .
The lowest order renormalization group ﬂows depend
˜
only on the scaling dimensions of tρ and vσ , and are un˜
x,z
aﬀected by the pseudospin τ . We ﬁnd

−1

.

(2.36)

Spin orbit interactions, and the T-Breaking Insulator

In this section we consider the role of spin orbit interaction terms which violate the conservation of spin Sz ,
but respect time reversal symmetry. We will argue that
such terms are irrelevant for the critical behavior of the
point contact when g > 1/2, but they are relevant for
g < 1/2 and drive the system at low energy to a time
reversal symmetry breaking insulator (TBI).
Time reversal symmetry allows the following terms in
the expansion about the CC ﬁxed point (2.19).
β

dτ
[vso cos ϕσ sin θσ + vsf cos(2ϕσ + ηsf )] .
0 τc
(2.42)
†
The ﬁrst term is a single electron process ψR↑ ψR↓ (Fig.
4(a)) in which an electron ﬂips its spin and crosses the
junction. The second term is a correlated tunneling pro†
†
cess ψR↑ ψL↑ ψR↓ ψL↓ (Fig. 4(b)), where a left and right
SO
SCC =

9
moving pair of up spins ﬂip into a left and right moving pair of down spins. Referring to Table I, it is clear
that both terms respect time reversal symmetry. ηsf is
allowed by time reversal symmetry, but violates both mirrors Mx and My . Higher order processes are also possible, though they will be less relevant perturbatively.
It is straightforward to determine the scaling dimensions of these perturbations. We ﬁnd,
1
1
−1
(gσ + gσ ) = (g + g −1 )
2
2
2
∆(vsf ) =
= 2g.
gσ

(a) vso

(b) vsf

or

FIG. 4: Tunneling processes in the CC limit allowed by spin
nonconserving spin orbit interactions. vso is a single particle process where a single spin is ﬂipped, while vsf is a two
particle process, ﬂipping two spins.

∆(vso ) =

(2.43)

For g = 1 the single particle spin orbit term, vso is always
irrelevant. However, vsf becomes relevant when g < 1/2.
At ﬁnite temperature these lead to corrections to the
conductance of the CC phase. To lowest order they do
not aﬀect GXX , GXY and GY Y . However we ﬁnd
√
−1
T g+g −2 g > 1/ 3
√
δGZZ ∝
(2.44)
T 4g−2
g < 1/ 3
Like GXY , GZX and GZY are zero unless higher order
irrelevant operators, which involve extra powers of ∂x ϕα
or ∂x θα , are included. We ﬁnd
δGZX ∝ T 2g
δGZY ∝ T

g+g−1

(2.45)
.

For weak interactions, g ∼ 1 these conductances vanish
for T → 0 as T 2 .
For g < 1/2 there are two relevant perturbations about
the CC limit. To study their eﬀects we consider a model
in which only the relevant perturbations appear. Since
these perturbations involve the commuting operators ϕσ
and θρ , it is useful to study the 0 + 1 dimensional ﬁeld
theory of those variables
0
SCC =
ωn

1
gσ
|ωn ||θρ (ωn )|2 +
|ωn ||ϕσ (ωn )|2 ,
2πgρ
2π
(2.46)

with
β

dτ
[vρ cos(2θρ + ηρ ) + vsf cos(2ϕσ + ηsf )] .
0 τc
(2.47)
The low temperature behavior of this theory can be studied by the duality arguments of section II.B.2. When vρ
and vsf are both large, (θρ , ϕσ ) will be stuck in the deep
minima of VCC (θρ , ϕσ ) shown in Fig. 5. In this phase,
the four terminal conductance is zero,
1
SCC =

GAB = 0.

(2.48)

This can be seen most simply by renaming the variables
θρ
ϕσ
ϕρ
θσ

→
→
→
→

θ1 + θ2
θ1 − θ2
ϕ1 + ϕ2
ϕ1 − ϕ2 .

2π
∼
tρ

(2.49)

∼
tσ

ϕσ
θρ
te

2π

FIG. 5: Minima of the potential V (θρ , ϕσ ) in 2.47. Large
vρ and vso deﬁne the time reversal breaking insulating phase.
˜
˜
Instanton processes tρ and tσ restore time reversal invariance.
They correspond to tunneling of spinless chargons or chargeless spinons.

The interpretation of θ1(2) and ϕ1(2) is simple. They are
the usual Luttinger liquid charge and phase variables for
the top (bottom) edges in Fig. 2(a,b,c). In the strong
coupling phase θ1 and θ2 are both pinned, so that any
current ﬂowing in from any lead is perfectly reﬂected back
into that lead. The four leads are completely decoupled.
This is the same perfectly reﬂecting phase that would
arise if we had a single particle backscattering term on
each edge vback (cos 2θ1 + cos 2θ2 ) = 2vback cos θρ cos ϕσ ,
which would be relevant for g < 1. However in our problem that term is forbidden by time reversal symmetry.
It is thus clear that time reversal symmetry is violated
by the strong coupling ﬁxed point. It is useful to see
this from Fig. 5. Note that since under time reversal
ϕσ → ϕσ + π. Thus pinning ϕσ violates time reversal.
There are two sets of minima of V (θρ , ϕσ ) which are interchanged by the time reversal operation.
At ﬁnite temperature tunneling processes between the
two sets of minima of V (θρ , ϕσ ) will restore time reversal
symmetry. These instanton processes correspond to tunneling of charge from one lead to another. Interestingly,
˜
˜
the lowest order instanton processes, denoted tρ and tσ
do not correspond to tunneling of electrons, but rather
spinless charge e “chargons”, or charge neutral “spinons”.
The scaling dimensions of these instanton processes
can be deduced from (2.46,2.47). We ﬁnd
˜
∆(tρ ) =

1
1
=
2gρ
2g

10
˜
∆(tσ ) =

1
gσ
=
.
2
2g

(2.50)

Thus, both processes are irrelevant for g < 1/2, and the
TBI phase is stable. These processes lead to power law
temperature behavior,
˜ρ
δGXX = c1 t2 T 1/g−2
˜σ
δGY Y = c1 t2 T 1/g−2 .

(2.51)

˜
When the tρ,σ processes dominate, there will be non
˜
trivial noise correlations in the current. The tρ process
involves transferring charge e/2 from lead 1 to lead 2
and another e/2 from lead 4 to lead 3. This leads to
correlations in the low frequency noise deﬁned by
Sij (ω) =

dteiωt Ii (t)Ij (0) + Ij (0)Ii (t) .

(2.52)

Consider the two terminal geometry IX = GXX VX . The
˜
current IX will be carried by the tρ processes, so that
I1 = I4 = IX /2. The shot noise correlations in the limit
ω → 0 will be
S11 = S44 = S14 = S41 = 2e∗ I1

(2.53)

with e∗ = e/2. Thus, the currents are all perfectly correlated, and the current in each lead is carried by fractional
charges, e/2.
III.

CRITICAL BEHAVIOR OF
CONDUCTANCE

In this section we describe the critical behavior of the
conductance at the pinch-oﬀ transition of the point contact. We will compute the critical conductance G∗ , the
g
critical exponent αg and the scaling function Gg (X) in
certain solvable limits. We will begin in section IIIA
with a discussion of the general properties of the scaling
function and a summary of our calculated results. Then
in the following sections we will√
describe in detail our
calculations for g = 1 − ǫ, g = 1/ 3 and g = 1/2 + ǫ.
A.

Scaling behavior and summary of results

The stability analysis of the previous sections leads to
the phase diagram as a function of g depicted in Fig. 6(a).
The top line depicts the CC phase and the bottom line
depicts the II phase, and the arrows denote the stability
associated with the leading relevant operators. Since the
II and CC phases are both stable for 1/2 < g < 2 they
are separated by an intermediate unstable ﬁxed point P,
denoted by the dashed central line. For g < 1/2 the II
and CC phases become unstable, and when spin is conserved the ﬂow is toward the IC phase. We will see in section III.D that the unstable critical ﬁxed point matches
smoothly onto the IC ﬁxed point at g = 1/2. Similarly,

the CI ﬁxed point is stable for g > 2, and connects to the
critical ﬁxed point at g = 2.
For 1/2 < g < 2 the unstable intermediate ﬁxed point
P describes the critical behavior of the pinch-oﬀ transition of the point contact. We will argue that this ﬁxed
point is characterized by a single relevant operator, which
allows us to formulate a single parameter scaling theory
for the pinch-oﬀ transition. If we denote u as the relevant operator, then the leading order renormalization
group ﬂow near the ﬁxed point has the form,
du/dℓ = αg u,

(3.1)

where αg is a critical exponent to be determined. By
varying a gate voltage VG it is possible to cross from
the region of stability of the II phase to the region of
stability of the CC phase. In the process one must pass
∗
through the ﬁxed point u = 0 at VG = VG . Near the
∗
transition, we thus have u ∝ ∆VG = VG − VG . Under a
renormalization group transformation in which energies
length and time are rescaled by b, we have u → ubαg and
T → T b. Invariance under this transformations requires
that physical quantities can only depend on u and T in
the combination u/T αg . Close to the transition we thus
have
lim

T,∆VG →0

GAB (T, ∆VG ) = 2

e2
∆VG
Gg,AB (c αg ),
h
T

(3.2)

where c is a nonuniversal constant and Gg,AB is a universal crossover scaling function which varies between 0 and
1.
We will argue that the critical point characterizing the
pinch-oﬀ transition has emergent spin conservation as
well as mirror symmetry, so that the only nonzero elements of the conductance matrix are GXX and GY Y .
Moreover, the duality considerations discussed in section
III.C require that Gg,Y Y (X) and Gg,XX (X) are related,
so that they are both determined by the same universal
scaling function,
Gg,XX (X) = Gg (X),
Gg,Y Y (X) = Gg (−X).

(3.3)

The scaling function Gg (X) has some general properties which are easy do deduce. First, the equivalence
between the CC theory at g with the II theory at 1/g
leads to the relation
G1/g (X) = 1 − Gg (−X).

(3.4)

Second, when T → 0 for ﬁxed ∆VG the system ﬂows to
either the CC or the II phase, where the temperature
dependence of the conductance is given by (2.24). The
behavior of the scaling function for large X then follows,
+

Gg (X → +∞) = 1 − a+ X −βg
g

Gg (X → −∞) = a− X −βg .
g
−

(3.5)

11
CC

~
t

P

In the following sections we compute properties of the
√
scaling function at g = 1 − ǫ, g = 1/ 3 and g = 1/2 + ǫ.
From (3.4) √ can deduce corresponding results at g =
we
1 + ǫ, g = 3 and g = 2 − ǫ. First consider the critical
conductance G∗ = Gg (X = 0). We ﬁnd,
g

CC

P

v (a)
IC

CI

~
v
t

II


 1/2 + O(ǫ3 ) g = 1 − ǫ
 √
√
∗
Gg = ( 3 − 1)/2 g = 1/ 3
 2
π ǫ
g = 1/2 + ǫ.

II

1/2

1

2

g

G*g

1

(b)
3− 3
2
3 −1
2

1/2

1/2 1/ 3

1

3 2

αg
(c)

.10
.05

1

3 2

g

FIG. 6: (a) Phase diagram for a point contact in a QSHI as
a function of the Luttinger liquid parameter g. The arrows
indicate the stability of the CC, II, CI and IC phases, as
well as the critical ﬁxed point P. This ﬁgure assumes spin
conservation. In the presence of spin orbit interactions, the
IC phase is unstable for g < 1/2. This leads to the TBI
phase discussed in section II.B.5. (b) Conductance G∗ of the
critical ﬁxed point as a function of g. The curve is a ﬁt,
which incorporates the data in (3.7). (c) Critical exponent
αg as a function of g. The curve is a ﬁt incorporating the
data in (3.8). g is plotted on a log scale in all three panels to
emphasize the g ↔ 1/g symmetry.

a±
g

The coeﬃcients
depend on the normalization of X,
′
but can be ﬁxed if we specify Gg (X = 0) = 1/2. The
exponents obey the relations
+
βg

=

√
(4g − 2)/αg
1/2 < g < 1/ 3
√
(g + g −1 − 2)/αg 1/ 3 < g < 1

−
βg = (g + g −1 − 2)/αg

1/2 < g < 1.

ǫ2 /2 g = 1 − ǫ
4ǫ
g = 1/2 + ǫ.

(3.8)

These results are summarized in Fig 6(c). The curve is
a polynomial ﬁt of α(log g). It is suggestive that in this
√
ﬁt αg exhibits a maximum near g = 1/ 3 with a value
α1/√3 = .123 ∼ 1/8. It is possible, however, that αg
√
exhibits a cusp at g = 1/ 3 analogous to the behavior
of βg in (3.6).
In sections III.C and III.D we compute the full scaling
function Gg (X) in the limits g = 1 − ǫ and g = 1/2 + ǫ to
lowest order in ǫ. For g = 1 + ǫ, ǫ → 0 we ﬁnd

.15

1/2 1/ 3

These results are summarized in Fig. 6(b). The curve is
a polynomial ﬁt of G∗ (log g) which incorporates the data
in Eq. (3.7) and the g ↔ 1/g symmetry. It is satisfying
that the curve is smooth and monotonic, which indicates
a consistency between the slopes at g = 1/2, 1 and the
√
value at g = 1/ 3.
We are able to deduce the critical exponent αg for g =
1 − ǫ and g = 1/2 + ǫ. We ﬁnd
αg =

g

(3.7)

(3.6)

±
The behavior of βg for 1 < g < 2 can be deduced using
(3.4).

G1 (X) =

1
2

X
1+ √
1 + X2

.

(3.9)

For g = 1/2 + ǫ, ǫ → 0
G1/2 (X) = θ(X)

X
.
1+X

(3.10)

The singular behavior near X = 0 in (3.10) is rounded
for ﬁnite ǫ. The perturbative analysis in Section III.D.1
shows that for |X| ≪ 1
G1/2+ǫ (X) =

X
.
1 − e−X/(π2 ǫ)

(3.11)

G1−ǫ (X) and G1/2+ǫ (X) are plotted in Figs. 7(a) and
7(b). For g close to 1 the pinch-oﬀ curve is symmetrical about G∗ = e2 /h. However, for stronger repulsive
interactions it becomes asymmetrical, as G∗ is reduced,
approaching 0 at g = 1/2.
The asymptotic |X| → ∞ behavior (3.5) of G1 (X)
and G1/2+ǫ can also be determined from (3.9,3.10),
though a separate calculation (see III.D.3) is required
for G1/2+ǫ (X → −∞). The results, which are consistent
with (3.6) are shown in Table II.

12

(a)

G1−ε(X)

1

G*=1/2

X
0
(b)

G1/2+ε(X)

1

G*= π2 ε

X
0

FIG. 7: The universal scaling function Gg (X) for (a) g = 1 − ǫ
(Eq. 3.9) and (b) g = 1/2 + ǫ (Eq. 3.10). In (b) the solid line
is ǫ → 0, and the dashed line shows the approximate behavior
for ǫ ∼ .02.
+
−
g
βg a+ βg
a−
g
g
1 − ǫ 2 1/4
2
1/4
1/2 + ǫ 1 1 1/(8ǫ) (2.75)1/(8ǫ)

TABLE II: Parameters in Eq. 3.5 for the asymptotic behavior
of the scaling function Gg (X) in the solvable limits g → 1,
g → 1/2.
B.

Quantum Brownian Motion Model, Duality and
√
g = 1/ 3

In this section we recast the Luttinger liquid model as
a model of QBM in a periodic potential. This mapping
elucidates the duality between the CC and II limits and
√
exposes an extra symmetry the problem at g = 1/ 3
which allows us to deduce the critical conductance at
that point. We begin with a brief review of the QBM
model and then derive its consequences for the scaling
function Gg (X) and G∗ √3 .
1/
1.

Quantum Brownian Motion Model

The QBM model28,29,30 was originally formulated as
a theory of the motion of a heavy particle coupled to
an Ohmic dissipative environment modeled as a set of
Caldeira Leggett oscillators32. Though the applicability

of this model to the motion of a real particle coupled to
phonons or electron-hole pairs has been questioned33,34 ,
it was later shown that this model is directly relevant
to quantum impurity problems. Speciﬁcally, the QBM
model in a one dimensional periodic potential is equivalent to the theory of a weak link in a single channel
Luttinger liquid16,18 . In this mapping the QBM takes
place in an abstract space where the “coordinate” of the
“particle” is the number of electrons that have tunneled
past the weak link. The periodic potential is due to the
discreteness of the electron’s charge. The low energy excitations of the Luttinger liquid play the role of the dissipative bath, and the strength of the dissipation is related
to the Luttinger liquid parameter g. The one dimensional
QBM model has two phases: a localized phase with conductance G = 0 stable for g < 1 and a fully coherent
phase with perfect conductance stable for g > 1.
The SLL model corresponds to a QBM model in a two
dimensional periodic potential, where the “coordinates”
are the spin and charge variables θρ,σ . This model is
richer than its one dimensional counterpart because it
admits additional ﬁxed points which are intermediate between localized and perfect. These ﬁxed points were ﬁrst
found in the Luttinger liquid model15,16 , and later formulated in terms of the QBM25 . For certain values of gρ
and gσ these intermediate ﬁxed points are related to the 3
channel Kondo problem25 and the 3 state Potts models24 .
However, those limits are not directly applicable to the
QSHI model, where gρ = 1/gσ = g. We will show that
√
when g = 1/ 3 the critical ﬁxed point of the QSHI point
contact corresponds to the intermediate point discussed
in Ref. 25 for a QBM model on a triangular lattice.
To formulate the QBM model we begin with the action
(2.19) and deﬁne new rescaled variables,
θα = π

2gα rα .

(3.12)

Then (2.19) takes the form,
S=

1
4πβ

n

|ωn ||r(ωn )|2 −

dτ
τc

vG e2πiG·r(τ ) .
G

(3.13)
The periodic potential is characterized by reciprocal lattice vectors G = m1 b1 + m2 b2 . The primitive reciprocal
lattice vectors b1,2 correspond to the single electron back
scattering processes, and are given by
1 √ √
b1 = √ ( gρ , gσ );
2

1 √
√
b2 = √ ( gρ , − gσ ). (3.14)
2

The Fourier components of the periodic potential are
vb1 = vb2 = ve eiηρ /4, vb1 +b2 = vρ /2 and vb1 −b2 = vσ /2.
The dual theory is obtained by expanding the partition
function for large vG in powers of instantons. When vG
is large, the potential has minima on a real space lattice
R = n1 a1 + n2 a2 . The primitive lattice vectors satisfy
ai · bj = δij and are given by
1
1
1
a1 = √ ( √ , √ );
gσ
2 gρ

1
1
1
a2 = √ ( √ , − √ ). (3.15)
gσ
2 gρ

13
rσ

because it provides evidence that the critical ﬁxed point
has emergent mirror and spin conservation symmetry.
When gρ = 1/2 + ǫρ and gσ = 3/2 + ǫσ the period
potential has triangular symmetry, which is slightly distorted if ǫσ = 3ǫρ . If we denote the relevant variables as
v1 = vb1 = vb2 = ve eiηρ /4 and v2 = vb1 +b2 = vρ /2, the
second order renormalization group ﬂow equations are16

(a)
tρ

1/ 2gσ
a1
a2

te

rρ
1/ 2gρ

1
∗
(ǫρ + ǫσ )v1 − 2v1 v2
2
2
dv2 /dℓ = 2ǫρ v2 − 2v1 .

dv1 /dℓ =

tσ
te

kσ
(b)
gσ/2

vσ

kρ

b1
b2

vρ
ve

gρ/2

ve

FIG. 8: (a) Minimia of the periodic potential V (r) in (3.13).
(b) Minima of V (k) in the dual theory (3.16). When gσ = 3gρ
both periodic potentials have triangular symmetry at the critical point, which implies the mobility µ∗ is isotropic. This
αβ
√
occurs at g = 1/ 3.

The expansion in instantons connecting these minima is
generated by the action
1
4πβ

dτ
tR e2πiR·k(τ ) .
τc
n
R
(3.16)
˜
This is equivalent to (2.26,2.27) with kα = π gα /2θα
and ta1 = ta2 = te eiηρ /4, ta1 +a2 = tρ /2, ta1 −a2 = tσ /2.
With the above normalizations for r and k the scaling
dimensions of the potential perturbations are
S=

|ωn ||k(ωn )|2 −

∆(vG ) = |G|2 ;

∆(tR ) = |R|2 .

(3.17)

Since operators are relevant when ∆ < 1, the most relevant potentials are those with the smallest lattice (reciprocal lattice) vectors |Rmin | (|Gmin |). As shown in
Refs. 16 and 25 there are ranges of gρ and gσ where both
|Rmin | and |Gmin | > 1, so that both phases are perturbatively stable. An unstable intermediate ﬁxed point must
therefore be present between them.
This ﬁxed point can be accessed perturbatively when
|Rmin | and |Gmin | are close to 1. While this does not occur in the regime gρ = 1/gσ relevant to the QSHI problem, it is instructive to study this perturbation theory

(3.18)

These equations describe an intermediate ﬁxed point
with a single unstable direction at v1 = ǫρ (ǫρ + ǫσ )/2
and v2 = (ǫρ + ǫσ )/4. Note that at the critical point v1
is real, so that ηρ = 0. Thus the critical point has an
emergent mirror symmetry even if the bare parameters
in the model do not. Moreover, the ﬂow out of the ﬁxed
point along the single unstable direction is also along a
line with v1 real. Thus the crossover between the intermediate ﬁxed point and the trivial ﬁxed point, which
determines the crossover scaling function also has emergent mirror symmetry. Mirror symmetry breaking is an
irrelevant perturbation at the critical ﬁxed point.
If ǫσ = 3ǫρ then the lattice vectors have a triangular
symmetry. In this case, the ﬁxed point is at v1 = v2 = ǫρ .
This means that the periodic potential at the ﬁxed point
has emergent triangular symmetry, even when the bare
potential does not. The unstable ﬂow out of the ﬁxed
point is also along the high symmetry line v1 = v2 .
It seems quite likely that the critical ﬁxed point and
unstable ﬂows connecting it to the trivial ﬁxed points
retain their high symmetry even outside the perturbative
small ǫ regime. This suggests that in general the critical
√
ﬁxed point has mirror symmetry, and that at g = 1/ 3
it has triangular symmetry. We will use this fact below
√
to determine the critical conductance at g = 1/ 3.
2.

Kubo conductance, mobility and duality relations

The spin and charge conductances in the Luttinger liquid model computed by the Kubo formula are given by
a retarded current-current correlation function. For the
present discussion it is useful to write this as an imaginary time correlation function, which can be analytically
continued to real time via iω → ω + iη before taking the
ω → 0 limit. Then
GK (iωn ) =
αβ

1
|ωn |

dτ eiωn τ Jα (τ )Jβ (0) ,

(3.19)

where the spin and charge currents are Jα = e∂t θα /π =
e[θ, H]/(iπ ). This may be expressed as
GK (ωn ) = 2
αβ

e2 √
gα gβ µαβ (ωn ),
h

(3.20)

where the mobility of the QBM model is
µαβ (ωn ) = 2π|ωn | rα (−ωn )rβ (ωn ) .

(3.21)

14
µαβ is normalized so that when vG = 0 µαβ = δαβ .
The conductance – or equivalently µαβ can also be
computed from the dual model. It is given by
µαβ = δαβ − µαβ ,
˜

(3.22)

where the dual mobility is
µαβ (ωn ) = 2π|ωn | kα (−ωn )kβ (ωn ) .
˜

(3.23)

(3.22,3.23) are obvious in the perfectly transmitting and
perfectly reﬂecting limits. They can be derived more
generally by starting with a Hamiltonian formulation of
the action, analogous to (2.13), which involves both r and
k. µαβ can then be computed either by ﬁrst integrating
out k to obtain (3.21) or ﬁrst integrating out r to obtain
(3.23).
Since gρ = 1/gσ = g, the dual theory depicted in Fig.
8(b) is identical to the original theory shown in Fig. 8(a)
with the identiﬁcation rρ ↔ kσ , rσ ↔ kρ . It follows that
the mobility µ∗ of the ﬁxed point satisﬁes
αβ
µ∗ = [σ x µ∗ σ x ]αβ .
˜
αβ

(3.24)

In addition, if u parameterizes the relevant direction at
the critical ﬁxed point, then under the duality u → −u.
It follows that slightly away from the critical ﬁxed point
we have
µαβ (u) = [σ x µ(−u)σ x ]αβ .
˜

(3.25)

Properties (3.22) and (3.25) imply that µρρ (u) = 1 −
µσσ (−u). Using (3.2,3.20,A15), this leads directly to the
property (3.3) of the crossover scaling function.
An additional set of relations follows from the equivalence between the theory characterized by g and the dual
theory characterized by 1/g. From this we conclude that
µg,αβ (u) = µ1/g,αβ (u).
˜

(3.26)

This, combined with (3.2,3.20,3.22,A15)) leads to (3.4).

It is well known that the physical conductance
measured with leads is not given by the Kubo
conductance35,36,37,38,39 . Rather, the Kubo conductance
needs to be modiﬁed to account for the contact resistance
between the Luttinger liquid and the leads. In appendix
A we review the relation between the physical four terminal conductance and the Kubo conductance. From (A19)
we conclude that
√
e2
GXX = GY Y = ( 3 − 1) .
h
C.

(3.30)

Weak interactions : g = 1 − ǫ

In this section we develop a perturbative expansion
for weak interactions to compute exactly the crossover
scaling function Gg (X) as well as the critical exponent
αg for g = 1 − ǫ. A similar approach was employed
by Matveev, Yue and Glazman27 to compute the scaling function for the crossover between the weak barrier
and strong barrier limits in a single channel Luttinger
liquid. In the single channel problem the transmission
for non interacting electrons is characterized by a transmission probability T . Weak forward scattering interactions lead to an exchange correction to T at ﬁrst order
in the interactions. This correction diverges for E → EF
as log |E − EF |. Matveev, Yue and Glazman27 used a
renormalization group argument to sum the log divergent
corrections to all orders, to obtain the exact transmission
T (E).
For non interacting electrons, the QSHI point contact
is characterized by a 4 × 4 scattering matrix Sij which
relates the incoming wave in lead i to the outgoing wave
in lead j,
|ψi,out = Sij |ψj,in .

(3.31)

In terms of Sij the four terminal conductance is
3.

√
Conductance at g = 1/ 3.

Gij =

√
When g = 1/ 3 the lattice generated by b1 and b2 has
triangular symmetry. In section III.B.1 we argued that
this means that at the critical ﬁxed point the periodic
potential also has triangular symmetry. The C6 rotational symmetry of the triangular lattice requires that
the mobility is isotropic:
µαβ = µ0 δαβ .

(3.27)

Combining (3.22), (3.24), and (3.27) requires that
µ0 =

1
.
2

(3.28)

It follows from (3.20) that the Kubo formula spin and
charge conductances are given by
GK =
ρρ

√ e2
3 ;
h

1 e2
GK = √
.
σσ
3 h

(3.29)

e2
(δij − |Sij |2 ).
h

(3.32)

Under time reversal Θ|ψi,out(in) = +(−)Qij |ψj,in(out) ,
where Q = diag(1, −1, 1, −1). This leads to the constraint S = −QS T Q. This combined with unitarity
S † S = 1 allows S to be parameterized as


0 t f
r


 t 0 r∗ −f ∗ 
S = U† 
(3.33)
 U,
 −f r∗ 0 −t∗ 
r f ∗ −t∗ 0

where Uij = δij eiχi is an unimportant gauge transformation. The complex numbers t and r describe the amplitudes for spin conserving transmission and reﬂection
across the point contact, while f describes the amplitude for tunneling across the junction, combined with a
spin ﬂip. f = 0 if spin is conserved. The conductance

15
can be expressed in terms of the transmission probabilities R = |r|2 , T = |t|2 and F = |f |2 , which satisfy
R + T + F = 1. We ﬁnd
2e2
(T + F )
h
2e2
(R + F )
=
h
2e2
(1 − F )
=
h
= 0 for A = B.

(a)

k,out

GXX =
GY Y
GZZ
GAB

(3.34)

(3.35)

where Tτ denotes imaginary time ordering. For non interacting electrons we have

∗ 
Sji
δij
1  z−z ′ z−¯′ 
z
Gij (z, z ′ ) =
(3.36)
 S
δij  .
ij
2πi
z −z ′ z −¯′
¯
¯ z

where z = τ + ix and z = τ − ix, and the a =in/out
¯
indices are displayed in matrix form.
We now compute the perturbative corrections to
Gout,in using the standard diagrammatic technique. For
ij
simplicity, we adopt a model in which u4 = 0, so that
†
†
the only interaction term involves u2 (ψin ψin )(ψout ψout ).
This considerably simpliﬁes the analysis because many
of the diagrams are zero. For instance, the exchange
diagram shown in Fig. 9(a), which was responsible for
the renormalization in the single channel Luttinger liquid problem is zero because it must involve Gin,out. This
kk
oﬀ diagonal Green’s function depends on Skk which is
zero due to the time reversal symmetry constraint. From
(2.9), g = (2πvF − λ2 )/(2πvF + λ2 ) ∼ 1 − λ2 /(2πvF ).
Thus for g = 1 − ǫ we may replace u2 by 2πvF ǫ. The
nonzero diagrams at second order in u2 are shown in

i,in
(d)

j,out
k,in

k,out

l,out

k,in

j,out
k,in

k,out

For a generic four terminal conductance device time
reversal symmetry guarantees only the reciprocity
relation42 Gij = Gji , (or equivalently GAB = GBA ). For
the QSHI point contact, the spin ﬁltered nature of the
edge states leads to additional constraints. First, the amplitude for an electron to be reﬂected back into the lead
it came from is Sii = 0. Thus Gii = e2 /h. A second
less obvious constraint is that G13 = G24 , which when
combined with reciprocity and unitarity is equivalent to
G12 = G34 and G14 = G23 . This leads to the vanishing
of the skew conductance GXY as well as GXZ and GY Z
even when mirror symmetries MX and MY are explicitly violated. This is a property of the non interacting
electron model and can be violated with electron electron
interactions if the mirror symmetries are absent.
In order to compute the renormalization of the S matrix due to interactions it is useful to study the perturbative expansion of the single electron thermal Green’s
function, which can be represented as a matrix in the
lead indices i, j as well as the channel labels a = in/out.
†
Gab (x, τ ; x′ , τ ′ ) = −i Tτ [ψi,a (x, τ )ψj,b (x′ , τ ′ )] ,
ij

i,in
(b)

i,in

l,in
l,in

j,out
k,out
k,in

j,out
k,in

i,in
(c)

l,out

i,in
(e)

l,in

l,out
l,out

j,out
k,out

l,in

FIG. 9:
Feynman diagrams for the single electron
Green’s function.
The dashed line is the interaction
†
†
u2 (ψin ψin )(ψout ψout ). The exchange diagram (a) vanishes because it involves Skk , and diagrams (b) and (c) cancel one
another. (d) and (e) lead to a logarithmic correction to the S
matrix given in (3.38).

Fig. 9(b-e). Evaluating the second order diagrams gives
a Green’s function of the form
Gout,in =
ij

′
1 Sij
2πi z − z ′
¯

(3.37)

with
′
Sij = Sij +

Λ
ǫ2
∗
Sij Sji Sji −
log
4
E

∗ ∗
Sik Skl Slk Skl Slj ,
kl

(3.38)
where Λ are E are ultraviolet and infrared cutoﬀs respectively. The ﬁrst term in the brackets was due to the
diagram in Fig. 9(d), while the second term was from
Fig. 9(e). Diagrams 9(b) and 9(c) cancelled each other.
Rescaling the cutoﬀ Λ → Λe−ℓ leads to a renormalization
group ﬂow equation for Sij ,
dSij
ǫ2
∗
Sij Sji Sji −
=
dℓ
4

∗ ∗
Sik Skl Slk Skl Slj .

(3.39)

kl

It is useful to rewrite this in terms of the transmission
probabilities T , R, F . The renormalization group ﬂow
equation then can be written in the form
dT /dℓ = ǫ2 T (T − T 2 − R2 − F 2 )
dR/dℓ = ǫ2 R(R − T 2 − R2 − F 2 )
dF /dℓ = ǫ2 F (F − T 2 − R2 − F 2 ).

(3.40)

The ﬂow diagram as a function of R, T and F is shown
in Fig. 10. There are seven ﬁxed points. The bottom
corners of the triangle are the stable ﬁxed points at R =
1, T = F = 0 (the II phase) and T = 1, R = F = 0 (the
CC phase). The third stable ﬁxed point at the top of the
triangle with F = 1, T = R = 0, corresponds to the case

16

F=1

P

II
R=1

T=R=1/2

We ﬁnd that the logarithmic renormalization to the S
matrix accounts for the only correction to the conductance to linear order in ǫ. In principle one must consider
a “RPA like” diagram for the conductance evaluated by
the Kubo formula. While this gives a correction for an
inﬁnite Luttinger liquid at ﬁnite frequency, the correction
is zero for a ﬁnite Luttinger liquid connected to leads in
the ω → 0 limit35,36,37,38 . Since the critical conductance
satisﬁes G∗ = 1−G∗ it follows that G∗ = 1/2+O(ǫ3 ).
g
1−ǫ
1/g

CC
T=1

FIG. 10: Renormalization group ﬂow diagram for the transmission probabilities T , R and F based on (3.40) represented
in a ternary plot. The CC, II and P ﬁxed points of interest
in this paper, which have F = 0 are on the bottom of the
triangle.

where an incident electron is transmitted perfectly with a
spin ﬂip. This is presumably diﬃcult to access physically.
On the edges of the triangle are unstable ﬁxed points
describing transitions between the diﬀerent stable phases.
The critical ﬁxed point P of interest in this paper is the
one on the bottom of the triangle at R = T = 1/2, F =
0. Note that at this ﬁxed point the spin non conserving
spin orbit processes, represented by F are irrelevant. At
the center of the triangle, at R = T = F = 1/3 is an
unstable ﬁxed point describing a multicritical point.
To describe the critical ﬁxed point P and the crossover
to the II and CC phases we now specialize to F = 0
and consider the ﬂow equation for the single parameter
T characterizing the point contact,
dT /dℓ = −ǫ2 T (1 − T )(1 − 2T ).

(3.41)

D.

g = 1/2 is at the boundary where the CC and II
phases become unstable and the IC phase becomes stable. We will show that when g = 1/2 + ǫ the critical
ﬁxed point describing the transition between the CC and
II phases approaches the IC ﬁxed point and can be accessed perturbatively using theory developed in Section
II.B.3. In addition, when g = 1/2, the marginal opera˜
˜
tors vρ cos 2θρ at the CC ﬁxed point and t cos θρ at the IC
ﬁxed point can be expressed in terms of ﬁctitious fermion
operators. This fermionization process allows the entire
crossover between the CC and IC phases to be described
using a non interacting fermion Hamiltonian. A similar fermionization procedure can be used to describe the
crossover between the II and IC phases, which connect
˜
˜
the marginal operators vσ cos θσ and tσ cos 2θσ . This will
˜
allow us to compute the full crossover scaling function
Gg (X) for g = 1/2 + ǫ.
We will begin by discussing the perturbative analysis of the IC ﬁxed point and then go on to describe the
fermionization procedure.

Eq. 3.41 can be integrated to determine the crossover
scaling function. If at ℓ = 0 T = T 0 , then,
T (ℓ) =

T 0 − 1/2

1
1+
2

.
(T 0 − 1/2)2 + T 0 (1 − T 0 )e−ǫ2 ℓ
(3.42)
As the gate voltage VG is adjusted through the pinch∗
oﬀ transition, T 0 passes through 1/2 at VG = VG , so
0
T − 1/2 ∝ ∆VG . At temperature T we cut oﬀ the renormalization group ﬂow at Λe−ℓ ∝ T . The conductance
is then given by GXX = 2(e2 /h)T (ℓ = log(Λ/T )). For
2
2
∆VG , T → 0 we deﬁne X = (2T 0 −1)eǫ ℓ/2 ∝ ∆VG /T ǫ /2
and write the conductance in the scaling form,
GXX (∆VG , T ) = 2

e2
∆VG
G1−ǫ (c αg ),
h
T

(3.43)

where c is a non universal constant, the critical exponent
is
α1−ǫ = ǫ2 /2,

(3.44)

and
G1−ǫ (X) =

X
1
1+ √
.
2
1 + X2

(3.45)

g = 1/2 + ǫ

1.

Perturbative Analysis

The IC ﬁxed point is described by (2.32, 2.33). When
˜
˜
˜
g = 1/2 + ǫ the perturbations tρ τ x cos θρ and tσ τ z cos θσ
both have scaling dimension ∆ = 1 − 2ǫ, so the IC ﬁxed
˜
point is weakly unstable. When vσ = 0, nonzero tρ is
˜
expected to drive the system to the CC phase, while for
˜
tρ = 0 nonzero vσ will drive the system to the II phase.
˜
˜
Thus, when both tρ and vσ are non zero there must be
˜
an unstable ﬁxed point which separates the two alternatives. This ﬁxed point can be described by considering
the renormalization group ﬂow equations to third order
˜
in vσ and tρ .
˜
˜
The ﬁrst order renormalization group equation for tρ
˜
is determined by the scaling dimension ∆(tρ ). The next
2
nonzero term occurs at order tρ vσ . To compute this term
it is suﬃcient to use the theory at ǫ = 0. Consider the
third order term in the cumulant expansion of the partition function, when fast degrees of freedom integrated
out:
1
2

dτ1 dτ2 { Tτ [Oρ (τ )Oσ (τ1 )Oσ (τ2 )]

>

17
− Oρ (τ )

>

Tτ [Oσ (τ1 )Oσ (τ2 )]

> }.

(3.46)

˜
˜
Here Oρ = (tρ /τc )τ x cos θρ and Oσ = (˜σ /τc )τ z cos θσ .
v
Tτ indicates time ordering, and · > denotes a trace over
degrees of freedom with Λ/b < ω < Λ, and we assume
˜
for simplicity b ≫ 1. Since θρ and θσ are independent
and commute with one another the other disconnected
terms all cancel. Moreover, the two terms in (3.46) will
cancel each other unless the time ordering of the τ x and
τ z operators leads to a relative minus sign between them,
Tτ [Oρ (τ )Oσ (τ1 )Oσ (τ2 )] > =
s± Oρ (τ ) > Tτ [Oσ (τ1 )Oσ (τ2 )]

>,

(3.47)

where s± = sgn(τ − τ1 )(τ − τ2 ). Thus the pseudospin
operators in (2.33) play a crucial role in the renormal˜
ization of tρ . Using the fact that Tτ [Oσ (τ1 )Oσ (τ2 )] > =
2
vσ /2(τ1 − τ2 )2 for ǫ = 0 we ﬁnd that the third order
˜
2
˜
˜
correction to tρ is δ tρ = −tρ vσ log b. This leads to the
˜
renormalization group ﬂow equation for tρ , along with a
corresponding equation for vσ ,
˜
˜
˜
˜ ˜2
dtρ /dℓ = 2ǫtρ − tρ vσ
d˜σ /dℓ = 2ǫ˜σ − vσ t2 .
v
v
˜ ˜ρ

(3.49)

The Kubo conductance GK at the ﬁxed point can be
ρρ
computed from (3.19) by identifying the current operator
˜
˜
Iρ = (tρ /τc ) sin θρ .

(3.50)

This leads to
GK =
ρρ

2

e 2 ˜2
π tρ .
h

(3.51)

˜ρ
It is useful to deﬁne Tρ = π 2 t2 . We will see in the following section that this can be interpreted as a transmission
probability for ﬁctitious free fermions that describe the
problem at g = 1/2. In terms of Tρ (noting that Tρ ≪ 1
in this perturbative regime) we may use (A19) to write
the physical conductance as
GXX =

e2
Tρ .
h

(3.52)

~
vσ

2ε

P
~
tρ

2ε

IC

vρ

CC

FIG. 11: Renormalization group ﬂow diagram characterizing
˜
the critical ﬁxed point P for g = 1/2 + ǫ. When vσ and tρ
˜
are small, the ﬂows are given by (3.48). On the axis vσ = 0
˜
the fermionization procedure outlined in section III.D.2 determines the entire crossover between the IC and CC ﬁxed
points. A similar theory describes the crossover between the
˜
IC and II ﬁxed points for tρ = 0.

where Rσ = π 2 vσ can similarly be interpreted as a re˜2
ﬂection probability for a diﬀerent ﬁctitious free fermion
at g = 1/2. At the critical ﬁxed point Tρ = Rσ = 2π 2 ǫ.
Thus,
∗
G∗
XX = GY Y = 2

G∗
XY = 0.

e2 2
π ǫ.
h
(3.54)

The behavior away from the critical point can be determined by integrating (3.48). To this end it is helpful
to rewrite (3.48) in terms of Tρ and Rσ in the form
d(Tρ − Rσ )/dℓ = 4ǫ(Tρ − Rσ )

d log(Tρ /Rσ )/dℓ = (2/π 2 )(Tρ − Rσ ).

(3.55)

If (Tρ , Rσ ) = (Tρ0 , R0 ) for ℓ = 0, then we ﬁnd
σ
Tρ (ℓ) =
Rσ (ℓ) =

1−

R0
σ
0
Tρ

1−

0
Tρ
R0
σ

(Tρ0 − R0 )e4ǫℓ
σ
exp −

0
Tρ −R0
σ
4ǫℓ
2π 2 ǫ (e

− 1)

(R0 − Tρ0 )e4ǫℓ
σ
exp −

0
R0 −Tρ
σ
4ǫℓ
2π 2 ǫ (e

. (3.56)
− 1)

∗
At the pinch-oﬀ transition VG = VG , R0 = T0 . Thus,
T0 − R0 ∝ ∆VG . At temperature T we cut oﬀ the renormalization group ﬂow at Λe−ℓ ∝ T . Thus, in the limit
∆VG , T → 0 we deﬁne X = (Tρ0 −R0 )e4ǫℓ /2 ∝ ∆VG /T 4ǫ.
σ
The conductance then has the form

e2
∆VG
Gg c αg
h
T
2
∆VG
e
GY Y (∆VG , T ) = 2 Gg −c αg
h
T

GXX (∆VG , T ) = 2

A similar calculation gives
GY Y =

tσ

(3.48)

The renormalization group ﬂow diagram is shown in√
Fig.
˜
11. There is an unstable ﬁxed point P at tρ = vσ = 2ǫ,
˜
with a single relevant operator. P separates the ﬂows
˜
to the CC and II phases for which which tρ or vσ grow.
˜
Note that spin orbit terms such as vso and vsf discussed
in Section II.B.5 are irrelevant at P (see Eq. 2.43). This
perturbative calculation provides further evidence that P
exhibits emergent spin conservation, as well as emergent
mirror symmetry. The critical exponent associate with
the single relevant operator a P is
α1/2+ǫ = 4ǫ.

II

e2
Rσ ,
h

(3.53)

,

(3.57)

18
with
G1/2+ǫ (X) =

X
.
1 − e−X/(π2 ǫ)

(3.58)

This perturbative calculation is only valid when
Tρ , Rσ ≪ 1. Thus (3.58) breaks down at low temperature, since as the energy is lowered either Tρ or Rσ
grows. (3.58) is valid as long as |X| ≪ 1. Note, however that when ǫ ≪ 1 we have G1/2+ǫ (X) = Xθ(X) when
ǫ ≪ X ≪ 1. In this regime, the smaller of T and R has
gone to zero. Thus we have
Tρ (ℓ) = (Tρ0 − R0 )e4ǫℓ
σ
Rσ (ℓ) = 0
Tρ (ℓ) = 0
Rσ (ℓ) = (R0 − Tρ0 )e4ǫℓ
σ

(Tρ0 − R0 ) > 0
σ
(R0 − Tρ0 ) > 0, (3.59)
σ

and the unstable ﬂow is either on the x or y axis of Fig.
11. In the next section we will solve the crossover exactly on these lines. This will allow us to compute the
G1/2+ǫ (X) exactly for all X.

˜
˜ ˜
where ψ = (ψR , ψL )T is a two component fermion operator describing right and left movers. Using the bosonization relation (2.3) we identify 2θρ = φR − φL and vf =
πvρ /v. The free fermion problem is solvable and characterized by a transmission probability Tρ = sech2 (vf /v).
The free fermion solution therefore connects the CC limit
(Tρ = 1) with the IC limit (Tρ = 0).
The Kubo conductance GK may be computed with the
ρρ
˜
˜
identiﬁcation Jρ = ∂t θρ /π = v ψ † σ z ψ, giving
GK =
ρρ

e2
Tρ .
h

Note that this is the same as (3.52), derived in the opposite limit near the IC ﬁxed point. When vρ is large,
˜
Tρ ≪ 1, and we can identify Tρ = (π tρ )2 . The physical conductance, measured with leads can be determined
following the analysis in appendix A. From (A19) we ﬁnd
GXX = 2

e2 Tρ
.
h 2 − Tρ

Fermionization

In this subsection we study the crossover between
the IC ﬁxed point and the CC and II ﬁxed points for
g = 1/2 + ǫ. There are two cases to consider. First, for
∆VG > 0 we will study the crossover between the IC and
CC on the horizontal axis of Fig. 11 with vσ = 0. This
˜
problem can be mapped to a single channel one dimensional fermi gas with weak electron electron interactions
proportional to ǫ. This allows us to use the method of
Matveev, Yue and Glazman27 to compute the crossover
scaling functions Gg,XX (X) and Gg,Y Y (X) for X > 0 exactly. For ∆VG < 0 the crossover between the IC and II
˜
ﬁxed points is on the vertical axis of Fig. 11 with tρ = 0.
This can be fermionized by introducing a diﬀerent set of
free fermions to compute the scaling functions for X < 0.
The latter calculation (which is virtually identical to the
former) is unnecessary, however, because we can use (3.3)
to deduce the scaling functions for X < 0. We will therefore focus on the IC to CC crossover.
The crossover between the IC and the CC ﬁxed points
can be described by the action in the CC limit
SCC =

1
β

n

1
|ωn ||θρ (ωn )|2 +
2πg

dτ vρ cos 2θρ . (3.60)

vρ ≪ 1 describes the CC phase. When vρ ≫ 1 the dual
theory, formulated as in section II.B.3 in terms of instan˜
tons with amplitude tρ , describes the IC phase. When
vσ = 0 at the IC ﬁxed point we can safely ignore the
˜
pseudospin, and set τ x = 1.
For g = 1/2 this model is equivalent to the bosonized
representation of a weak link in a single channel non interacting fermion with weak backscattering.
˜
˜
˜
˜
Hf = −iv ψ † ∂x σ z ψ + vf ψ † σ x ψδ(x).

(3.61)

(3.63)

Since vσ = 0 in (3.60), we have
GY Y = 0.

2.

(3.62)

(3.64)

For g = 1/2 + ǫ the IC ﬁxed point becomes slightly
unstable, while the CC ﬁxed point becomes slightly stable. In this case the free fermion problem includes a weak
attractive interaction
int
˜† ˜
˜† ˜
Hf = −uf (ψL ψL )(ψR ψR ),

(3.65)

with uf = 2πvǫ. This leads to a logarithmic renormalization of Tρ , which drives a crossover to the CC limit. The
correction to Tρ occurs at ﬁrst order in uf , and is due to
the exchange diagram, shown in Fig. 9(a). The analysis
is exactly the same as that performed by Matveev, Yue
and Glazman. As in section III.C the result can be cast
in terms of a renormalization group ﬂow equation for Tρ .
dTρ /dℓ = 4ǫTρ (1 − Tρ ).

(3.66)

Integrating (3.66) gives
Tρ (ℓ) =

Tρ0 e4ǫℓ
,
1 + Tρ0 (e4ǫℓ − 1)

(3.67)

where Tρ0 = Tρ (ℓ = 0). The scaling function for ∆VG > 0
then follows by using the initial condition from (3.59),
so that Tρ0 ∝ ∆VG . Then, for ∆VG , T → 0 we deﬁne
X = Tρ0 e4ǫℓ /2 ∝ ∆VG /T 4ǫ . Using (3.63), (3.64) and
(3.67), the conductance has the scaling form for X > 0
X
X +1
GY Y,1/2+ǫ (X) = 0.

GXX,1/2+ǫ (X) =

(3.68)

Using (3.3), we may deduce the corresponding behavior
for ∆VG < 0 (or X < 0). The scaling function then has
the form
X
.
(3.69)
G1/2+ǫ (X) = θ(X)
X +1

19
Note that for X ≪ 1 G1/2+ǫ (X) = Xθ(X), in agreement
with the limiting behavior of (3.58) for X ≫ ǫ. These
two expressions can thus be combined to give
X
,
G1/2+ǫ (X) =
X + 1 − e−X/(π2 ǫ)

(3.70)

which reproduces (3.58) when |X| ∼ ǫ ≪ 1 and (3.69)
when |X| ≫ ǫ. This function is plotted in Fig. 7(b).
Note, however, that this formula does not correctly capture the leading behavior for X < 0 when |X| ≫ ǫ.
In particular, it misses the X → −∞ behavior, which
(3.5) and (3.6) predict is proportional to |X|−1/(8ǫ) . This
regime is analyzed in the following section.

3.

(3.71)

0
Hσ = Hσ + vσ τ + eiφ− δ− /π + τ − e−iφ− δ− /π cos θσ
˜

(3.75)
where τ ± = τ z ±iτ y . The renormalization of vσ can then
˜
easily be determined for arbitrary δ− . We ﬁnd

˜
˜
Hρ = −iv ψ σ ∂x ψ + tf τ ψ σ ψδ(x).
x

˜† x

(3.72)

0
Hσ

is the σ part of (2.13), and we explicitly account
Here
for the pseudospin τ x . Eq. 3.72 can be rebosonized by
ﬁrst replacing ψ2 (x) → ψ2 (−x), which transforms the
non chiral fermions to chiral fermions, eliminating the
σ z in the ﬁrst term, but leaving the second term alone.
˜ ˜
˜ ˜
Then we perform a SU(2) rotation (ψ1 , ψ2 ) → (ψe , ψo ),
˜
which changes σ x in the second term into σ z . ψe(o) describe the even (odd) parity scattering states characterized by scattering phase shifts δe = −δo that specify
˜
˜
ψe(o) (x > 0) = e2iδe(o) ψe(o) (x < 0). We next bosonize
√
˜
ψe,o → eiφe,o / 2πxc and deﬁne φ± = φe ± φo . Then
v
v
(∂x φ+ )2 + (∂x φ− )2 +
δ− τ x (∂x φ− )δ(x),
8π
2π
(3.73)

2ǫ −

δ−
π

2

vσ .
˜

(3.76)

˜
˜
For small tρ , δ− = π tρ , and (3.76) reproduces (3.48).
However, (3.76) remains valid to lowest order in ǫ for all
Tρ .
We now integrate (3.55) to a scale ℓ0 where from (3.56)
2
Tρ (ℓ0 ) = 2X0 and Rσ (ℓ0 ) = 2X0 e−X0 /(π ǫ) is small.
0
0 4ǫℓ0
(Here X0 = (Tρ − Rσ )e
/2.) We then use that as
an initial value for (3.76), which we integrate assuming
Tρ (ℓ) is given by (3.67) and is unaﬀected by the small
Rσ . Expressing (3.67) in terms of (3.74) we have
δ− (ℓ) = tan−1 δ− (ℓ0 )e2ǫ(ℓ−ℓ0 )

(3.77)

√
where δ− (ℓ0 ) = sin−1 Tρ (ℓ0 ) ∼ 2X0 . As before, we
deﬁne X = (Tρ0 − R0 )e4ǫℓ /2. We may express GY Y =
σ
(e2 /h)Rσ with Rσ = π 2 vσ . Integrating (3.76) we then
˜2
ﬁnd
GY Y (X) = 2

e2
Xe−F (X)/ǫ,
h

(3.78)

where
1
F (X) = 2
π

and

(3.74)

δ− can be eliminated from (3.73) by the canonical
transformation U = exp[iτ x δ− φ− (x = 0)/(2π)], which
shifts φ− → φ− + sgn(x)δ− τ x . This transformation also
rotates τ z in (3.71), which becomes

Rebosonization

0
Hσ = Hσ + vσ τ z cos θσ
˜

Hρ =

Tρ = sin2 δ− .

d˜σ
v
=
dℓ

We now analyze the leading behavior of G1/2+ǫ (X) for
X < 0 and |X| ≫ ǫ when ǫ is small. Equivalently, we
consider G1/2+ǫ,Y Y (X) for X > 0. This requires extending the renormalization group ﬂow equation for vσ given
˜
˜
in (3.48) to all tρ (or equivalently Tρ ). This can be done
˜
˜
by using the fermionized representation of tρ τ x cos θρ in
(2.33). The key point is that the presence of the pseudospin operator τ x means that the operator vσ τ z cos θσ
˜
changes the sign of the transmission amplitude for the
˜
fermions ψ. This results in an X ray edge like contribution to the renormalization of vσ . This can be com˜
puted by a method analogous to that used by Schotte
and Schotte40 to solve the X ray edge problem, which involves transforming the non interacting fermions to even
and odd parity scattering states and then rebosonizing.
This approach was used to study the X ray edge problem
in a Luttinger liquid in Ref. 41.
We begin by writing (2.33), H = Hσ + Hρ with

˜† z

where φ± obey, [φ± (x), φ± (x′ )] = 2πisgn(x − x′ ). δ− =
δe − δo is related to the transmission probability by

√

2X

0

dx
tan−1 x
x

2

.

(3.79)

Thus, for X < 0, |X| ≫ ǫ and ǫ → 0 we ﬁnd
G1/2+ǫ (X) = |X|e−F (|X|)/ǫ.

(3.80)

The asymptotic behavior F (X) = X/π 2 for |X| ≪ 1
reproduces (3.58) when |X| ≫ ǫ. For |X| ≫ 1 we ﬁnd
F (X → ∞) =

1
7ζ(3)
.
log 2X −
8
4π 2

(3.81)

where ζ(3) = 1.20 is the Riemann zeta function. This
gives the asymptotic behavior
G1/2+ǫ (X → −∞) =
which is quoted in Table II.

e14ζ(3)/π
2|X|

2

1
8ǫ

,

(3.82)

20
IV.

DISCUSSION AND CONCLUSION

In this paper we have examined several novel properties of a point contact in a QSHI. We showed that the
pinch-oﬀ as a function of gate voltage is governed by
a non trivial quantum phase transition, which leads to
scaling behavior of the conductance as a function of temperature and gate voltage characterized by a universal
scaling function. We computed this scaling function and
other properties of the critical point in certain solvable
limits which provide an overall picture of the behavior as
a function of the Luttinger liquid parameter g.
In addition, we showed that the four terminal conductance has a simple structure when expressed in terms of
the natural variables, GAB , and that at the low temperature ﬁxed points, the leading corrections to the diﬀerent components of GAB can have diﬀerent temperature
dependence. In particular, we showed that the skew conductance GXY vanishes as T γ with γ ≥ 2.
Finally, we showed that for strong interactions, g <
1/2, the stable phase is the time reversal breaking insulating phase. Transport in that phase occurs via novel
fractionalized excitations that have clear signatures in
noise correlations.
There are a number of problems for future research
that our work raises. We will divide the discussion into
experimental and theoretical issues.
A.

Experimental Issues

The QSHI has been observed in transport experiments
on HgTe/HgCdTe quantum well structures. A crucial
issue is the value of the interaction parameter g. A simple estimate can be developed based on the long range
Coulomb interaction43 . First consider the limit ξ ≫ w,
where w is the quantum well width and ξ is the evanescent decay length of the edge state wavefunction into the
bulk QSHI. We model the edge state as a two dimensional charged sheet with a charge density proﬁle proportional to θ(x) exp(−2x/ξ), a distance d above a conducting ground plane. The long range interaction then
leads to u2 = u4 = (2e2 /ǫ) log(4eγ d/ξ), where ǫ is the
dielectric constant and γ = .577 is Euler’s constant. As
a second model, assume ξ ≪ w, and model the edge state
as a uniformly charged two dimensional strip of width w
perpendicular to a ground plane a distance d away. This
gives u2 = u4 = (2e2 /ǫ) log(2e3/2 d/w). The intermediate regime ξ ∼ w can be solved numerically, and we ﬁnd
that it is accurately described by a simple interpolation
between the above limits with 4d/(ξe−γ + 2we−3/2 ) in
the log. This leads to44 .
g = 1+

2 e2
log
π ǫ vF

7.1d
ξ + 0.8w

−1/2

.

(4.1)

For ǫ = 15, vF = .35eVnm, ξ = 2 vF /Egap ∼ 30nm
(Egap is the gap of the bulk QSHI), w = 12nm and d =

150nm45 this predicts g ∼ 0.8. The critical exponent
governing the temperature dependence of the pinch-oﬀ
curve (1.1) is then αg ∼ .02. In the CC and II phase the
conductance vanishes as T δ with δg = g + g −1 − 2 ∼ .05.
The good news is that since g is close to 1 the low temperature scaling behavior should be accurately described
by the scaling function (3.9) computed in the limit g → 1.
The bad news, is that the smallness of αg and δg mean
that it will be diﬃcult to see much dynamic range in the
conductance as a function of temperature. Nonetheless,
it may be possible to observe logarithmic corrections to
the conductance as a function of temperature, and by
comparing pinch-oﬀ curves at diﬀerent temperatures it
may be possible to observe the predicted sharpening of
the transition as temperature is lowered.
The skew conductance GXY is predicted be zero for
non interacting electrons, and with weak interactions
vanishes as T 2 . This is a consequence of the unique edge
state structure of the QSHI, and remains robust when
the interactions are weak.
To probe the critical behavior of the pinch-oﬀ transition, as well as the more exotic strong interaction phases
it would be desirable to engineer structures with smaller
g. Perhaps this could be accomplished by modifying either the dielectric environment or the bare Fermi velocity of the edge states. Maciejko et al.12 have suggested
that this may be possible using InAs/GaSb/AlSb type-II
quantum wells46,47 .

B.

Theoretical Issues

Our work points to a number of theoretical problems
for future study. It would be very interesting if the powerful framework of conformal ﬁeld theory can be used to
analyze the intermediate critical ﬁxed point as well as the
crossover √
scaling function. Perhaps the ﬁrst place to look
is g = 1/ 3. Maybe it is possible to take advantage of
the triangular symmetry of the QBM problem to develop
a complete description of the critical ﬁxed point, analogous to the mapping to the 3 channel Kondo problem25
and the 3 state Potts model24 that apply in a diﬀerent
regime. In the absence of an analytic solution, this problem is amenable to a numerical Monte Carlo analysis
analogous to the calculation of the resonance crossover
scaling function performed in Ref. 19.
In addition, there are a number of other ﬁxed points
which we did not analyze in detail in this paper. (Recall
for g = 1 − ǫ we found seven). It would be of interest
to develop a more systematic classiﬁcation of all of the
ﬁxed points, analogous to the analysis of three coupled
Luttinger liquids performed by Oshikawa, Chamon and
Aﬄeck and Hou39,48 .

21
Acknowledgments

It is a pleasure to thank Claudio Chamon and Eun-Ah
Kim for introducing us to their work and Liang Fu for
helpful discussions. This work was supported by NSF
grant DMR-0605066.
APPENDIX A: FOUR TERMINAL
CONDUCTANCE

The electrical response of the point contact can be
characterized by a four terminal conductance,
Gij Vj ,

Ii =

where Ii is the current ﬂowing into lead i and Vj is the
voltage at lead j. In this appendix we will develop a
convenient representation for Gij . Section 1 shows that
Gij can be characterized by a 3 × 3 matrix, whose entries have a clear physical meaning. This representation
allows constraints due to symmetry to be expressed in
a simple way, which reduces the number of independent
parameters characterizing the conductance. Finally, in
section 3 we show how Gij is related to the conductance
of the SLL model computed by the Kubo formula.
Conductance matrix

The 4 × 4 matrix Gij is constrained by current conservation to satisfy i Gij = j Gij = 0. In the absence of
any symmetry constraints, there are thus 9 independent
parameters characterizing Gij . In this section we will
cast these 9 numbers as a 3 × 3 matrix, in which each
of the entries has a clear physical meaning. In this representation constraints due to symmetry have a simple
form.
Since the four currents Ii satisfy i Ii = 0, they are
determined by three independent currents, which we deﬁne as IA = (IX , IY , IZ ), and satisfy
Ii =

MiA IA ,

(A2)

α

where the 4 × 3 matrix MiA is


1 1 1
1  −1 1 −1 


M= 
.
2  −1 −1 1 
1 −1 −1

(A3)

IX = I1 + I4 is the total current ﬂowing from left to right
along the Hall bar, whereas IY = I1 + I2 is the current
ﬂowing from top to bottom. The third current IZ =
I1 + I3 is the current ﬂowing in on opposite leads (1 and
3) and ﬂowing out in leads 2 and 4. Similarly, the voltages

T
MBj Vj .

VB =

(A4)

j

VX biases leads 1 and 4 relative to leads 2 and 3, VY
biases leads 1 and 2 relative to leads 3 and 4, and VZ
biases leads 1 and 3 relative to leads 2 and 4.
The new currents and voltages are then related by a
3 × 3 conductance matrix
GAB VB .

IA =

(A5)

β

(A1)

j

1.

Vi , which are deﬁned up to an additive constant, deﬁne
three independent voltage diﬀerences Vβ = (VX , VY , VZ ),
with

The 9 elements of GAB determine the four terminal conductance matrix,
T
MiA GAB MBj .

Gij =

(A6)

AB

The elements of GAB have a simple physical interpretation. GXX is the “two terminal” conductance measured
horizontally in Fig. 1 by applying a voltage to leads 1
and 4 and measuring the current I1 + I4 . Similarly GY Y
is a two terminal conductance measured vertically. GZZ
describes a two terminal conductance deﬁned by combining the opposite leads 1 and 3 together into a single
lead (and similarly for leads 2 and 4). GXY is a “skew”
conductance describing the current I1 + I4 in response to
voltages applied to leads 1 and 2. The other oﬀ diagonal
conductances can be understood similarly.

2.

Symmetry Constraints

The form of GAB simpliﬁes considerably in the presence of symmetries.

a.

Time Reversal Symmetry

In the presence of time reversal symmetry the four
terminal conductance obeys the reciprocity relation42 ,
Gij = Gji . This implies GAB = GBA . Thus, with time
reversal symmetry the conductance has 6 independent
components.

b.

Spin Rotational Symmetry

When the spin Sz is conserved the current of up and
down spins ﬂowing into the junction must independently
be conserved. It follows that
I1,in + I3,in = I2,out + I4,out
I2,in + I4,in = I1,out + I3,out .

(A7)

22
Since in the Fermi liquid lead (where the interactions
have been turned oﬀ) we have Ii,in = (e2 /h)Vi , this implies that
I1 + I3 = −I2 − I4 =

e2
(V1 + V3 − V2 − V4 ).
h

(A8)

It then follows that
GZZ = 2e2 /h
GZX = GZY = 0.

(A9)

Thus, which spin conservation the conductance is characterized by 3 components: the two terminal conductances
GXX , GY Y and the skew conductance GXY .
The quantization of GZZ and vanishing of GZB are
therefore a diagnostic for the conservation of spin.
Though spin orbit terms violating Sz conservation are
generically present, we will argue that at the low energy
ﬁxed points of physical interest the conservation of spin
is restored.
c.

GXY = 0.

(A10)

Though mirror symmetry is not generically present in a
point contact we will argue that that symmetry is restored in the low energy ﬁxed points of interest. Moreover, the crossover between the critical ﬁxed point and
the stable ﬁxed point described by (1.1) is also along a
line with mirror symmetry. Thus the crossover conductance is characterized by two parameters, GXX and GY Y ,
which are simply the two terminal conductances.
Critical conductance

At the transition, where the point contact is just being
pinched oﬀ the two terminal conductances must be equal,
GXX = GY Y ≡ G∗ .

Relation to Kubo conductance

In this section we relate the conductance matrix GAB
to the conductances of the SLL model, which can be computed with the Kubo formula. There are two issues to

(A12)

Similarly, deﬁne charge and spin voltages
Vρ = (V1 + V4 − V2 − V3 )/2
Vσ = (V1 − V4 + V2 − V3 )/2.

(A13)

These are related by the conductance matrix.
Iα = Gαβ Vβ ,

(A14)

where α, β = ρ, σ. By comparing (A5) and (A14) it is
clear that
GXX = Gρρ
GY Y = 2e2 /h − Gσσ
GXY = Gρσ = −Gσρ .

(A15)

Gαβ can be computed using the Kubo formula using
the model in which the interactions are turned oﬀ for
x > L. It is useful, however, to relate this to the Kubo
conductance GK of an inﬁnite Luttinger liquid. This
αβ
can be done by relating the voltage Vα=ρ,σ of the Fermi
¯
liquid leads with gρ = gσ = 1 to the voltage Vα of the
incoming chiral modes of the Luttinger liquid with gρ = g
and gσ = 1/g. By matching the boundary conditions at
x = L this contact resistance has the form

(A11)

In addition, we will argue that this ﬁxed point also has
spin rotational symmetry and mirror symmetry. Thus,
the critical four terminal conductance Gij depends on a
single parameter G∗ .
3.

Iρ = I1,in + I4,in − I1,out − I4,out
Iσ = I1,in − I4,in + I1,out − I4,out .

Mirror Symmetry

If the junction has a mirror symmetry under interchanging leads (1, 2) ↔ (3, 4) or (1, 4) ↔ (2, 3), it follows
that

d.

be addressed. First is to translate GAB into the spin and
charge conductances of the SLL model. Second, we must
relate the physical conductance measured with leads to
the conductance computed with the Kubo formula. The
Kubo conductance describes the response of an inﬁnite
Luttinger liquid, where the limit L → ∞ is taken before
ω → 0. This does not take into account the contact resistance between the Luttinger liquid and the electron
reservoir where the voltage is deﬁned. An appropriate
model to account for this is to consider a 1D model for
the leads in which the Luttinger parameter g = 1 for
x > L35,36 .
In this section we assume time reversal symmetry and
that spin is conserved. In this case we may deﬁne the
charge and spin currents in the Fermi liquid leads (x > L)
to be,

c
˜
Vα − Vα = Rαβ Iβ

(A16)

with
c
Rαβ =

h gα − 1
δαβ .
e2 2gα

(A17)

The Kubo formula with inﬁnite leads relates Iα =
¯
GK Vβ . Eliminating Vα from (A16) and (A17) gives the
αβ
39
matrix relation
Gαβ =

I − Rc GK

−1

GK

αβ

.

(A18)

23
When there is mirror symmetry, so that GXY = µρσ =
0, the conductance matrix is diagonal, so that (A18) simpliﬁes. In that case we ﬁnd
GXX =

1

2
3
4
5

6

7

8
9
10

11
12
13
14
15
16
17

18
19
20

21
22

23
24

GY Y = 2

e2
GK
σσ
−
.
h
1 − Rσσ GK
σσ

(A19)

GK
ρρ
1 − Rρρ GK
ρρ

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