Fractional Helical Liquids and Non-Abelian Anyons in Quantum Wires

2 MB·English

by  Yuval Oreg

Fractional Helical Liquids and Non-Abelian Anyons in Quantum Wires
Yuval Oreg,1 Eran Sela,2 and Ady Stern1
1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, 76100, Israel
2 Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel Aviv, 69978, Israel
We study one dimensional wires with spin-orbit coupling. We show that in the presence of
Zeemanfieldandstrongelectron-electroninteractionacleanwiremayformfractional helical liquid
stateswithphenomenologysimilartofractionalquantumHallliquids. Mostnotably,thewire’stwo
terminalconductanceispredictedtoshowfractionalquantizedconductanceplateausatlowelectron
density. When the system is proximity-coupled to a superconductor, fractional Majorana bound
states may be stabilized. We discuss how disorder destabilizes these fractional phases. Possible
experimental realizations of similar states in double wire systems are discussed.
3
1 PACSnumbers: 74.78.Na,03.67.Lx,73.63.Nm,74.78.Fk
0
2
Introduction: Two-dimensional topological insulators comes then much richer.
n
have one dimensional helical modes at their edges. In
a
J these edges counter-propagating modes are related by
time reversal [1, 2]. These systems are most easily un-
0 Ν(cid:61)1
3 derstood when viewed as integer-quantum-spin-Hall sys-
tems,inwhichelectronswithoppositespinsareatinteger O1 I1
l] quantum Hall states with filling ν =±1. When properly E 0 O3I2 S I3O2 V
-e subjected to proximity coupling to superconductors and 2kF I4 O4
r ferromagnets these edges may host localized zero energy Ν(cid:61)13
t 1 2 3 4
s Majorana modes [3].
spin(cid:45)up spin(cid:45)down
. (cid:144)
t
a Similar phenomenology occurs in one-dimensional
m (1D) wires where electrons are subjected to spin- 0 kSO 0113 15 1 2
- orbit coupling and an external Zeeman field [4]. The k G e2h
d two-terminal conductance of such a wire varies non-
n monotonically as a function of increasing chemical po- FIG.1: Leftpanel: spin-orbitsplitbandsinaq(cid:64)ua(cid:144)n(cid:68)tumwire,
o and a small energy gap opening due to magnetic field near
tential, changing from zero (for a fully depleted wire),
c the band-crossing point. Right panel: a sketch of the zero
[ to a 2e2/h plateau, then to a e2/h plateau at which the temperature conductance dI/dV versus gate voltage. Inset:

wireishelical,andbackto2e2/h(seeFig.1)asobserved two probe measurement setup where the fractional state is
1
v in a recent experiment [5]. Furthermore, it was shown stabilized in the central region between two non-interacting
5 theoretically[4,6]andexperimentally[7–9]thatinprox- leads;dashedlinesrepresentthescatteringpatternofbosons
3 imity to a superconductor the helical state gives rise to for n=0.
3 end modes that are Majorana modes.
7
1. Intwo-dimensions(2D)interactionsbetweenelectrons Furthermore, in 2D counter-propagating edge modes
0 may give rise to fractionalized phases, whose edges may occur in fractionalized phases as well. Examples
3 are gapless Luttinger liquids. In this work we study range from fractional topological insulators, that are
1 how this fractionalization may be reflected in quantum presentlytheoretical,tographeneelectron-holebi-layers,
:
v wires. Considering the same setting used for the one- which are experimentally feasible. It was shown theo-
i dimensional helical quantum liquids in [4, 6], namely a retically that in proximity to superconductors and fer-
X
single-mode wire where electrons are subjected to spin- romagnets the counter-propagating edge modes may be
r
a orbit coupling and a Zeeman field, we examine the con- gapped to host localized fractionalized Majorana modes,
ditions under which one-dimensional “fractional helical that are non-abelian anyons of a new type [10–13]. We
liquids” may be formed, their properties and their sensi- show here that proximity-coupling of a 1D fractional he-
tivity to disorder. The crucial dimensionless number for lical liquid to a superconductor gaps its spectrum, and
our analysis is the ratio of the wire’s Fermi momentum givesrisetofractionalizedMajoranamodesakintothose
kF to the momentum characterizing spin-orbit coupling, foundin[10–13]. Theresultinggappedstateisnotwithin
kSO. Wefindthatwhenthisratioisanoddinteger2n+1 the list of possible gapped one-dimensional phases clas-
afractionalhelicalliquidstatemayform(nisanonneg- sified in Refs. 14, 15. It then comes as no surprise that
ative integer). In such a liquid the two-terminal con- these states are not stable to disorder.
ductance of the wire is e2 2 . The non-monotonic Our analysis benefits greatly from the study of the
h 1+(2n+1)2
dependenceoftheconductanceonchemicalpotentialbe- two-dimensional quantum Hall effect in terms of a set
2
of one-dimensional coupled quantum wires. In this ap- action term
proach, carried out in [16] and [17], the 2D plane is seen
(cid:16) (cid:17)n (cid:16) (cid:17)n
as made of an array of coupled wires. The spectrum of OnB =gB ψ1†ψ2 ψ2ψ3† ψ3†ψ4 . (2)
eachwireisparabolic,andtheparabolasareshiftedwith
respect to one another in momentum space by the mag- Forn=0thistermreducestothemixingofmodes2and
netic field. The number of wires is proportional to the 3 at the band crossing point, OB = g ψ ψ†. For n = 1
0 B 2 3
sample’swidth,andapproachesinfinityinthethermody- this process takes 3 particles across the 4 Fermi points,
namic limit. As we show below, our system corresponds 2→1, 2→3, 4→3; see arrows in Fig. (1). Such a term
to the system considered in Ref. 16 when the number of (togetherwithitshermitianconjugate)isoddundertime
wires is two. reversal, hence it requires magnetic field to be present.
While the main focus in this manuscript is on a single It is generated at second order in the bare interaction
wirewithspin-orbitcouplingthatshiftsthefreeelectron strength gB ∝BU22kF, with Uq the interaction potential.
spectrumparabolas,asimilareffectmayoccurinapairof
The combined Hamiltonian is most conveniently
parallelwiressubjectedtoamagneticfieldperpendicular
treated by bosonization. The four chiral bosonic fields
to their plane. In these wires, spin polarization by the
corresponding to the four fermionic fields are ψ =
l
meaacghnwetiircefimeladyasntdabailipzreopfrearcttiuonnianlghoeflitchael Fpehramseisl.evSeulcihn eiφl; ψl†ψl = (−1)l21π∂xφl. The bosonic Lagrangian
density is
wires were experimentally realized by means of cleaved
edge overgrowth [18]. 1 (cid:88) (cid:88)
L = − (−1)l∂ φ ∂ φ − ∂ φ U ∂ φ
Model: Ourmodelofinterestconsistsofaspinfull,sin- 4π t l x l x l lj x j
l lj
gle band wire, parallel to the xˆ direction, with Rashba
+ g cos{φ −φ +n[(φ −φ )−(φ −φ )]+kx}
spin-orbit (SO) coupling u, and with a magnetic field B B 3 2 3 4 2 1
1 (cid:88)
nonparellel to the SO quantization axis. The Hamilto- − E (−1)lφ . (3)
2π l
nian is
l
k2 ThepositivedefinitesymmetricmatrixU,describingthe
H= −µ+uσˆyk+Bσˆz, (1)
2m barevelocitiesofthefieldsandthelowmomentumtrans-
ferinteraction,isnotuniversal,herek =k −k +n[(k −
with σˆi, i = x,y,z being the Pauli matrices (here and 3 2 3
k )−(k −k )]. When k ≈ 0, the term proportional to
4 2 1
henceforth we work with a system of units in which
g gaps out two of the four modes in the spectrum. Us-
(cid:126) = 1), and k being the momentum. The SO coupling B
ing the parabolic spectrum when B (cid:28) ∆ = mu2/2
SO
causes a shift of the momenta k of the free electrons
we find that k = 0 for k = k /(2n + 1). With
F SO
parabolic spectrum E(k). The spin up electronic dis-
the relation ρ = 2k /π, where ρ is the density of the
F
persion (up with respect to a direction yˆ set by the SO
spinfull wire, we find that a gap will open for filling
coupling)isshiftedtotheleftbyamomentumk =mu
SO ν = kF = πρ = 1 . Notice that a component of
whilespindownisshiftedtotherightbykSO (seeFig.1). the ZkeSeOman 2fimelud in 2tnh+e1SO coupling direction (yˆ) will
The two parabolas intersect at zero momentum so that
change the conditions for k = 0, the generalization is
an application of uniform (Zeeman) magnetic field (not
straightforward.
parallel to yˆ) opens a gap in the spectrum near zero mo-
mentum. Two electron helical modes, one propagating The g term in the Lagrangian (3) motivates us to
B
to the left and the other to the right, remain free. introduce a new set of chiral bosonic fields, denoted by
(cid:80)
Away from the gap there are four Fermi points [with ηl = jMljφj, given explicitly by:
wave vector k = ±(k ±k )] and therefore we define
l SO F     
4 fermionic fields ψl with l = 1,2 corresponding to the η1 n+1 −n 0 0 φ1
leftandrightmovingmodesofσy =1andl=3,4corre- η2 = −n n+1 0 0 φ2 . (4)
sponding to the left and right moving modes of σy =−1 η3   0 0 n+1 −n φ3 
(see Fig. 1). η4 0 0 −n n+1 φ4
InteractionsaffectthisHamiltonianintwoways. Small
momentuminteractionsareaccountedforbyintroducing Notice that for n=0 we have η =φ . In terms of the
l l
Luttinger liquid parameters, as usual. Large momentum η fields, the Lagrangian density becomes,
interactions need to be elaborated on for our discussion.
AspointedoutbyKaneetal. inRef. 16inthecontextof L = − 1 (cid:88) (−1)l ∂ η ∂ η −(cid:88)∂ η U˜ ∂ η
thefractionalquantumHall(FQH)effect, multi-electron 4π 2n+1 t l x l x l lj x j
l lj
processesthatinvolvelargemomentumtransfermaylead
to an opening of an energy gap at proper values of the + g cos(η −η )− 1 E(cid:88) (−1)l η , (5)
chemical potential. Following [16] we focus on the inter- B 3 2 2π 2n+1 l
l
3
with U˜ =(M−1)TUM−1. A second pair of equations is obtained by realizing that
For B (cid:29)∆SO the description in terms of four bosonic fields η1 and η4 propagate freely through the interacting
fields ceases to be valid (as k2 and k3 do not exist any- wire implying η1,4(−L/2) = η1,4(L/2). In terms of the
more). However, similar to the case for n = 0 when the original fields the equations for η1 and η4 become
system is fully gapped (e.g. by proximity to a supercon-
ductor) the gap is not necessarily closed when B (cid:38)∆ (n+1)O1−nO2 = (n+1)I1−nO2,
SO
and a phase transition would not necessarily occur then. (n+1)I −nO = (n+1)O −nI , (8)
4 3 4 3
We now turn to discuss several properties of the wire
respectively. This set of four equations can be written in
that follow from the analysis of Lagrangian (5) in the
amatrixformasAO(cid:126) =BI(cid:126)whereA,B are4×4matrices,
limit of large g .
B
giving the desired result S = A−1B. Substituting this
Two terminal conductance: In a two terminal con-
result for S in (6) we find the two terminal conductance:
ductance measurement one connects two noninteracting
reservoirs to the wire. We model this situation by a cen-
e2 2 e2 2ν2
tral interacting section connected to the reservoirs via G= = . (9)
h 1+(2n+1)2 h ν2+1
two non interacting sections. By assumption, the transi-
tion between the sections is smooth on the scale of k so
F For n = 0 we find G = e2 as expected, for n = 1
that a 2k scattering does not occur. The DC conduc- h
tance canFthen be found by studying a scattering prob- (ν =1/3)G= 51eh2 andsoon. Ingeneral,unliketheFQH
lem in which electron currents incident from the reser- effecttheconductanceisnotequaltothefillingfactorν.
voirs scatter in the central region as shown in the inset Similar to the two modes on the edges of a FQH 2D sys-
to Fig. 1. tem,inthepresentcasetwomodesremainuntouchedby
the massive term that mixes η and η . However, unlike
To solve the scattering problem it is useful to define 2 3
the 2D system were the free modes on different edges
incoming and outgoing current vectors. The incoming
currents are I(cid:126) = (I ,I ,I ,I )T, with I = e ∂ φ | emanate from different reservoirs and propagate freely,
1 2 3 4 l 2π t l x=xl in the present case the free propagating modes in the in-
and l = 1,...,4. The points x are at +∞ for the left
l
teractingsectionaremixedattheboundarybetweenthe
moversl=1,3and−∞fortherightmoversl=2,4(see
interactingandthenoninteractingsections. Thephysical
Fig. 1). A similar definition holds for the outgoing cur-
rent O(cid:126) = (O ,O ,O ,O )T, with x → −x . Assuming proximity of the two counter-propagating modes makes
1 2 3 4 l l them susceptible to disorder, as we will discuss below.
nowthatincomingleftmovers(1,3)bothemanatefroma
WenotethatasinthecaseofaLuttingerliquiddescrib-
reservoiratpotentialV andtherightmovers(2,4)froma
ing quantum Hall effect or quantum wires, the low mo-
reservoir at potential zero, we have I(cid:126)= e2V (1,0,1,0)T,
and O(cid:126) = SI(cid:126), with S being the scatterinhg matrix. The mcoenndtuucmtanincteer[1a9ct–i2o3n]s. Ulj do not affect the two terminal
conductance G is then given by:
Refs. 16, 17 show that in the limit of an infinite num-
berofspinlesswireswhoseparabolicdispersionisshifted
GV = I +I −O −O =−I −I +O +O
1 3 2 4 2 4 3 1
by a magnetic field the two-terminal conductance ap-
G/(e2/h) = 2−(0,1,0,1)S(1,0,1,0)T
proaches the expected value of the quantized Hall con-
= (1,0,1,0)S(1,0,1,0)T. (6) ductivity. Thecasewestudycorrespondstotwowires. In
the appendix we study numerically the case of N wires,
Next we should find the scattering matrix S. In the free
and show that the deviation of the two terminal con-
caseg =0andthescatteringmatrixisequaltothe4×4
B ductance from the quantized Hall conductivity decreases
identity matrix. Simple algebra then gives: G = 2e2/h. very quickly with N, and is the order of e−N for N ≥5.
For the n = 0 case with g → ∞ the chiral field φ is
B 2 Thisreflectsthefactthatthetunnelingbetweentheedge
fully reflected to φ (see inset to Fig. 1), and this leads
3 modesisexponentiallysmall. Allthevaluesthatwefind
to a scattering matrix S equal to a matrix in which the
0 for the two terminal conductance of N wires at filling
second and the third rows of the rank 4 identity matrix
ν =k /k =1/(2n+1) conform to the expression
F SO
are interchanged. For that case G=e2/h as expected.
Determination of the scattering matrix for n (cid:54)= 0 G (ν+1)N −(ν−1)N
N =ν . (10)
follows directly from the structure of the Lagrangian e2/h (ν+1)N +(ν−1)N
Eq. (5). For g → ∞ the cosine term implies that
B
η2(x)=η3(x)for−L/2<x<L/2whichistheregionof Measurements of individual elements of the S-matrix:
thewirewheretheelectronsinteract. Usingthisequation The individual elements of the S-matrix may be mea-
at x=−L/2 and at x=+L/2 gives two relations suredinasetupsuchasusedbyAuslaenderet. al [18]. In
this setup two parallel wires are subjected to a magnetic
(n+1)I2−nO1 = (n+1)O3−nI4, field perpendicular to their plane. If the spins are po-
(n+1)O −nI = (n+1)I −nO . (7) larized and tunneling between the wires is enabled, this
2 1 3 4
4
system maps onto the system we study here, with the n a kink in the the η fields carries a charge [16]
wire index playing a role of a pseudospin. The two wires
e
areconnectedtofourseparatereservoirs,allowingasepa- C =± =±νe.
2n+1
ratecontroloftheincomingcurrentsI ...I . Forexam-
1 4
ple,theapplicationofavoltagedifferenceV betweenthe We note, however, that in 1D those quantized charges
wirescorrespondstoanincomingcurrentI(cid:126)=(V,V,0,0), can not be isolated since the free modes (composed of η
1
and results in a net current flowing between the wires. and η ) will screen those charges in a way that depends
4
This current is Is =GsV =(0,0,1,1)S(V,V,0,0). Using on the low momentum interaction Ulj.
the above S matrix this current is found to be equal to
thechargecurrentthatflowswhenvoltageisappliedbe- Fractional Majorana modes: So far we considered a
tween the right and left moving modes, i.e., G =G. In- compressible helical liquid, in which the fractionalized
s
terestingly, even when the current from wire 1 emanates nature of the wire reflects itself in the fractional two ter-
equally from the two reservoirs the wire is connected to, minalconductance. TheinteractiontermgBcos(η3−η2)
the magnetic field between the wires results in a cur- gapstwoofthefourgaplessmodesinthewire,butleaves
rentpartitioninginwire2thatisnotsymmetricbetween the other two gapless. By coupling the wire to a super-
the left and right drains. Rather, the difference between conductor through the proximity effect, the wire may
the currents flowing to the left and right in wire 2 is be fully gapped. In the n = 0 case a coupling to su-
(0,0,1,−1)S(V,V,0,0)= e2 2(2n+1) V. perconductor leads to the formation of Majorana end
h 1+(2n+1)2 modes [4, 6]. We will show now that in the n ≥ 1 case
Disorder: Disorder that leads to scattering between proximity to a superconductor leads to the formation of
the free left and right movers spoils the quantized con- fractional Majorana modes.
ductance. This is obvious in the n = 0 case, where in-
The proximity induced superconducting mean field
teractions are absent. For n (cid:54)= 0 single electron tun-
Hamiltonian projected to the four Fermi points is H =
neling is of the form ψl†ψl(cid:48). In the bosonized form it ∆(cid:82) dx(ψ (x)ψ (x)+ψ (x)ψ (x)+H.c.). Bosonizing∆, we
is λ cos(φ − φ ). To find their relevance we write 1 4 2 3
ll(cid:48) l l(cid:48) find H ∼ ∆cos(φ +φ )+∆cos(φ +φ ). Both op-
them in terms of the η’s. Since η is pinned to η ∆ 1 4 2 3
2 3 erators are irrelevant for n ≥ 1 since they contain an
by the g term, η + η fluctuates wildly. As a con-
B 2 3 exponent of η +η which is conjugate to a pinned field
sequence, a term that contains η + η is irrelevant, 2 3
2 3 and hence its correlation function decays exponentially.
while the combination η −η can be approximated as
2 3 We thus look for an operator that couples to the super-
a constant. These considerations make the λ and λ
14 23 conductor but does not contain η + η . The leading
the only relevant single particle terms. These terms 2 3
operator of this form is
translatetocos[α(η −η )+β(η −η )],where(α,β)=
1 4 2 3
2n1+1(n+1,n) for λ14 and (α,β) = 2n1+1(n,n+1) for On∆ =g∆ψ1†(ψ1†ψ2)n(ψ3ψ4†)nψ4† ∼g∆cos(η1+η4). (11)
λ . Both terms scatter charge between the two com-
23
pressible modes η1 and η4, and affect the two terminal (In the wire the operator ∝ cos(η1 −η4) does not con-
conductance. servemomentumandhenceisabsentinatranslationally
invariant system.) This process is shown in Fig. 2b. For
Generally, the scaling dimensions for these two terms
n = 1 (at filling factor ν = 1/3) it is generated at first
makes them relevant. For repulsive interactions, these
order in ∆ and in the interaction, g ∝ ∆U2 . Thus
dimensionsareboundedfromaboveby2α2/(1+ν2), see ∆ 2kF
weseethatwhentheproximitycouplingisstrongenough
Eq. (32) in the appendix. However, the divergences as-
thistermwillpinη +η andhencethelowenergytheory
1 4
sociated with the relevance of impurities are cut-off here
will become gapped. A similar many body process will
by the finiteness of the wire, as well as by finite voltage
also generate a term cos(η +η ) which competes with
2 3
and temperature.
theg term,seeFig.2a. Aslongasthelatterisstronger,
B
however, we may neglect this term.
Multi-particle scattering by impurities may be even
more relevant than single particle scattering. For exam-
Wenowclaimthattheresultinggappedphasehasfrac-
ple,4k scatteringoftheformcos(φ −φ +φ −φ )=
F 1 2 3 4 tional Majorana modes at the ends of the wire. To treat
cosν(η −η −η +η ) leads to a scaling dimension
1 4 2 3 the boundaries of the finite length (L) wire we use the
bounded by 2ν2/(1 + ν2). This backscattering opera-
usual unfolding transformation [24] which results in a
tor corresponds to a quasiparticle tunneling in the FQH
boundary condition
2D limit.
φ (x)=φ (x), φ (x)=φ (x), (x=0,L). (12)
1 2 3 4
Fractional charges: In the n = 0 case a soliton (i.e.
a kink between 0,2π in the η2 −η3 field) describes an In the new variables the boundary conditions become
electronorholewithchargeC =±e. Since(cid:80) (−1)lφ =
l l
1 (cid:80) (−1)lη it is readily understood that for general η (x)=η (x), η (x)=η (x), (x=0,L). (13)
2n+1 l l 1 2 3 4
5
n n n n same fermion parity. Systems that host four zero modes
may be double wire systems, or a single wire divided to
segments of phases that are topologically different. In
1 2 3 4 1 2 3 4
the present context, we may imagine constructing such
segments by having regions where the g ,g are opera-
B ∆
tive,alternatingbetweenregionswhereg isputtozero,
∆
andtheη ,η modesaregappedbyaperturbationofthe
1 4
form
(cid:90) b
g cos(η −η )dx (16)
B(cid:48) 1 4
(cid:68) (cid:68) a
a cosΗ2(cid:43)Η3 b cosΗ1(cid:43)Η4
where a,b are the end-points of the segment. Physically
FIG.(cid:72)2(cid:76): ((cid:64)a)Su(cid:68)perconductingan(cid:72)a(cid:76)log(cid:64)ofthe(cid:68)processinFig. 1. this term corresponds to a spatially oscillating magnetic
(b) Many particle process inducing a superconductivity gap field with wave vector 4k . Alternatively, four zero
SO
in the fractional helical liquid. modes may be constructed in systems of two wires with
properly-tunedcouplingsthatallowalsoforatransferof
charge between the wires.
Disorder has a destabilizing influence on the
It proves convenient to define a theory with a single
gapped superconducting phase and its degener-
left moving field ξ (x) and a single right moving field
L
acy. Indeed, a backscattering term of the form
ξ (x) on a ring of circumference 2L, according to
R
cos[α(η −η )+β(η −η )] will lift the degeneracy,
1 4 2 3
(cid:40) as now for example the states with η − η = 0 and
η (x), 0≤x≤L, 3 2
ξ (x)= 2(3) (14) η −η =2π have different energies.
R(L) η (2L−x), L≤x≤2L, 3 2
1(4) Relevance and irrelevance: The fractional helical
states,aswellasthesuperconductingphases,canbesta-
see Fig. (3). The magnetic and superconducting pertur-
bilized if both g and g are large enough. But yet a
B ∆
relevant question is under which conditions it is possible
3
1 2 to trigger those instabilities even when the bare pertur-
B bations g and g are small. We will use the renormal-
2 B ∆
0 X X L izationgroup(RG)methodtoaddressthisquestion. We
3 2L note, however, that for finite one-dimensional wires, the
(cid:68)
1
4 4 RGflowiscut-offbythewire’slength,anddoesnotnec-
essarily get to its asymptotic value. Therefore, the rel-
0 x L
evance of perturbations to actual experimental systems
will strongly depend on the bare values of couplings.
FIG. 3: Unfolding transformation on the spinfull finite wire
In an interacting Luttinger liquid described by charge
toaringwhoseupperpartiscoupledtothemagneticpertur-
bationandwhoselowerpartiscoupledtothesuperconducting andspininteractionparametersKc andKs,respectively,
perturbation. Fractional Majorana modes form at x=0 and we find that the operators OB [in Eq. (2)] and O∆ [in
n n
x=L. Eq. (11)] have scaling dimension
1
x = (K−1+(2n+1)2K ),
bations to the Hamiltonians now act in different regions, B 2 s c
1
(cid:90) 2L x∆ = 2(Kc−1+(2n+1)2Ks), (17)
δL= dxg (x)cos(ξ −ξ )+g (x)cos(ξ +ξ )(,15)
B R L ∆ R L
0 respectively. For small magnetic field we have K = 1
s
where g is finite in the region 0 ≤ x ≤ L and g is duetoanSU(2)symmetryofmodel(1)realizedinterms
B ∆
finite in the region L ≤ x ≤ 2L. For n = 0 the interface of fermions ψl(cid:48)(x)=eislxkSOψl(x) with s1 =s2 =−s3 =
between two gapped regions dominated by cos(ξ −ξ ) −s =1 [25].
R L 4
and cos(ξ +ξ ), binds Majorana zero modes. For n (cid:54)= Consider first the magnetic perturbation g . If the
R L B
0, the interfaces bind fractional Majorana modes as was density is fixed at ν = 1/(2n + 1), then the instabil-
recently demonstrated in [10–13]. ity into the fractional helical state occurs for infinitesi-
In principle, such zero modes lead to a degeneracy of mal magnetic field when the scaling dimension is equal
the ground state. However, this degeneracy is apparent or smaller than the space-time dimension, namely at
only when the system hosts at least four zero modes, K ≤ 3/(2n+1)2. If k deviates from the special value
c F
such that there are several ground states that share the k /(2n+1)thenanonzerobarevalueofg isessential
SO B
6
to stabilize the helical phase through a commensurate- coupling and proximity coupling to a super-conductor
incommensurate (C-IC) transition. As known from the mayallowforsuchcomputationtobecarriedoutinclean
study of C-IC transitions [24] at the transition the di- quasi-1D systems as well made of several wires.
mension xB takes the value one. Repulsive interactions We acknowledge discussions with J. Alicea, O. Aus-
reduce the value of the critical (bare) gB [26]. laender, M. Barkeshli, E. Berg, D. Clarke, L. Fidkowski,
A rough estimate of Kc in wires shows that it may M.Freedman,N.Lindner,andA.Yacoby. Theworkwas
actually be rather small. To linear order in the inter- supported by a BSF, ISF, DFG, Minerva and Microsoft
action Uq the charge Luttinger parameter is given by grants.
(cid:16) (cid:17)−1/2
K = 1+ 4Uq . Since the Coulomb interaction
c π(cid:126)vF
U divergeslogarithmicallywiththedistancetothegates
q
and near the band bottom the density of states 1/v di-
F APPENDIX
verges as well, their combination leads to a small K .
c
The superconducting perturbation g does not oscil-
∆
Generalization to N parallel wires
lateinspaceandhencedoesnotrequireaspecialdensity.
However, except for n = 0, it is irrelevant, x > 2. In-
∆
deed, even in the n=0 case it was pointed out that the Kane et. al. [16] studied the 2D FQH system as many
superconducting gap is reduced due to interactions [27]. parallel spinless wires where a term of the form Eq. (2)
Thus, strong proximity coupling is necessary. coupleseachpairofneighboringwires. Wewillnowfocus
We note that apart from the couplings g and g , the on such a system with a finite number of wires, N. This
B ∆
systemmaybecomeunstablewithrespecttospindensity analysis will bridge between our system of interest, of a
wave formation O = ψ†ψ ψ ψ†. Note that there is spinfull wire with SO coupling (N =2) and the 2D limit
SDW 1 2 3 4
no special filling condition for this operator to conserve (N →∞).
momentum and this operator is exactly marginal for all Each wire j =1,...,N is described by left (φ ) and
2j−1
valuesofK . Thestrongcouplingpictureofthisinstabil- right(φ )bosonfields. Thechiralfermionoperatorsare
c 2j
itycorrespondstoanantiferromagneticspinarrangement given by ψj(x) ∝ eiφj(x). The Lagrangian under consid-
with wave vector 2kF. This instability can be inhibited eration is L = L0 + δL, where we assume that U˜lj of
by applying a magnetic field parallel to the yˆaxis, shift- Eq. (5) is diagonal and
ing the up-spin and down-spin bands vertically. When
tthuamt hsianpcpeeants tthhee SchDeWmictaerlmpodtoenestianlotthceonsspeirnveupmoFmeremn-i L0 = −4π1(cid:88)2N (cid:0)(−1)j∂tφj∂xφj +vF(∂xφj)2+2Eφj(−1)j(cid:1).
momentum k = (k −k )/2 is not equal to the spin j=1
F↑ 2 1
down Fermi momentum k = (k −k )/2 and the cor- (18)
F↓ 4 4
responding SDW phase can not be stabilized.
By a straightforward generalization of Eq. (4) we define
Summary: In this manuscript we showed theoretically
η fields,j =1,...,2N,intermsofwhichtheperturbation
that clean spinfull wires with spin-orbit coupling and j
δL reads
strongelectron-electroninteractionmaygiverisetonovel
fractional helical liquids. The two terminal conductance
N−1
(cid:88)
of these liquids is quantized to a fractional value. We δL= g cos(η −η ). (19)
B 2j 2j+1
argued that in proximity to a superconductor fractional
j=1
Majorana modes may be stabilized. We discussed the
influence of disorder on these wires and pointed out that The fields η1 and η2N will form the low energy sector
alternative systems such as the two coupled wires (that providing the edge theory.
do not have substantial spin-orbit effects) may give rise
to a similar fractionalization.
Weexpectthattheideasputforwardhereforfraction-
Two-terminal conductance
alized one-dimensional systems coupled to a supercon-
ductor can be extended, by studying several wires con-
nectedinparallel,tonoveltwo-dimensionalfractionalized We now consider the two-terminal conductance in the
superconducting phases. Furthermore, it was shown in case that the interactions act in a finite region −L/2 ≤
Ref. 17 that two dimensional non-abelian quantum Hall x ≤ L/2. The analysis below is a direct generalizations
systemswhosequasi-particlesallowforuniversaltopolog- of Eqs. (6), (7), and (8). Consider a situation where
icalquantumcomputationmaybeviewedasconstructed the non-interacting lead to the right has chemical poten-
out of an array of parallel wires, with properly tuned tial V, and the lead to the left is at chemical potential
inter-wiretunnelingandinteractions. Weenvisionthata 0. We now describe the scattering problem in terms of
combinationofsuchinteraction,togetherwithspin-orbit incoming fields I (i = 1,...,N) and outgoing fields O
i i
7
(i=1,...,N). The conductance is given by For an incoming pulse (soliton) at I , we have as out
1
going currents O(cid:126) = 1(4,2,−2,1). The negative current
N 5
G V = (cid:88)(I −O ) inO3 isnotforbiddenasitcorrespondstoanantisoliton.
N 2l−1 2l
l=1
= NV −(0,1,0,1,...)S(V,0,V,0,...)
Low energy action
N
(cid:88)
= (−I +O )
2l 2l−1
We now derive the low energy effective action. This
l=1
= (1,0,1,0,...)S(V,0,V,0,...). (20) derivation will lead explicitly to the two freely propa-
gating chiral modes as specific linear combinations of η
1
The 2N × 2N scattering matrix O(cid:126) = SI(cid:126) needs to be and η2N, see Eq. (28). For finite N this effective theory
determined by a set of 2N equations. Generalizing the is different from the edge theory of a FQH state, as dic-
procedure in the main text, we get (2N −2) equations tated by the parameter KN derived below, determining
by setting η =η at x=−L/2 and at x=L/2, the commutation relations of the chiral fields.
2i 2i+1
In terms of the η fields the free Lagrangian becomes
(n+1)I −nO = (n+1)O −nI ,
2i 2i−1 2i+1 2i+2
2N
(n+1)O −nI = (n+1)I −nO , (21) (cid:88) ν Eν
2i 2i−1 2i+1 2i+2 L = − [(−1)j ∂ η ∂ η + η (−1)j
0 4π t j x j 2π j
where i=1,2,...,N −1. Two additional equations orig- j=1
v C v D
inate from the free propagation of η1 and η2N, + F (∂ η )2+ F ∂ η ∂ η ], (24)
8π x j 4π x 2j−1 x 2j
(n+1)I −nO = (n+1)O −nI ,
1 2 1 2 with C(D) = 1±ν2. We see that for ν (cid:54)= 1 there is
(n+1)I −nO = (n+1)O −nI .(22)
2N 2N−1 2N 2N−1 coupling between fields η and η that will medi-
2j−1 2j
ate a coupling between η and η . The effective action
ThissetofequationscanbewritteninmatrixformAO(cid:126) = 1 2N
for g → ∞ is obtained after integrating out the fields
BI(cid:126),givingS =A−1B,andthetwoterminalconductance B
{(η +η ), j =1,...,N −1} and setting η =η
is found using Eq. (20). 2j 2j+1 2j 2j+1
(j =1,...,N−1). WritingEq.(24)intermsofη +η ,
2j 2j+1
The result of this calculation is that the conductance
and setting η −η =0 we find
exponentially tends to e2 . Explicitly, we find that 2j 2j+1
h(2n+1)
the result is given by Eq. (10) for all ν and N, which is L = − (cid:88) (cid:2)(−1)j ν ∂ η ∂ η + Eνη (−1)j
plotted in Fig. 4 for n=1 (ν =1/3). 0 4π t j x j 2π j
j=1,j=2N
0.5G0 N + v8FπC(∂xηj)2(cid:3)
Log GN-13 N−1
0.45 --840 − 1v6Fπ (cid:88) ∂x(η2l+η2l+1)Ulj∂x(η2j +η2j+1)
0.40 -@1¨2 (cid:144)¨D N l,j=1
5 101520
0.35 v D
− F (∂ (η +η )∂ η +∂ (η +η )∂ η ),
0.30 8π x 2 3 x 1 x 2N−2 2N−1 x 2N
0.25 where the (N −1)×(N −1) matrix U is given by
N
0 5 10 15 20  C D/2 0 ...
D/2 C D/2 
FfoIrGn. =4:1T(νw=o t1e/r3m).inal conductance on an array of N wires U = 0 D/2 C . (25)
 ... ... 
We remark that the sum of each column of the S Togettheinteractionbetweenη andη wecompleteto
1 2N
matrix is equal to unity, implying current conservation. asquarethetermscontaining∂ (η +η )and∂ (η +
x 2 3 x 2N−2
However,S canhavenegativeelements. Forexample,for η ) and obtain the effective Lagrangian
2N−1
N =2,
ν
L [η ,η ]=− (∂ η ∂ η −∂ η ∂ η )
 4 2 −2 1  eff 1 2N 4π t 2N x 2N t 1 x 1
S = 51−22 41 14 −22. (23) − E2πν(η2N −η1)+ 1v6Fπ(cid:2)(C+F1)(∂x(η1+η2N))2
1 −2 2 4 + (C+F )(∂ (η −η ))2(cid:3), (26)
2 x 1 2N
8
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