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PHYSICA L R EVIEW LET T ERS

Junctions of Three Quantum Wires and the Dissipative Hofstadter Model
Claudio Chamon,1 Masaki Oshikawa,2 and Ian Affleck1
1

2

Department of Physics, Boston University, Boston, Massachusetts 02215, USA
Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551 Japan(Received 25 April 2003; published 13 November 2003)
We study a junction of three quantum wires enclosing a magnetic flux. This is the simplest problem of
a quantum junction between Tomonaga-Luttinger liquids in which Fermi statistics enter in a nontrivial
way. We present a direct connection between this problem and the dissipative Hofstadter problem, or
quantum Brownian motion in two dimensions in a periodic potential and an external magnetic field,
which in turn is connected to open string theory in a background electromagnetic field. We findnontrivial fixed points corresponding to a chiral conductance tensor leading to an asymmetric flow of
the current.
DOI: 10.1103/PhysRevLett.91.206403

Electric conduction in quantum wires is of much current interest. From a practical viewpoint, continuing advance in electronic technology is now reaching a level
which requires understanding electric conduction in thequantum limit. From a basic science perspective, conduction in low-dimensional systems exhibits many interesting properties due to strong correlation effects. A seminal
example is discussed by Kane and Fisher (KF) [1], who
predicted that backscattering due to a single impurity,
however small, makes the conductance vanish at low
temperature in the presence of repulsive interactions.
This is a clear manifestation of the non-Fermi-liquidbehavior of interacting electrons in one dimension, which
can generally be regarded as a Tomonaga-Luttinger liquid(TLL). A TLL is nothing but a field theory of free bosons
in 1  1 dimensions, and the effect of the impurity can bestudied as a boundary interaction in the field theory;
renormalization-group (RG) fixed points can be identified
with conformally invariant boundary conditions. Study ofboundary conditions in conformal field theory is an active
research area in itself, with applications to string theory
and statistical mechanics.
In order to make any nontrivial circuit with quantum
wires, one needs a junction of three or more wires, which
generalizes the KF problem. This turns out to be a surprisingly rich problem [2 –8] with a number of open
questions, even for three wires. One of the novel aspectsof the multilead problem is that, unlike the junction of
two wires (KF problem), electron fermion statistics play a
crucial role [2].
Here we study a junction of three quantum wires withan enclosed magnetic flux , as shown in Fig. 1. Using a
low-energy limit Dirac fermion formulation, the physicaljunction becomes equivalent to electron hopping between
wires, represented by the boundary term
3
X
HB   ei=3 jy j1  H:c:  r jy j :
(1)
j1

Here

j is the electron annihilation operator for the wire j

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PACS numbers: 71.10.Pm, 05.30.Fk, 11.25.Hf

at the junction (with 0
3 ).  is the (real) hopping
amplitude, r a backscattering potential, and  is essentially the dimensionless flux through the junction (forweak hopping). r has no effect on the conductance (inthe zero frequency, zero temperature limit), with the
exception of the noninteracting case, so we henceforth
ignore it. For simplicity, we consider spinless fermions,while considering arbitrary strength bulk interactions. In
a TLL, the interaction strength is essentially contained ina single parameter g, which determines various physicalquantities and exponents. g < 1 and g > 1 correspond,
respectively, to repulsive and attractive interactions, whileg  1 is the Fermi liquid.
In the presence of a magnetic flux  through thejunction, the combination of the quantum phases due tothe flux and fermion statistics is mapped to a ‘‘magnetic
field’’ for the free boson field at the boundary. Such atheory has been discussed from the viewpoint of open
string theory [9]. Moreover, the phase diagram in thepresence of both interactions and the magnetic field at
the boundary was studied in a beautiful paper by Callanand Freed (CF) [9]. We will show that a low-energy RG
fixed point of our junction problem is given by the ‘‘magnetic’’ boundary condition of CF, which leads to a chiralconductance tensor corresponding to an asymmetric flowof the current. The magnetic boundary condition exhibitsa nonmonotonic dependence on the TLL interaction

3
1

φ
2

FIG. 1. A quantum junction of three wires containing amagnetic flux .

 2003 The American Physical Society

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parameter g, unlike in other applications of TLL. In
particular, the magnetic fixed point is stable for 1 <g < 3. This means that an arbitrarily small magnetic
flux  leads to a substantial breaking of time-reversal
symmetry for this range of g.
The electron annihilation operator in wire j is represented as [1]pi’j = 2 ;(2)jjewhere j is the so-called Klein factor satisfying
f j ; k g  2jk , which is necessary to ensure the anticommutation relations of the fermion operators in different wires. The Klein factors may be represented by thePauli matrices. In the low-energy limit, the (imaginarytime) effective action for the three disconnected quantum
wires is given by the three component free boson
S

Z

d dx

3
X
g
j1

4

@ ’j 2 ;(3)

defined over the half-line x > 0, with the junction atx  0. Neumann boundary conditions @’j =@x  0 are
satisfied.
The boundary interactions lead to renormalization to
an infrared fixed point with a different boundary condition. However, current conservation at the junction
requires the ‘‘center-of-mass’’ field
1(4)
0  p ’1  ’2  ’3 
3to always obey Neumann boundary conditions. Thus theremaining degrees of freedom at the boundary comprise a
two component bosonfield ~   1 ; 2  defined
by
p
p
1  ’1  ’2 = 2,2  ’1  ’2  2’3 = 6. Using
this new basis, the hopping term is written as3Z
XiK~ a  ~  H:c:;SB  iei=3 d
(5)
a e
a1

p
p
where
K~ 1  1; 3=2; K~ 2  1;  3=2; K~ 3 
1; 0. The scaling dimension of the hopping term in the
disconnected limit (  0) is calculated by standard
methods [1] as 1=g, which is not affected by the Klein
factors nor the magnetic flux. However, if one attempts a
perturbation theory in the hopping amplitude , a phasefactor appears in each term. This makes the problem
rather different from the standard free boson field theory,and hence it is difficult to predict the nature of the strong
hopping limit. In particular, for g > 1, when the disconnected wire fixed point is unstable, it is difficult to
identify the infrared fixed point.
Each order of the expansion of the partition function
contains an integral over the correlation function~~~

heiL~ 1  1  eiL~ 2  2     eiL~ n  n  i0
X   X
1
 K
L~ j exp
L~ j  L~ k lnjj  k j2 : (6)g j>kj206403-2

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where K L~   1 if L~  0~ and K L~   0 otherwise, L~ j is
one of the six vectors K~ 1;2;3 , and h i0 denotes the expectation value with the Neumann boundary condition on ~ .
~Thus, identifying the vertex operator eiL~ j  with thedisplacement by L~ j in the two-dimensional plane, a nonvanishing contribution (6) corresponds to a closed loop
on a triangular lattice spanned by K~ 1 and K~ 2 . The perturbation series is given by the multipoint correlation
function (6) to any order, in principle. However, in thepresent problem, we pick up extra phase factors due to the
Klein factors (and the flux .) The phase factor for agiven loop is given by the product of phase factors ofelementary triangular loops. Referring to the two inequivalent triangles in a triangular lattice as ‘‘up’’ and‘‘down,’’ the phase factor for the counterclockwise loopon the up triangle is determined to bei 3 ei=3 i 2 ei=3 i 1 ei=3  ei ;(7)while that for the counterclockwise loop on the down
triangle isi 3 ei=3 i 2 ei=3 i 1 ei=3  ei :(8)

Namely, loops on the triangular lattice pick up a phase asif there is a staggered magnetic flux of  and    ineach elementary triangle.
Fortunately, we shall see that the above perturbative
expansion coincides with the ‘‘generalized Coulomb gas’’studied by CF [9] in the context of the dissipative
Hofstadter model (DHM) (quantum motion of a single
particle under a magnetic field and a periodic potential,
subject to dissipation). The ‘‘free’’ action reads:
1
S0 X~  2Z d!
 !X !;
 j!j!  "!#! X
!
22
(9)

where ; !  1; 2 and #12  #21  1, #11  #22  0.
and " are related to the dissipation and the magnetic field,
respectively. This determines the propagator:
D!   hX X! 0i2  "2ln2 !  i"
2  "2sgn#! :(10)

We now introduce a potential term, which is somewhat
different from the ‘‘rectangular’’ one in Ref. [9], asSV X~   Vei=3

Z

d

3
X

eiK~ a X~  c:c:;

(11)

a1

where V and  are chosen to be real. Expanding thepartition function in powers of V, a nonvanishing contribution is given by an integral over the correlation
function
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VOLUME 91, N UMBER 20

heiL~ 1 X~ 1  eiL~ 2 X~ 2     eiL~ n X~ n  i0  K

X  
L~ j expjwhere L~ j is again one of the six vectors K~ 1;2;3 .
The perturbation series in the DHM can be made tomatch that in the quantum wire model, including theKlein factors and flux, for appropriate choice of and". The absolute value of the correlation functions agree
exactly if
1(13)
2  "2  :g
The phase factor for any loop in the DHM is determinedby that for two elementary triangles:
p
"
"
3
 2
 ; (14)K~ 1  K~ 2   
2
2
2
"
 "2
where  and  signs apply to the up and down triangles,
just as in the quantum wire problem, (7) and (8). Thus, if
one chooses
p
"
3 2
 2n  1
(15)
 "2
with an integer n, there is an appropriate  which makes
the two expansions based on (6) –(8) and (12) coincide
exactly including the phases. Because the phases are only
defined modulo 2, there is actually an infinite number ofdifferent choices of and " labeled by the integer n. Eachchoice defines a quite different theory with respect to the
dynamical variable X~ , but gives the identical generalized
Coulomb gas.
Now let us consider the case of g > 1. Then, as we havealready discussed, the electron hopping is a relevant
perturbation at the disconnected limit. In the DHM, for
any choice of n, V is a relevant perturbation for g > 1. We
would like to find the infrared stable fixed point reachedin the low-energy limit. A simple guess is that it occurs at
V ! 1. The stability of the V ! 1 fixed point can be
determined using the instanton method [9]. Namely, inthe strong potential limit, the X~ field is pinned at one of
the minima of the potential (11). The leading perturbation
to this limit is given by a tunneling between the neighboring minima, represented by an ‘‘instanton.’’ A calculation
following Ref. [9] gives the scaling dimension of suchperturbation to betimes the distance between thenearest minima squared whereis determined by(13) –(15). For a generic value of , minima of the potentialp (11) form a triangular lattice with lattice constant2= 3. Thus the dimension of the most relevant operator at
this V ! 1 fixed point is
4g
n 
:
(16)
3  2n  12 g2
This implies that the V ! 1 fixed point is unstable forany value of g for all choices of n except for n  0 or 1,
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PHYSICA L R EVIEW LET T ERS
X

L~ j  L~ k lnjj  k j2
 " j>kX"~ j  L~ k sgnj  k  ; i 2L "2 j>k
2

2

(12)

where it is stable for 1 < g < 3. In this range of g we
expect that the infrared limit of the junction model
corresponds to the V ! 1 limit of the DHM for n  0
or 1. The infrared fixed point for other values of n mustcorrespond to nontrivial intermediate V fixed points, forwhich we do not know explicit solutions. The infrared
fixed point for each value of n should give the same
physical behavior for the junction model. In principle
we could study this using any choice of n. However,clearly the n  0 or 1 choices are preferred because
only in those cases can we explicitly analyze the fixed
points. Let us denote the RG fixed points corresponding
to the strong potential limit in these cases as & and & ,respectively, for n  0 and 1. Since & have the samestability, at this stage we cannot determine which of these
two fixed points is realized in the infrared limit for ageneric value of .
On the other hand, in the special cases   =2
which maximally break the time-reversal symmetry,the choice of the fixed point is unique. For   =2,
the choice n  1 (" positive) gives   0, and the potential minima forms a triangular lattice as usual. However,
the other choice n  0 (" negative) gives   , forwhich the potential minima form a honeycomb latticewith minimum distance 2=3, making the strong potential
limit unstable for all g. Similarly, for   =2, n  0(" negative) is the unique choice to give a stable fixed
point. This suggests that these fixed points & reflect the
breaking of time-reversal symmetry due to the magnetic
flux . Indeed, the conductance at these fixed points
exhibits a chiral behavior breaking the time-reversal invariance, as we show below. On the other hand, changing
the flux corresponds to an irrelevant perturbation at these
fixed points. Thus we conjecture that the flow from small
V goes to the V ! 1 fixed point in the n  1 representation, but not in the n  0 representation, where we expect
the flow to be to a nontrivial fixed point, for 0 <  < .
Conversely, for  <  < 0, we conjecture that the infrared fixed point is given by the V ! 1 one in the n  0
representation, not n  1.
The nature of the & fixed points is reflected in the
conductance at the junction, which is an experimentallymeasurable quantity. The conductance is expressed as atensorX
Ij  Gjk Vk ;(17)kwhere Ij is the total current flowing into the junction fromwire j and Vk is thePvoltage applied to wire k. From
current conservation j Gjk  0. It is also clear that thecurrents depend only on voltage differences. According to206403-3

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the Kubo formula, the conductance tensor is given by1@2 lnZ;(18)
Gjk   lim!!0 ! @Aj !@Ak !
where ! is the Matsubara frequency, and Aj and Vj are
related in real time t by dAj =dt  Vj . Assuming thatthe voltage drops only across the junction, its effect canbe represented as an extra time-dependent phase factor(vector potential) eiAj Ak  in the electron hopping termfrom wire k to wire j. It can be conveniently included in
the CF representation as
3ZX
SV X~   Vei=3 d
eiK~ j X~ a~   c:c:;
(19)
P

j1

~
j;k;l K j #jkl Ak  Al =3. Thus the conductance

where a~ is related to2@2 lnZ
4 X
 ! @ lnZ
#jm #kn Km
Kn  ! ;@Aj @Ak
9 m;n;;!
@a @a

(20)

where #12  #23  #31  1, #21  #32  #13  1, and#jj  0, ; !  1; 2 refers to the components of twodimensional vectors a~ and K~ j . In fact,@2 lnZ(21)
 h. !.! !i@a !@a! !
is the correlation function of the generalized Coulomb
gas density as defined in Ref. [9], which we use to determine the conductance tensor. In the strong potential limitof the CF representation, the X~ field is pinned and thecorrelation function is given exactly byh. !.! !i  j!j!  "!#! :(22)
From (18) –(22), we obtain the conductance tensor
G
jk  G3jk  1  g #jk =2;

(23)

with G  e2 =h4g=3  g2  and  corresponding to the
fixed points & . The term proportional to #jk represents
an asymmetric conduction which is allowed by the Z3
symmetry of the Hamiltonian once time-reversal symmetry is broken by the flux.
For example, when a voltage V1 > 0 is applied towire 1 while V2  V3  0, the current flowing fromwire 1 to the junction is I1  GV1 , which is independentof the asymmetry. On the other hand, the currents flowing
out from the junction to wires 2 and 3 are given by I2 
G1  gV1 =2 and I3  G1  gV1 =2. In fact, forg > 1, the current for fixed point & (& ) rather flows
in towards the junction from wire 3 (wire 2), opposite to
the voltage drop, when the voltage V1 > 0 is applied only
on wire 1.
The asymmetric conduction is forbidden under time
reversal but is possible in the presence of magnetic flux .
In the limit of g ! 1, which corresponds to noninteracting electrons, we find, for example, G11  G12  1,
G13  0 for & . This asymmetric conduction of noninteracting electrons can be simply understood as a boundinary condition jout  j1. At g  1, there is in fact acontinuous family of boundary conditions characterized
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P
by a 3  3 S matrix S as jout  k Sjk kin . Our & correspond to special cases of these free fermion boundary
conditions in the g ! 1 limit.
A remarkable aspect of the fixed points & is that the
conductance (23), as well as the scaling dimension (16),exhibits nonmonotonic dependence on the interaction
parameter g, unlike in other known applications ofTLL. Making the electron interaction more attractive
can decrease the conductance.
In summary, we studied a junction of three quantum
wires of interacting spinless electrons with a magnetic
flux. The fermionic statistics, together with the magnetic
flux, bring nontrivial phases into the problem, making it
different from the standard boundary problem of freeboson field theory. It is shown to be equivalent to a certain
generalized Coulomb gas introduced by CF. Using the
mapping, we have shown that the chiral fixed points &
exhibit an asymmetric conduction and nonmonotonic dependence on the interaction parameter g. The fixed points& are stable for 1 < g < 3, and we expect the junction
with   =2 (  =2) to be renormalized to the
fixed point & (& ), since it is the unique stable fixed
point obtained with the CF representation at V ! 1. We
conjecture that the system renormalizes to these fixed
points (for 1 < g < 3) for all values of  except thosewhich respect time reversal: = integer.
We note in passing that our results also apply to an
equivalent problem if we interpret ~ as the coordinate ofa neutral spin-1=2 particle (3) and (5), describe its propagation in a periodic textured Zeeman field with components Ba  ~   4 sinK~ a  ~  =3, and Hamiltonian
HZ  B~  ~   S~, subject to dissipation.
We thank P. Degiovanni, S. Rao, H. Saleur, and X.-G.
Wen for useful discussions. This work is supported in partby NSF Grants No. DMR-0203159 (I. A.), No. DMR9876208, and No. DMR-0305482 (C. C.), the A. P. Sloan
Foundation (C. C.), and a Grant-in-Aid for scientific research from MEXT of Japan (M. O.).

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(1992); Phys. Rev. B 46, 15 233 (1992).
[2] C. Nayak, M. P. A. Fisher, A.W.W. Ludwig, and H. H.
Lin, Phys. Rev. B 59, 15 694 (1999).
[3] S. Lal, S. Rao, and D. Sen, Phys. Rev. B 66, 165327(2002).
[4] S. Chen, B. Trauzettel, and R. Egger, Phys. Rev. Lett. 89,
226404 (2002).
[5] K-V. Pham, F. Piechon, K-I Imura, and P. Lederer, condmat/0207294.
[6] I. Safi, P. Devillard, and T. Martin, Phys. Rev. Lett. 86,4628 (2001).
[7] Hangmo Yi, Phys. Rev. B 65, 195101 (2002).
[8] J. E. Moore and X.-G. Wen, Phys. Rev. B 66, 115305(2002).
[9] See C. G. Callan and D. E. Freed, Nucl. Phys. B374, 543(1992), and references therein.

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