Flux-Stabilized Majorana Zero Modes in Coupled One-Dimensional Fermi Wires

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by  Chun Chen

Flux-StabilizedMajoranaZeroModesinCoupledOne-DimensionalFermiWires
Chun Chen,1,∗ Wei Yan,2,3 C. S. Ting,4 Yan Chen,2,3 and F. J. Burnell1
1SchoolofPhysicsandAstronomy, UniversityofMinnesota, Minneapolis, Minnesota55455, USA
2DepartmentofPhysicsandStateKeyLaboratoryofSurfacePhysics,FudanUniversity,Shanghai200433,China
3Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China
4TexasCenterforSuperconductivityandDepartmentofPhysics,UniversityofHouston,Houston,Texas77204,USA
(Dated:January10,2017)
Onepromisingavenuetostudyone-dimensional(1D)topologicalphasesistorealizetheminsyntheticma-
terials such as cold atomic gases. Intriguingly, it is possible to realize Majorana boundary modes in a 1D
number-conservingsystemconsistingoftwofermionicchainscoupledonlybypair-hoppingprocesses[1]. It
iscommonlybelievedthatsignificantinterchainsingle-particletunnelingnecessarilydestroystheseMajorana
7 modes,asitspoilstheZ2fermionparitysymmetrythatprotectsthem. InthisLetter,wepresentanewmecha-
1 nismtoovercomethisobstacle,bypiercinga(synthetic)magneticπ-fluxthrougheachplaquetteoftheFermi
0 ladder. Using bosonization, we show that in this case there exists an exact leg-interchange symmetry that is
2 robusttointerchainhopping,andactsasfermionparityatlongwavelengths. Weutilizedensitymatrixrenor-
malizationgroupandexactdiagonalizationtoverifythattheresultingmodelexhibitsMajoranaboundarymodes
n
uptolargesingle-particletunnelings,comparabletotheintrachainhoppingstrength. Ourworkhighlightsthe
a
unusualimpactsofdifferenttopologicallytrivialbandstructuresontheseinteraction-driventopologicalphases,
J
andidentifiesadistinctroutetostabilizingMajoranaboundarymodesin1Dfermionicladders.
7

PACSnumbers:67.85.−d,71.10.Pm,03.67.Lx,74.90.+n
]
l
e
-
The classification of topological phases in one dimension obstacle: We begin with a different band structure, in which
r
t [2–4] revealed an intriguing array of possible new states of each plaquette of the ladder has a flux of π. We show that
s
. matter, with a variety of types of protected gapless bound- with pair hopping this model also has an interacting topo-
t
a ary modes. Of these, one class that has generated consider- logicalregimehostingMajoranaboundarymodes—whichin
m
ableexcitementrecentlyistheone-dimensional(1D)topolog- this case are protected by a symmetry preserved even in the
- ical superconductors first described by Ref. [5]. These have presence of finite t . We bolster our theory with numerical
d ⊥
protectedboundaryMajoranazeromodes,whichharbornon- evidence of Majorana boundary modes over a wide range of
n
o Abelian statistics [6–8] and hence are promising candidates t⊥. Ourfindingsthusfurnishanappealingmechanismtoen-
c fortopologicalquantumcomputing[9,10]. gineer topological boundary modes in a particle-conserving
[
In practise, to obtain long-range superconducting order in andstronglyinteractingsystemwithouttheneedtofine-tune

1 1Dsystemsrequiresinducingsuperconductivityviacoupling single-particletunneling.
v to a 3D bulk superconductor [11–21]; experimental progress Fermionicfluxladdermodel.—Motivatedbytheseconsid-
4 in this direction has been made in several distinct solid-state erations,westudyaninteractingtwo-legladdermodelofspin-
9
systems [22–26]. Interestingly, however, it is also possible less fermions in a perpendicular magnetic field described by
7
1 tohosttopologicalboundarymodesintruly1Dplatforms,in thefollowingnumber-conservingHamiltonian,
0 spiteofthefactthatthesesystemsdonotsupportlong-range
H =H +H , (1)
. superconductingorder,andareinfactgapless[1,27–31].This K W
1
70 oppheansessuipntshyenpthoestsiicbimlitayteorfiasltsusduycinhga1sDcofledrmatioomniicctgoapsoelsog[3ic2a–l HK =−L(cid:88)−2[(t(cid:107)eiφ2c†n,0cn+1,0+t(cid:107)e−iφ2c†n,1cn+1,1)+H.c.]
1 39],offeringanattractivearchitecturewithincreasedtunabil- n=0
v: ity. L(cid:88)−1
− (t c† c +H.c.), (2)
i One concrete model in this category was proposed by ⊥ n,0 n,1
X
n=0
Ref. [1], who showed numerically that even starting from a
r L−2
a latticeHamiltonianwithatopologicallytrivialbandstructure, H =+(cid:88)(Wc† c† c c +H.c.), (3)
aregimebearingthehallmarksofMajoranaboundarymodes W n,0 n+1,0 n,1 n+1,1
n=0
canbeaccessedinanatomictwo-legladderbyintroducingan
interlegpair-hoppinginteraction. Theseboundarymodesare wherec(†) isthefermionicannihilation(creation)operatorat
n,(cid:96)
protectedbytheconservedfermionparityofoneofthewires, rung n on the leg (cid:96)=0,1. The intraleg and interleg single-
and are therefore robust provided that the single-particle in- particle tunneling strengths are t and t , respectively, and
(cid:107) ⊥
terleg tunneling (t⊥), which breaks this symmetry explicitly, W (whichwetaketobenegativethroughoutthiswork)isthe
is sufficiently small. A number of proposals [1, 29–31] for pair-hoppingstrength. Twoessentialingredientsoftheabove
suppressing t⊥ such that this regime may be experimentally model are the synthetic Peierls phase φ∈[0,π] per plaque-
realizedensued. tte,andtheinterchainpair-hoppinginteractionH . Previous
W
In this paper we propose a distinct route to overcome this workshavedemonstratedtheexistenceofMajoranaboundary
2
modes in this Hamiltonian at φ=0 and t =0 based on a Bosonizationandrenormalizationgroupanalysis.—Toun-
⊥
preservedfermionnumberparityP(cid:96):=(−1)N(cid:96), whereN(cid:96) is derstandthespecialroleoftheZ2 symmetryLs,webosonize
theparticle-numberoperatorofasingleleg(cid:96)[1,29–31].Here the model (1). For φ=π, the kinetic Hamiltonian H has
K
(cid:113)
wewillshowthatwhenφ=π,theseboundarymodespersist bandenergiesE =± t2 +4t2sin2(ka),withbandgap
hgr/lwr ⊥ (cid:107)
uptot ofordert . Notethatwithoutpairhopping,thebare
⊥ (cid:107) 2t . Hereweconsiderasystematlessthanhalf-filling,with
bandH in(2)istopologicallytrivial, sothatthemodel(1) ⊥
K theinteractionscaleW smallrelativetothebandwidth,such
requires interactions to realize the topological regime. This
thatwecanprojectouttheunoccupiedbandandfocusonlyon
isnecessarilythecaseforisolated1Dsystems,wheretheto-
theprocessesinsidethelowerbandwhentreatingH . Here
W
talfermionnumberisconserved—incontrasttomodelsbased
the Fermi energy intersects the lower band at four separate
one.g.,Kitaevorspin-orbit-coupledwires[40–42],wherethe
Fermipoints,asshowninFig.1.
Majorana modes may originate from nontrivial Bogoliubov–
Linearizing about these four Fermi points results in two
deGennesbandstructures,asCooperpairscanbeexchanged
right-moving(R)andtwoleft-moving(L)fermionoperators,
withabulk3Dsuperconductor.
whichwedistinguishusingavalleyindex(IorII).Bosonizing
Symmetry analysis.—Though our system is gapless, its
theseintheusualway,wehaveψκ,ν∼eiϕκ,ν,withκ=R(L)
topologicalboundarymodescanbeunderstoodusingthesym-
andν=I(II). Wedefinethenonchiralbosonicfields
metry classification of 1D gapped fermionic phases [3]. We
therefore begin with a discussion of the underlying symme- 1 1
θ = √ {θ +θ }, θ = √ {θ −θ }, (4)
c I II s I II
tries. BelowwearguethattheMajoranaboundarymodesare 2 2
protectedbyaunitaryZ leg-interchangesymmetryL which 1 1
2 s
φ = √ {φ +φ }, φ = √ {φ −φ }, (5)
takes c → (−1)n+1c , c → (−1)n+1c , where n c I II s I II
n,0 n,1 n,1 n,0 2 2
indexes the site along the chain. This is a symmetry only
for φ = π; we will see that its action is equivalent to that where θν = √1 (ϕR,ν −ϕL,ν), φν = √1 (ϕR,ν +ϕL,ν),
2 2
of an emergent fermion parity operator that we will describe such that ϕ = 1[(φ +κθ ) + ν(φ +κθ )]. Af-
κ,ν 2 c c s s
presently using bosonization. Additionally, for any flux φ, ter including appropriate Klein factors, the only nontriv-
thereisanantiunitarytime-reversalsymmetrythatcombines ial commutators among these nonchiral bosonic fields are
complex conjugation with interchanging the two legs of the [θ (x),φ (x(cid:48))]=−2iπΘ(x(cid:48)−x), [θ (x),φ (x(cid:48))]=2iπΘ(x−
c c s s
ladder,obeyingT2=+1. Inprinciple,thisenableseightdis- x(cid:48)), [θ (x),φ (x(cid:48))]=−2iπ,whereΘ(x)istheHeavisidestep
s c
tincttypesofboundarymodes[3],thoughnumericallyweob- function. The corresponding density and current operators
serve only one of these, with a single Majorana zero mode are: ρ =(1/π)∂ θ , ρ =(1/π)∂ θ , J =(1/π)∂ φ , J =
c x c s x s c x c s
at each boundary. Finally, there also exists an overall U(1) (1/π)∂ φ .
x s
fermion number conservation. In our case its chief signifi- Bosonizing the interaction term H produces multiple
W
canceisthat,unlikethesituationconsideredbyRef.[3],only four-fermionterms,ofwhichonlyslowly-varyingtermscon-
atexactlyhalf-fillingisamicroscopic(ratherthanemergent) tribute in the continuum limit. When φ = π, in addition to
particle-holesymmetrypossible. Ift =0,themodelhasan the usual momentum-conserving processes, this allows for
⊥
additional exact Z symmetry, corresponding to the fermion intervalley umklapp scattering, since the Fermi points obey
2
parityofasinglelegoftheladder,givenbyP . Itisthissym- k =−k , k =−k , k +k =π/a,suchthat
(cid:96) L,I R,II R,I L,II L,II R,II
metrythatprotectsthetopologicalboundarymodesobserved
2π
byRef.[1]atzeroflux. k +k −k −k = . (6)
L,II R,II L,I R,I a
Eq. (6) is valid independent of the chemical potential within
thelowerband,butitwillnotholdifφ(cid:54)=π.
TheresultingbosonizedformofHdecouplesintoagapless
chargesectorandagappedspinsector:
)
a
(kwr H (cid:39)Hc+Hs, (7)
El (cid:90) 1 u
kL,I kR,I kL,II kR,II Hc = 2π{ucKc(∂xφc(x))2+ Kc (∂xθc(x))2}, (8)
x c
(cid:90) 1 u
H = {u K (∂ φ (x))2+ s (∂ θ (x))2}
s 2π s s x s K x s
x s
2g (cid:90) 2g (cid:90)
+ um cos(2φ (x))− bs cos(2θ (x))
ka (2πa)2 s (2πa)2 s
x x
FIG. 1: Linearization of the lower band (blue solid line) at φ = − 2gmx (cid:90) cos(2θ (x))·cos(2φ (x)), (9)
π. Theobtainedfourchiral-fermionbranchescanbeseparatedinto (2πa)2 s s
x
valley-I (light magenta) and valley-II (light orange) that generalize
(cid:82) (cid:82)
the original chain degrees of freedom. Specifically, we depict the where ≡ dx and the associated coupling constants are
x
twotypesofumklappprocessesthatobeyEq.(6). given by g = −aWcos2(k a)(sin4ξ +cos4ξ), g =
um π2 R,II 2 2 bs
3
−aWsin2(k a)sin2ξ, and g = −aW(sin4ξ +cos4ξ). which amounts to sending the fermion mass outside the
2π2 R,II mx 2π2 2 2
Herethewavefunctioninthelowerbandatk ,hastheform system to +∞ on both legs of the ladder [44]. This
R,II
(cosξ, sinξ) in the leg (i.e., 0,1) basis [43]. Note that the fixes θ (x ∈ (−∞,−L)) = ±π +2nˆ(1)π and θ (x ∈
2 2 s 2 2 θs s
gum-termfavorsCooperpairingwhilethecompetinggbs-term (L,∞)) = ±π+2nˆ(2)π. In the gapped bulk of the ladder,
favorsavalleydensity-wave-typeorder. Moreover, owingto 2 2 θs
φ (x∈(−L,L))=−π+nˆ π,wherenˆ(1,2), nˆ areinteger
the density-density interactions, the velocities and Luttinger s 2 2 2 φs θs φs
operators. Thetwodomain-wallMajoranaoperatorsarethen:
parameters are renormalized. Particularly, for the spin chan-
nel γL (cid:39) eiπ(nˆ(θ1s)+nˆφs), γR (cid:39) eiπ(nˆ(θ2s)+nˆφs). These satisfy the
(cid:113) Majorana relations γL†/R=γL/R, γL2/R=1, {γL,γR}=0,
us= (u+g)2−4g2cos2(2kR,IIa), (10) and define an emergent fermion parity Pv = iγLγR.
(cid:115) This acts on the spin fields within the gapped system via
u+g+2gcos(2k a)
Ks= u+g−2gcos(2kR,IIa), (11) PvφsP−v1=φs−π, PvθsP−v1=θs. In particular, when gum
R,II dominates the RG flow, P maps between the system’s two
v
whereu=12vF≡12dEdklwr(cid:12)(cid:12)k=kR,II>0andg=2(a2Wπ)3sin2ξ. ctrliavsisailclyalognrothuendtwsotactelass;swichalengrgobusnddosmtaitnesa.tes the flow, it acts
Asthechargesectorisgapless,wewillconcentrateonthe
Strictly speaking the operator P does not correspond to
v
spin sector, which is responsible for the topological Majo-
any microscopic symmetry for |t | > 0. (For t = 0, it
⊥ ⊥
ranaboundarymodes.Todeterminewhichofthesine-Gordon
can be interpreted as the fermion parity of one of the lad-
termsinH dominates, weusetheone-looprenormalization
s der’s two legs.) However, at φ = π its action is equivalent
group(RG)flowequations:
tothatofthemicroscopicZ symmetryL ,whichactsonthe
2 s
dKs(l) = yu2m(l) − yb2s(l)Ks2(l), (12) bosonizedfieldviaLsφs(x)L−s 1=−φs(x)(mod2π). Specif-
dl 2 2 ically,sincegum ispositive,bothPv andLs mapbetweenthe
dy (l) two classical minima |±π/2(cid:105) of the sine-Gordon potential,
udml =(2−2Ks−1(l))yum(l), (13) whereφs(x)|±π/2(cid:105)=(±π/2mod2π)|±π/2(cid:105). Thuswithin
theground-statemanifold,theactionofP isequivalenttothat
dy (l) v
bdsl =(2−2Ks(l))ybs(l), (14) of the exact symmetry of Ls. Indeed, in a finite-size system,
due to the proliferation of instanton processes [27, 45], the
dy (l)
mdxl =(2−2Ks(l)−2Ks−1(l))ymx(l), (15) groundstatesofHs-Garesplitinto|±(cid:105)=√12(|π/2(cid:105)±|−π/2(cid:105))
whoseenergydifferenceisexponentiallysmallinthesystem’s
whereyum=g2uumπ, ybs=2gubsπ,andymx=g2muπx arethedimen- size. The lowest two eigenstates therefore obey Pv|±(cid:105) =
sionlesscouplingconstants. Theg -channel,involvingboth
mx ±|±(cid:105), L |±(cid:105) = ±|±(cid:105), and can be labeled by their eigen-
s
θ and φ , is power-counting irrelevant for any value of K
s s s valuesunderthemicroscopicsymmetryL .
s
and can be neglected. Accordingly, the gap-opening compe-
We note that for flux φ (cid:54)= π, the umklapp scattering
tition is between g and g . If we take W to be negative,
um bs leading to Eq. (16) is no longer an effective zero (lattice)-
thenfor3π/(4a)>k >π/(2a)wehaveK >1andg is
R,II s um momentum transfer process, and cos(2φ (x)) is replaced by
s
theonlyrelevantcoupling.Inthisregimethelong-wavelength
cos(2φ (x)−δx), where δ parameterizes the deviation of
s
effectiveHamiltonianisthereforeexpectedtobe
the flux from π. In this case we expect a commensurate-
H ∼H =(cid:90) L2 dx(cid:26)u K (∂ φ (x))2+ us (∂ θ (x))2 ibnocuonmdmaryenmsuordaetes.tArasnsdietisocnribwedhiicnhthdeesSturopypslemtheenttaolpMolaotgeirciaall
s s-G 2π s s x s K x s
−L2 s [43],numericalevidencesuggeststhatthistransitionleadsto
g (cid:111)
+πuam2 cos(2φs(x)) . (16) apChoasmepianrwinhgicφh=θsπiswliothckφed=. 0.—Asnotedabove, thetopo-
Note that for W < 0, g > 0 and the coefficient of the logical state found in Ref. [1] at φ=0, t =0 is protected
um ⊥
cos(2φ ) term is positive, contrary to the usual sine-Gordon by the symmetry P corresponding to the conserved fermion
s (cid:96)
model. As explainedbelow, this signflip, which arisesfrom parity of one of the ladder’s legs. For t = 0, the operator
⊥
theparticularumklappprocessesthatoccurinthebandstruc- P constructedfromtheMajoranaboundarymodesisexactly
v
ture for φ=π, is crucial to ensuring that the staggered leg- equal to P , and therefore corresponds to an exact symme-
(cid:96)
interchange symmetry L acts like a fermion parity operator. trythatprotectstheresultingtopologicalboundarymodes. In
s
Indeed, for W > 0 and 3π/(4a) > k > π/(2a) it is the comparison, at φ=π we have found that P ’s action on the
R,II v
negative coupling g that is most relevant; this phase shows classicalgroundstatesisidenticaltothatoftheleg-exchange
bs
no numerical signatures of topological boundary modes, as symmetry L , which is not violated by finite t . Hence it
s ⊥
discussedintheSupplementalMaterial[43]. is the particular action of L at this point which gives the
s
Z symmetry and Majorana operators.—Following topological boundary modes their enhanced stability. This
2
Ref. [21] we can further derive the bosonized Majorana- result highlights the fact that different choices of topologi-
boundary-mode operators by specifying the vacuum Hamil- cally trivial band structures can have profound implications
tonian density Hvac(x) = −2Mπa∞sin(θs(x))cos(θc(x)), forinteraction-driventopologicalphases.
4
In addition to these symmetry considerations, introducing model (1) at φ = π and t = 0.5t . The numerical out-
⊥ (cid:107)
t at φ = 0 has a significantly different impact on the band comesprovidestrongevidencesupportingourtheoreticalpre-
⊥
structureandumklappprocessesthandoingsoatφ=π,mak- dictions. Fig.2(a)demonstratesthatinthelow-populationre-
ing the latter state more stable. For φ = t = 0, the two gion(N/L=1/3),whenpairhoppingisstrong(W=−1.7t ),
⊥ (cid:107)
chains’bandsareidentical. Thisoverlapfavorstheumklapp- the energy gap between the ground state and the 1st excited
scatteringprocessesthatleadtothetopologicalphase. How- state closes exponentially as the ladder’s size L increases.
ever,increasingt atφ=0separatesthetwobandsandcre- Thisisincontrastwithapower-lawgapclosingresultingfrom
⊥
ates a Fermi-velocity mismatch, which disfavors these pro- thegaplesschargesector. Asanticipated,thesetwonearlyde-
cesses,ultimatelycausingthespingaptocloseatasmallbut generateeigenstatesaredistinguishedbytheireigenvaluesof
finitevalueoft [1]. Bycontrast, whenφ=π, thetwoval- L : The ground (1st excited) state has eigenvalue +1 (−1).
⊥ s
leyswherethechains’bandscrosstheFermisurfacearemax- Theresultingground-statemanifoldisfurtherseparatedfrom
imallyseparatedbyawavevectorπ/a, andremainsymmet- the rest of the spectrum by a gap which only decreases in-
ricwithafinitet . InthiscasetheFermi-velocitymismatch versely with L (Fig. 2(b)). Moreover, the topological Majo-
⊥
is absent, and the spin gap persists up to large values of t . rana boundary modes can be characterized via the nonlocal
⊥
Thus the different nontopological band structures play a key correlations[1,30,31]inthesingle-particleGreenfunctions
roleinmakingthetopologicalMajoranamodesrobust(frag- G := (cid:104)c† c (cid:105) = (cid:104)c† c (cid:105). These are apparent for a
mn m,0 n,0 m,1 n,1
ile) against t at φ=π (φ=0), consistent with the fact that rangeoft valuesinthetopologicalregime(Figs.2(c)–(d)).
⊥ ⊥
t breaksP butpreservesL . The presence of the edge states also gives rise to a two-fold
⊥ (cid:96) s
degeneracyintheentanglementspectrum(ES)[2,48]onthe
centralbond(Fig.2(f)). TheseDMRGresultshavebeencon-
0.20 1/12 firmedbyEDforsmallsystemsizes.
ln(E1 E0) E2 E1
5 0.15 As discussed in the Supplemental Material [43], adding a
1/24
10 0.10 1/32 smallLs-symmetry-breakingperturbationto(1)causesthede-
0.05 1/601/48 generacy in the ES to lift, and the energy gap of E1−E0
15 (a) 1/72 (b) to deviate from the exponentially small splitting observed in
12 24 36 48 60 72 0.00 0.03 0.06 0.09
L 1/L Fig.2(a). Thissupportsourclaimthatthetopologicalbound-
a8
00..0150 tL4=80N.51t6|| ||GG01nn|| (c) nt Spectr6 aryTmoofudlelysacrheaprarcotteercitzeedtbhyeLtospaotloφg=icπalarnedgitm⊥e(cid:54)=,w0.ealsoshow
me4 evidenceofaphasetransitionoutofthetopologicalregimeat
e
0.000 15 30 45 ntangl2 (f) tn⊥o,ncl≈oc2a.l5cto(cid:107)r.rFeliagt.io2n(es)isnhGowsaavraeluaebste⊥n>t. tF⊥u,cr,thfeorr,washischhowthne
E mn
00..0150 tL4=81N.01t6|| ||GG01nn|| (d) 0 0.5 1.0 1.5t /t||2.0 2.5 3.0 ibsyliFftiegd.2a(nfd),tthheeldoewgeesntelreavceyliinscthleearElySnaopnpdareegnetnfeorartte⊥a<tt⊥t⊥=,c
3.0t . Our numerics suggest that the transition is toward a
0.00 0.18 (cid:107)
0.20 15 30 45 state with density-wave order at large t⊥: For t⊥ > t⊥,c,
t =3.0t|| |G0n| (e) n (t=0.5t||) the fermion-density profile ρn := (cid:104)c†n,0cn,0(cid:105) = (cid:104)c†n,1cn,1(cid:105) in
0.1 L48N16 |G1n| 0.15 n (t=1.0t||) Fig.2(g)evolvesfromauniformdistributiontowardanoscil-
n (t=3.0t||) (g) latory pattern. Additionally, for t⊥ ≥3.0t(cid:107), the two lowest
0.00 15 n 30 45 0 15 n 30 45 energy eigenstates split and both have Ls eigenvalues of +1
[43]. Alltheaboveobservationssuggestthatoncet >t ,
FIG.2: Numericalsignaturesofthetopologicalphase. (a)and(b): ⊥ ⊥,c
theMajoranaboundarymodesdisappear.
Scaling of energy gaps as functions of L from DMRG. (a) shows
thattheenergydifferencebetweenthefirsttwolowest-lyingeigen- Experimental feasibility.—In cold atom laboratories, cur-
statesof(1)decaysexponentiallywithL.Theprotectedground-state rently a great deal of effort has been devoted to simulat-
manifoldisseparatedfromtherestofthespectrumbyagapthatde- ing strong synthetic magnetic fields in optical lattices by
creasesinverselywithL,asshownin(b). HereW=−1.7t(cid:107), t⊥= Raman-assisted tunnelings and lattice-shaking techniques,
0.5t , φ=π, N/L=1/3. (c)–(g):Transitionoutofthetopological
(cid:107) which mimic standard Peierls substitutions through attach-
phaseatlarget forfixedW =−1.7t , φ=π, L=48, N=16.
⊥ (cid:107) ing an Aharanov–Bohm-like complex phase to the nearest-
(c)–(e) demonstrate the edge mode via the nonlocal correlations
neighbor single-particle hopping [49–56]. The remarkable
[1, 30, 31] in single-particle Green functions. At t =3.0t , the
⊥ (cid:107)
edgemodedisappearsindicatingthetransitiontoatrivialstate.This tunabilityoftheseartificialgaugefluxeshaspavedanexper-
isinaccordancewith(f)and(g)whichshowthecorrespondingevo- imental avenue to exploring the Bose–Einstein condensation
lutions of entanglement spectra and local fermion densities as the in Harper–Hofstadter model of 2D bosonic gases [55]. Chi-
transitionisapproached. ral edge states due to particle’s cyclotron motion stirred by
uniformfluxeshavealsobeendetectedinquasi-1Dfermionic
Numericalverification.—Simulationsbasedondensityma- laddersandHallribbons,whereasizablefluxφcanreachup
trixrenormalizationgroup(DMRG)[46]andexactdiagonal- to 1.31π per plaquette by utilizing the technology of optical
ization (ED) [47] have been performed to solve the lattice atomicclocks[53,54]. InRef.[1]anatomicschemehasalso
5
beenoutlinedforcreatingthemoreintricatepair-hoppingin- 111,186805(2013).
teraction. Thesedevelopmentsmaketheprospectofrealizing [20] B. Braunecker and P. Simon, Phys. Rev. Lett. 111, 147202
flux-stabilizedMajoranazeromodesapossibilityinthenear (2013).
[21] D.J.Clarke,J.Alicea,andK.Shtengel,Nat.Commun.4,1348
future.
(2013).
To summarize, we have shown that by threading π-flux
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SupplementalMaterial
Bandstructure
HereweprovidemoredetailsonthebandstructureofH ,andshowhowitentersthebosonizedformofthefullHamiltonian.
K
ThenoninteractingkineticHamiltonian,
 (cid:16) (cid:17) 
HK =(cid:90)−∞∞ 2dπk (cid:16)ψk†,0 ψk†,1(cid:17)−2t(cid:107)co−st⊥ka+ φ2 −2t(cid:107)co−s(cid:16)t⊥ka− φ2(cid:17)(cid:18)ψψkk,,10 (cid:19)
(cid:90) ∞ (cid:16) (cid:17)(cid:18)E (k,φ) 0 (cid:19)(cid:18) ψ (cid:19)
= dk ψ† ψ† + k,+ ,
k,+ k,− 0 E (k,φ) ψ
−∞ − k,−
canbediagonalizedbyintroducingtheunitarymatrixM(k,φ),
 √−uk,φ √ 1  (cid:18) (cid:19)
M(k,φ)=(cid:0)|χ+(cid:105) |χ−(cid:105)(cid:1)= √1+1u2k,φ √1u+k,uφ2k,φ , M†H(k,φ)M= E+(0k,φ) E (0k,φ) ,
−
1+u2 1+u2
k,φ k,φ
wherethecomponentinthematrixelementsassumes
 (cid:115) 
1  φ (cid:18) φ(cid:19)2
u = 2t sin(ka)sin + t2 + 2t sin(ka)sin .
k,φ t (cid:107) 2 ⊥ (cid:107) 2
⊥  
Fromnowon,wefocusonthecaseofφ = π,withthechemicalpotentialinthelowerband. Linearizingthebandstructure
aboutthefourFermipointsinthelowerbandgivesrisetofourindependentbutmutuallyinteractingchiral-fermionbranches,
whichwelabel(L,I), (R,I), (L,II),and(R,II)(seeFig.1inthemaintext). Thefermionoperatorinrealspaceisrelatedto
thesefourchiralfermionsvia:
ψx,(cid:96) (cid:39)eikL,Ix(cid:104)(cid:96)|χ−(kL,I)(cid:105)ψ˜L,I(x)+eikR,Ix(cid:104)(cid:96)|χ−(kR,I)(cid:105)ψ˜R,I(x)
+eikL,IIx(cid:104)(cid:96)|χ−(kL,II)(cid:105)ψ˜L,II(x)+eikR,IIx(cid:104)(cid:96)|χ−(kR,II)(cid:105)ψ˜R,II(x),
wherexisthepositionalongtheladder,and(cid:96)=0,1indexestheleg. Herewehavedefined(cid:104)(cid:96)=0|≡(1 0),(cid:104)(cid:96)=1|≡(0 1),
 √ 1  (cid:32)cosξ(k) (cid:33)
and|χ−(k)(cid:105)≡|χ−(k,φ=π)(cid:105)= √1u+k,uπ2k,π = sinξ(2k) .
1+u2k,π 2
TheHamiltonianinthemaintextisobtainedbysubstitutingtheseexpressionsintothepair-hoppingtermH ,andperforming
W
astandardbosonizationanalysisoftheresult. Followingthesecalculationsthrough,onefindsthatthematrixelements(cid:104)(cid:96)|χ (cid:105)
−
entertheresultingLuttingerparameters,aswellasthesine-Gordoncouplingsg , g , g . Therelevantexpressionsinthe
um bs mx
spinsectoraregiveninthemaintext;inthechargesectortheassociatedLuttingerparametersaredefinedby
(cid:114)
(cid:112) u−g
u = (u−g)·(u−5g), K = ,
c c u−5g
wherethematrixelemententersu andK viag= aW sin2ξ.
c c 2(2π)3
Itfollowsthatthemicroscopicparametersinourbosonizedmodelaresensitivetothepreciseinitialbandstructure, aswell
astotheinitialvalueofthechemicalpotentialwithinthelowerband. Inparticular,thewrongchoiceofchemicalpotentialcan
leadtoK <1,andanRGflowthattendstotakethesystemawayfromthetopologicalregime.
s
7
Topologicalquantumphasetransition
Though at φ = π our model is robust to the single-particle interchain tunneling up to t ≈ t , Figs. 2(c)–(e) in the main
⊥ (cid:107)
text suggest that by t = 3t , the system has undergone a transition to a nontopological regime. In particular, for this value
⊥ (cid:107)
we see that the nonlocal correlation in the single-particle Green function, an indicator of the presence of Majorana boundary
modes,disappears. Further,thedoubledegeneracyintheentanglementspectrum,anotherindicatorofthetopologicalboundary
modes,disappearsbyt = 2.5t . Finally,thespatialprofileofthelocalfermiondensity(showninFig.2(g)inthemaintext)
⊥ (cid:107)
showsclearevidenceofafermiondensity-wave-typeorderfort = 3t . Incomparison, thebulkfermiondensityisuniform
⊥ (cid:107)
fort ≤t ,wherethenonlocalcorrelationindicatesthepresenceofMajoranaboundarymodes.
⊥ (cid:107)
As further evidence of this transition, Fig. 3 shows the low-energy spectrum of the π-flux ladder as a function of t /t at
⊥ (cid:107)
fixed pair-hopping strength W. At approximately t = 2.8t a level crossing occurs between two states with opposite Z
⊥ (cid:107) 2
quantumnumbersunderthestaggeredleg-interchangesymmetryL . Forsmallert thetwolowestenergystateshaveopposite
s ⊥
Z quantum numbers; whereas for larger t both have the same (+1) Z eigenvalue. As explained in the main text this Z
2 ⊥ 2 2
quantumnumberisassociatedwiththefermionparityinthetopologicalregion, suggestingthatthislevelcrossingindicatesa
transitionoutofthetopologicalregime. Thislevelcrossingisconsistentwithwhatweexpectforatransitionfromatopological
regimetoafermion-density-waveorder: Inthelattercaseweexpectthelowesttwostatestoberelatedbytranslation,andhence
tohavethesameparityundertheleg-exchangesymmetryL .
s
0.4
W = 1.7t||
0
E
En 0.3 =
a:
ctr L12N4ED
e
p
S
y 0.2
g
r
e
n
e
n
e
g
Ei 0.1
0.0
0 1 2 3 4 5
t /t||
FIG.3: ExactdiagonalizationresultsshowtheevolutionoftherescaledeigenenergiesofH: E −E ,asweincreaset . Herewefixed
n 0 ⊥
W = −1.7t , φ = π, L = 12, N = 4, andsett = 1.0astheenergyunit. WemonitortheL-quantumnumberforthelowestthree
(cid:107) (cid:107) s
eigenstatesandusedifferentcolorstodistinguishtheZ eigenvaluesofL: Bluedenotes(cid:104)L(cid:105) = +1,whilereddenotes(cid:104)L(cid:105) = −1. When
2 s s s
0≤t /t (cid:46)2.5,thelowesttwoeigenstatescanbedifferentiatedbytheirquantumnumbersofL. However,whent /t (cid:38)3,theypossess
⊥ (cid:107) s ⊥ (cid:107)
thesamequantumnumber(cid:104)L(cid:105)=+1astheconsequenceoflevelcrossingbetweenthe1stand2ndexcitedstates.
s
Thepi-fluxladderforpositiveW
TheRGanalysisinthemaintextsuggeststhatforfinitet whenW >0, φ=π,themostrelevantgappingchannelofH for
⊥ s
low fermion density (3π/(4a)>k >π/(2a)) is the backscattering term g . In this section we present the corresponding
R,II bs
DMRGresultstoillustratethattheresultingphaseexhibitsnosignaturesoftopologicalprotectionatfinitet .
⊥
In particular, we evaluate the entanglement spectra (ES) for W = +1.7t at several different values of t . The results in
(cid:107) ⊥
Fig. 4(a) demonstrate the significant splitting of the ES two-fold degeneracy at the lowest level even for the smallest nonzero
valuesoft ≈ 0.1t . ThisisinsharpcontrasttothesituationofW = −1.7t , wheretheevendegeneracyinESpersistsup
⊥ (cid:107) (cid:107)
8
8
a
ctr6
e
p
S
nt
e
m4
e
gl
n
a
nt
E2
(a)
0
0.0 0.2 0.4 t /t|| 0.6 0.8 1.0
0.18
n,0 (t /t||=0.0) n,1 (t /t||=0.0)
0.16 n,0 (t /t||=0.5) n,1 (t /t||=0.5)
(b)
0 15 n 30 45
FIG.4: DMRGresults: (a)showsthedegeneracysplittingsintheentanglementspectraasafunctionoft forthegroundstateoftheladder
⊥
systemwiththefixedparameters: W =+1.7t , φ=π, L=48, N =16. Twoprototypicalfermion-densityprofileshavebeenillustrated
(cid:107)
inpanel(b),whereforthetrivialstatethereexistweakdensitymodulationsalongthechain.
tot ≈ 2.0t (seeFig.2(f)inthemaintext). Further,forparameterswherethisdegeneracyislifted,thelocalfermion-density
⊥ (cid:107)
profileρ exhibitsweakmodulationsalongthechain,ascanbeseenfromFig.4(b). Herewedefineρ =(cid:104)c† c (cid:105), ρ =
n,(cid:96) n,0 n,0 n,0 n,1
(cid:104)c† c (cid:105)fortheupperandlowerchains,respectively.
n,1 n,1
Leg-interchange-symmetry-breakingperturbations
Here,wedemonstratenumericallythattheZ leg-exchangesymmetryandthefluxφ = π arebothcentraltoprotectingthe
2
topologicalfeaturesofourmodel. Specifically, weinvestigatethreedifferentwaysofbreakingtheleg-interchangesymmetry,
andshowthatsmall(butfinite)valuesofallsymmetry-breakingperturbationsliftthedegeneraciesintheESassociatedwiththe
topologicalboundarymodes. Inaddition,weperformananalysisoftheeffectofperturbingthefluxawayfromφ = π,which
liftsthedegeneraciesintheentanglementspectrumandleadstoafermion-density-waveorder.
Makingtheintrachainhoppingsdifferent
Wefirstconsidertheresultofexplicitlybreakingtheleg-interchangesymmetrybychoosingt tobedifferentonthetwolegs
(cid:107)
oftheladder. WethusmodifytheHamiltonianinthefollowingway:
L−2 L−1
H(cid:48) =−(cid:88)(cid:104)(cid:16)i(cid:0)t −δ (cid:1)c† c −i(cid:0)t +δ (cid:1)c† c (cid:17)+H.c.(cid:105)−(cid:88)(cid:16)t c† c +H.c.(cid:17)
(cid:107) t n,0 n+1,0 (cid:107) t n,1 n+1,1 ⊥ n,0 n,1
n=0 n=0
L−2
(cid:88)(cid:16) (cid:17)
+ Wc† c† c c +H.c. , (17)
n,0 n+1,0 n,1 n+1,1
n=0
wherewehavesetφ=π,andtheaddedδ -termsexplicitlybreaktheL symmetry.
t s
9
8
a
ectr6
p
S
nt
e
m4
e
gl
n
a
nt
E2
(a)
0
0.00 0.05 0.10 t/t|| 0.15 0.20
0.3
n,0 ( t/t||=0.05)
0.2 n,1 ( t/t||=0.05)
n,0 ( t/t||=0.2)
0.1
(b) n,1 ( t/t||=0.2)
0.0
0 7 14 n 21 28 35
FIG.5: DMRGresults: (a)showstheevolutionofES(forasystemofL = 36, N = 12)asafunctionofL-breakingperturbationδ /t
s t (cid:107)
(seeEq.(17)). TheevendegeneracyofESsurvivesonlyuptosmallvaluesofδ /t ≈ 0.125. Panel(b)illustratesthelocalfermion-density
t (cid:107)
profilesforthetwodifferentvaluesofδ /t .HereW =−1.7t , t =t , φ=π.
t (cid:107) (cid:107) ⊥ (cid:107)
TheargumentsgiveninthemaintextassertthatbreakingL shouldliftthedegeneracyassociatedwiththetopologicalbound-
s
arymodes,sincethereisnolongeranyreasonforthefermionparityevenandfermionparityoddstatestobeproximateinenergy.
Thisisapparentintheevolutionoftheground-stateentanglementspectra(showninFig.5(a)forasystemofL=36, N =12)
asafunctionofδ /t .WeobservethatforfixedW =−1.7t , t =t ,thetwo-folddegeneracyinthelowestleveloftheEShas
t (cid:107) (cid:107) ⊥ (cid:107)
completelydisappearedbyδ =0.15t . ThissuggeststhatthesymmetryL is,asclaimed,integraltoprotectingthetopological
t (cid:107) s
boundarymodes. Fig.5(b)furthersupportsthis, byshowingthatforsufficientlylargeδ (δ = 0.2t ), thefermiondensityon
t t (cid:107)
thelowerchaintendstodevelopweakspatialmodulations,similartothoseobservedforpositiveW intheprevioussection. For
smallervalues (δ = 0.05t ), the density profilesappearto remainuniformin thebulk, though therelativepopulations ofthe
t (cid:107)
twochainsaregenericallydifferentwhentheleg-interchangesymmetryisbroken.
It is somewhat surprising that for the smallest values of δ /t the degeneracy in the ES and the uniform fermion density
t (cid:107)
persist,suggestingthatthetopologicalboundarymodesareonlydestroyedatfinitevaluesofthesymmetry-breakingparameter.
AsimilareffectwasobservedinRef.[1],wheretheauthorssawsignaturesoftopologicalfeaturesintheφ = 0chainforsmall
butfiniteinterchainsingle-particlehopping—therelevantsymmetry-breakingperturbationinthatcase. Thismaybebecausethe
spinsectorisgappedandcompletelydecoupledfromthegaplesschargesector;henceitsbehaviorisrobusttosufficientlysmall
perturbations.
10
8
a
ctr
pe6
S
nt
e
m
e4
gl
n
a
nt
E
2
(a)
0
0.0 0.1 h/t|| 0.2 0.3
0.3
n,0 (h/t||=0.1)
0.2 n,1 (h/t||=0.1)
n,0 (h/t||=0.3)
0.1
(b) n,1 (h/t||=0.3)
0.0
0 7 14 n 21 28 35
FIG.6: DMRGresults: (a)showstheevolutionofES(forasystemofL = 36, N = 12)asafunctionofL-breakingperturbationh/t
s (cid:107)
(seeEq.(18)). TheevendegeneracyofESsurvivesuptosmallvaluesofh/t ≈ 0.225. Panel(b)showsthedevelopmentofweakdensity
(cid:107)
modulationsalongthelowerchainash/t increases.HereW =−1.7t , t =t , φ=π.
(cid:107) (cid:107) ⊥ (cid:107)
Makingimbalancedlocalpotentials
Asasecondtest,weconsideraddingachemicalpotentialthatisimbalancedbetweenthetwochainsoftheFermiladder. We
thustaketheHamiltoniantobe
L−2 L−1
(cid:88)(cid:104)(cid:16) (cid:17) (cid:105) (cid:88)(cid:16) (cid:17)
H(cid:48) =− it c† c −it c† c +H.c. − t c† c +H.c.
(cid:107) n,0 n+1,0 (cid:107) n,1 n+1,1 ⊥ n,0 n,1
n=0 n=0
L−2 L−1
(cid:88)(cid:16) (cid:17) (cid:88)(cid:16) (cid:17)
+ Wc† c† c c +H.c. + hc† c −hc† c , (18)
n,0 n+1,0 n,1 n+1,1 n,0 n,0 n,1 n,1
n=0 n=0
wherethelasth-termsalsobreaktheL symmetryexplicitly.
s
Via DMRG, we find that the effect of h-terms (see Fig. 6) is similar to that of the δ -terms, both of which tend to destroy
t
the topological Majorana boundary modes as indicated by the splitting (or nondegeneracy) in the lowest level of ES. This
perturbation also has a similar effect on the fermion-density profile, first changing the relative occupancies of the two chains,
and at higher values leading to density modulations in the higher density chain. The two perturbations differ quantitatively,
however: Inthiscase, thetwo-foldESdegeneracysurvivesuptoh/t betweenabout0.2and0.25. Theotherparametersare
(cid:107)
keptthesameasthatinFig.5: W =−1.7t , t =t , φ=π, L=36, N =12.
(cid:107) ⊥ (cid:107)