A junction of three quantum wires: restoring time-reversal symmetry by interaction
X. Barnab´-Th´riault,∗ A. Sedeki, V. Meden, and K. Sch¨nhammer
e
e
o

arXiv:cond-mat/0411612v2 [cond-mat.mes-hall] 9 Mar 2005

Inst. f. Theoret. Physik, Universit¨t G¨ttingen, Friedrich-Hund-Platz 1, D-37077 G¨ttingen, Germany
a
o
o
We investigate transport of correlated fermions through a junction of three one-dimensional quantum wires pierced by a magnetic ﬂux. We determine the ﬂow of the conductance as a function of
a low-energy cutoﬀ in the entire parameter space. For attractive interactions and generic ﬂux the
ﬁxed point with maximal asymmetry of the conductance is the stable one, as conjectured recently.
For repulsive interactions and arbitrary ﬂux we ﬁnd a line of stable ﬁxed points with vanishing
conductance as well as stable ﬁxed points with symmetric conductance (4/9)(e2 /h).

Electronic transport through quasi one-dimensional
(1d) systems is of current experimental and theoretical
interest. In one spatial dimension correlations play an
important role and the physical properties of interacting fermions show distinctive non-Fermi liquid features.
Generically such systems can be described as TomonagaLuttinger liquids (TLLs) characterized by a vanishing
quasi-particle weight and power-law scaling of correlation functions.[1] For spinless fermions, on which we focus here, the characteristic exponents are expressible in
terms of the interaction dependent TLL parameter K.
For repulsive interactions 0 < K < 1, while K > 1 in
the attractive case. In TLLs inhomogeneities have a dramatic eﬀect as can be inferred from the singular behavior
of response functions of homogeneous models.[2, 3]
In an important ﬁrst step the conductance G through
a TLL with a single impurity was understood.[4, 5]. For
vanishing energy scale s (e.g. temperature) and 0 < K <
1, G tends towards 0 following a power-law. The lowenergy physics is governed by the “decoupled chain” ﬁxed
point (FP). The scaling dimension of a hopping term connecting two open ends is 1/K and leads to G ∝ s2(1/K−1) .
One can understand this behavior in a simple picture.
Due to the interaction the self-energy develops long range
oscillatory behavior and the scattering oﬀ the resulting
eﬀective impurity potential leads to the power-law suppression of G.[6, 7, 8] For K > 1, the conductance approaches the impurity free limit. Close to the “perfect
chain” FP an impurity has scaling dimension K and
the correction to the impurity free conductance scales
as s2(K−1) . In this case the eﬀective impurity potential
leads to a resonance at the chemical potential.
Recently junctions of several quasi 1d quantum wires
were realized experimentally with single-walled carbon
nanotubes.[9, 10] They might form the basis of electronic
devices. Already the physics of the three wire junction
(Y-junction) is considerably richer than the one of a single impurity. Taking into account correlations transport
through such systems was investigated theoretically in
Refs. 11 and 12. These studies posed a number of interesting questions. In Ref. 12 a symmetric triangular Yjunction pierced by a magnetic ﬂux φ (measured in units
of the ﬂux quantum hc/e) was studied. In this geometry
time reversal symmetry is broken. In the non-interacting

case this generically leads to an asymmetry of the conductance from wire ν to wire ν ′ and vice versa: Gν,ν ′ = Gν ′ ,ν .
For K > 1 at ﬂux φ = ±π/2 one FP was found applying
an exact method adopted from boundary conformal ﬁeld
theory. The FP corresponds to the case of maximal asymmetry of Gν,ν ′ . We here consider TLL wires connected to
semi-inﬁnite Fermi liquid (FL) leads with TLL-FL contacts that are modeled to be free of fermion backscattering. The results of Ref. 12 obtained for semi-inﬁnite
TLLs can be extended to our modeling. Then maximal
asymmetry is given by Gν,ν ′ = 1(0) and Gν ′ ,ν = 0(1) in
units of the conductance quantum e2 /h.[13] The scaling
dimension of the most relevant operator at this FP is
∆ = 4K/(3 + K 2 ) and the correction to the FP conductance scales as s2(∆−1) .[12] For 1 < K < 3, 2(∆ − 1) > 0
and the “maximal asymmetry” FP is attractive. This
implies that independently of the junction parameters at
low energy scales the conductance is 1 from ν to ν ′ and
0 from ν ′ to ν or vice versa. It was conjectured that for
1 < K < 3 and all φ, except |φ|/π being integer, this FP
is the only stable one.
We investigate the same physical problem considering
arbitrary ﬂuxes and attractive as well as repulsive interactions. An approximation scheme based on the functional renormalization group (fRG) method is used. It
was earlier applied successfully to transport problems in
inhomogeneous TLLs. By comparison with numerical results and exact scaling relations this scheme was shown
to be reliable for 1/2 ≤ K ≤ 3/2. In particular, the
scaling dimensions of the two FPs of the single impurity problem come out correctly to leading order in the
interaction U .[7, 8] For the Y-junction we here conﬁrm
the conjectured stability of the “maximal asymmetry”
FP for all φ, with |φ|/π not integer, and reproduce the
scaling dimension ∆ to leading order in U . For U > 0
and arbitrary φ a line of stable FPs with G = 0 is identiﬁed. In cases with symmetric conductance we suppress
the indices on G. We surprisingly ﬁnd additional “perfect junction” FPs with symmetric conductance G = 4/9
that for U > 0 are stable and have a scaling dimension
not discussed before. In a Y-junction of non-interacting
wires without ﬂux G = 4/9 is the value maximally allowed by symmetry.[11] Although time reversal symmetry is explicitly broken, for systems that ﬂow towards the

2
“perfect junction” FPs the electron correlations cause the
conductance – an observable that is commonly believed
to indicate this breaking of symmetry – to behave as if
the symmetry is restored. In a certain sense (see below)
this also holds for systems that ﬂow towards the line of
G = 0 FPs and thus for all parameters.
2

tY
1

V

φ

t∆
3

FIG. 1: Symmetric junction of three quantum wires.

Our wire Hamiltonian is
∞

Hν = −

c† cj+1,ν + h.c.
j,ν
j=1
N −1

+
j=1

Uj nj,ν −

1
2

1
2

nj+1,ν −

(1)

in standard second-quantized notation. The diﬀerent
wires are indicated by an index ν = 1, 2, 3. The amplitude of the nearest neighbor hopping is set to 1. The
nearest neighbor interaction Uj is allowed to depend on
the position. It is set to zero for j > N , i.e. the wires of
length N are connected to non-interacting leads. Close
to the contacts the interaction is switched oﬀ (spatially)
smoothly to avoid any fermion backscattering.[8] The
bulk value of the interaction is denoted by U . We here
consider the half-ﬁlled band case. To prevent depletion
of the interacting region we shifted the operator nj,ν by
the average density 1/2. The model with interaction U
across all bonds (not only the ones within [1, N ]) shows
TLL behavior for |U | < 2 with[14]
K=

U
2
arccos −
π
2

−1

.

(2)

The Y-junction sketched in Fig. 1 is described by
3

3

HY = −tY
−t△

c† c0,ν + h.c. + V
1,ν
ν=1
3

n0,ν
ν=1

eiφ/3 c† c0,ν+1 + h.c.
0,ν

,

(3)

ν=1

where we identify the wire indices 4 and 1.
Our starting point to calculate the linear response con3
ductance of ν=1 Hν + HY is an exact hierarchy of differential ﬂow equations for the self-energy ΣΛ and higher
order vertex functions (in the presence of the interaction, the leads, and the junction), with an infrared energy cutoﬀ Λ as the ﬂow parameter. It is derived using the fRG. The hierarchy is truncated by neglecting

n-particle vertices with n > 2, and the 2-particle vertex is parametrized by a renormalized nearest neighbor
interaction U Λ . This implies that terms of order U 2
are only partly taken into account and that ΣΛ is frequency independent. The important spatial dependence
of ΣΛ , is however fully kept. The details of this procedure are given in Refs. 7 and 8. At the end of the ﬂow (at
Λ = 0), the self-energy can be regarded as an eﬀective,
N -dependent impurity potential Σ with non-vanishing
matrix elements Σj,j and Σj,j+1 , where j is restricted to
the interacting region. Due to the symmetry of the junction the matrix elements are the same for the three TLL
wires. We here focus on the zero temperature case for
which the ﬂow equations can numerically be solved for
up to N = 107 lattice sites.[7, 8] Generically Σj,j and
Σj,j+1 oscillate around an average value with an amplitude that decays slowly with increasing distance from the
junction.
The conductance Gν,ν ′ = |tν,ν ′ |2 can be calculated
from the eﬀective transmission tν,ν ′ (at the chemical potential µ = 0), which in turn can be expressed in terms
of real space matrix elements of one-particle Green functions of the system. Using single particle scattering
theory,[8] the conductance can be written as
2

Gν,ν ′ =

4 (Im g) e−iφ − g

2

|g 3 − 3g + 2 cos φ|2

,

(4)

with g = (−V − t2 G1,1 )/|t△ |. The Green function G is
Y
obtained by considering Σ as an eﬀective potential for a
single wire setting tY = 0 and G1,1 denotes its diagonal
matrix element taken at site j = 1. It is evaluated at energy ε + i0 with ε → 0. Eq. (4) holds if ν, ν ′ are in cyclic
order. Gν ′ ,ν follows by replacing φ → −φ. For symmetry
reasons we only have to consider 0 ≤ φ ≤ π/2. In particular Eq. (4) can be applied for U = 0 and shows that
the conductance can be parameterized by the ﬂux and
a single complex parameter g. Via the ﬂow of the selfenergy G1,1 for U = 0 develops an additional dependence
on (tY , t△ , V ), U , and most importantly on N . The energy scale δN = πvF /N (with the Fermi velocity vF ) is
a natural infrared cutoﬀ of our problem.[8] To obtain a
comprehensive picture of the low-energy physics we investigate the ﬂow of g as a function of δN and use Eq.
(4) to calculate Gν,ν ′ for a given g. In Fig. 2 the ﬂow of
g is shown for diﬀerent φ (and U = −1). Each line is obtained for a ﬁxed set of junction parameters (tY , t△ , V )
with N as a variable. As Im g has the opposite sign of
Im G1,1 < 0 it is restricted to positive values.
Eq. (4) allows for three special conductance situations
which turn out to be the FPs of the ﬂow. On the real
axis (green in Fig. 2), the conductance G vanishes except
at special, φ-dependent points. They are given by the
crossings of the real axis with the set C(φ) (red in Fig. 2),
on which the reﬂection R = 1 − |tν,ν ′ |2 − |tν ′ ,ν |2 takes a
local minimum. For φ = 0, C(0) is given by the circle

3
2

2

a)

φ=0

1
0.5

−2

−1

0
Re g

1

2

0
−3

3

2

c)

φ=0.4π

1
0.5
0
−3

−2

−1

0
Re g

1

d)

2

3

φ=0.5π

1.5
Im g

1.5
Im g

1
0.5

0
−3
2

φ=0.1π

1.5
Im g

Im g

1.5

b)

1
0.5

−2

−1

0
Re g

1

2

3

0
−3

−2

−1

0
Re g

1

2

3

FIG. 2: (color online) Flow of g for diﬀerent φ. Arrows indicate the direction for U < 0. For U > 0 it is reversed. For details
see the text.

(Re g + 1/2)2 + (Im g)2 = (3/2)2 . For φ > 0, C(φ) has a
more complex analytical form not presented here. At the
crossings the conductance is symmetric with G = 4/9.
They are located at g = −2 cos (φ/3), 2 cos ([π ± φ]/3)
and in Fig. 2 are indicated as blue circles and a yellow
diamond. As a peculiarity of φ = 0 on C(0) one ﬁnds
G = 4/9 for all Im g, not only Im g = 0. A situation with
G = 0 is also reached for |g| → ∞. For 0 < φ ≤ 1/2
and g = eiφ Eq. (4) yields Gν,ν ′ = 1 and Gν ′ ,ν = 0. At
this point, indicated by cyan squares in Fig. 2, maximal
asymmetry of the conductance is achieved.
We next discuss the FP scenarios depicted in Figs. 2 a)d). For U = 0 the general form of the ﬂow diagrams
is independent of the absolute value and sign of U . In
Figs. 2 a)-d) results for U = −1 are shown. At φ = 0
we ﬁnd two “perfect junction” FPs with G = 4/9 and a
line of “decoupled chain” FPs with G = 0 (green line).
For all U < 0 the “perfect junction” FP at g = 1 (yellow diamond) is the only stable FP. All trajectories are
attracted towards C(0) and reach this FP following C(0).
For all U > 0 it turns unstable and the line of “decoupled
chain” FPs is stable. In addition the “perfect junction”
FP at g = −2 (blue circle) is stable. The basin of attraction of g = −2 is given by C(0). Increasing φ from 0 the
“perfect junction” FP at g = 1 splits up into three FPs
– the two “perfect junction” FPs at g = 2 cos ([π ± φ]/3)
and the “maximal asymmetry” FP at g = eiφ . For all

U < 0 the latter is the only stable FP but becomes unstable for U > 0. A third “perfect junction” FP is located
at g = −2 cos (φ/3). For all U > 0 each of the three
“perfect junction” FPs has a basin of attraction given by
one of the three parts of C(φ) which are separated by the
“maximal asymmetry” FP. In addition for U > 0 the line
of “decoupled chain” FPs is stable. This scenario holds
up to φ = π/2, at which the “perfect junction” FPs are
√
at g = ± 3, g = 0, and the “maximal asymmetry” FP
is at g = i. For φ = π/2 and U > 0 there is a single
trajectory that runs along the imaginary axis to inﬁnity
(leading to G = 0) and does not bend back towards the
real axis as all other trajectories at U > 0 do. In the
mapping of the complex plane onto the Riemann sphere
the g = ∞ FP (north pole) is part of the projected line
of “decoupled chain” FPs and shows the same stability
properties and scaling dimension.
To obtain the scaling dimensions of the FPs we generically analyze the scaling of Gν,ν ′ as a function of δN
with respect to the FP conductance which is 1, 4/9, or
0 depending on the FP studied. For suﬃciently large
N (in some cases up to 105 sites are required) we ﬁnd
power-law scaling close to all FPs. In cases were the FPs
are not stable this scaling does not present the asymptotic behavior, but exponents smaller than 0 can still
be read oﬀ at intermediate N . Approaching the “perfect junction” FPs along C(φ) for φ > 0 Eq. (4) yields

4
Gν,ν ′ − 4/9 ∝ Im g, i.e. Gν,ν ′ − 4/9 and Im g scale with
the same exponent. The scaling dimensions of the “perfect junction” FPs at φ = 0 cannot be read oﬀ from the
conductance as G = 4/9 on C(0) and we use the scaling
of Im g.

exponent

0.8
fRG, "max. asym." FP
2(∆-1)
fRG, "perf. junct." FP
U/(3π)

0.4

0

-0.4
-1.5

-1

-0.5

U

0

0.5

1

1.5

FIG. 3: Scaling exponents close to FPs.

The scaling exponent of the “maximal asymmetry” FP
is independent of the ﬂux φ > 0 and the direction from
which it is approached (U < 0) or in which it is left
(U > 0). Its U -dependence is shown in Fig. 3 (squares)
and agrees to leading order in U with the prediction
2(∆ − 1) (solid line),[12] with ∆ = 4K/(3 + K 2 ) and
K given in Eq. (2). In our scheme terms of order U 2 are
only partly included and we cannot expect agreement to
higher order. The non-monotonic behavior of 2(∆ − 1) is
not reproduced and in our approximation the “maximal
asymmetry” FP stays attractive for all U < 0. For small
to intermediate |U | with TLL parameters 1/2 ≤ K ≤ 3/2
we conﬁrm the conjecture of Ref. 12.
For the line of “decoupled chain” FPs we ﬁnd for all φ
the same scaling exponent βs , as found for the single impurity problem close to the respective “decoupled chain”
FP, applying the fRG. Its dependence on U is shown in
Ref. 8 and agrees to leading order with 2(1/K − 1).
As discussed above the φ = 0 “perfect junction” FP
at g = 1 [yellow diamond in Fig. 2 a)] has properties
diﬀerent from those of the other “perfect junction” FPs
(blue circles in Fig. 2). The scaling exponent γ of the
latter FPs is independent of φ and shown in Fig. 3 (circles). To leading order it is given by γ ≈ U/(3π) (dashed
line). This form does not coincide with the expansion
of any of the above K-dependent exponents [after using
K ≈ 1 − U/π; see Eq. (2)]. We thus ﬁnd a new scaling
dimension. Since higher order terms are only partly included we cannot determine the functional dependence
of γ on K. To derive such an expression presents a challenge for methods that do not require approximations in
the strength of the interaction. For the φ = 0 “perfect
junction” FP at g = 1 the scaling exponent (of Im g) is,
up to our numerical accuracy, equal to −γ. For junction parameters that initially do not fall onto C(0), but
are close to it, one can read oﬀ an exponent from the
conductance that describes how C(0), with G = 4/9, is

approached (U < 0) or left (U > 0). It is equal to the
fRG exponent βw of the “perfect chain” FP in the single
impurity problem. Its U -dependence is presented in Ref.
8 and agrees with 2(K − 1) to leading order.
The appearance of stable FPs with symmetric conductance G = 4/9 at U > 0 is a surprising result.
Even though time reversal symmetry is explicitly broken at φ > 0, due to correlations the conductance of
systems with parameters on C(φ) behaves as if time reversal symmetry is restored. It is remarkable that close to
the line of “decoupled chain” FPs the relative diﬀerence
β /2
|Gν,ν ′ − Gν ′ ,ν |/(Gν,ν ′ + Gν ′ ,ν ) scales as δNs and thus
vanishes if U > 0. This implies that Gν,ν ′ and Gν ′ ,ν become equal faster than they go to zero. In that sense also
on this line of FPs and thus for all junction parameters
time reversal symmetry is restored if U > 0.
Using the fRG we determined the complete renormalization group ﬂow for a TLL Y-junction pierced by a magnetic ﬂux. Besides uncovering a new type of low-energy
physics we demonstrated the power of our approximation
scheme. Usually ﬁeld theoretical models are used to investigate the transport properties of inhomogeneous TLL
applying methods that are speciﬁc to such models. This
way exact results for either fairly simple geometries (single impurity[5]) or restricted parameter regimes[12] were
obtained. Our method can be applied to microscopic
models with arbitrary junction parameters and provides
results for the conductance that are accurate for small to
intermediate |U |. It can also be used to study transport
on intermediate and large energy scales.[8] Furthermore,
the technique can be applied to investigate the transport
through more complex networks of TLLs that might become important in future nano-electronic applications.
We thank C. Chamon, S. Andergassen, T. Enss,
W. Metzner, and Th. Pruschke for discussions. V.M.
and K.S. are grateful to the Deutsche Forschungsgemeinschaft (SFB 602) for support.

∗

[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]

Xavier B.-T. passed away in a tragic traﬃc accident on
August 15, 2004.
For a review see K. Sch¨nhammer in Interacting Eleco
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5
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