Topological phase transitions on a triangular optical lattice with non-Abelian gauge fields

1.3 MB·

by  M. Iskin

Topologicalphasetransitionsonatriangularopticallatticewithnon-Abeliangaugefields
M. Iskin
DepartmentofPhysics, Koc¸ University, RumelifeneriYolu, 34450Sarıyer, Istanbul, Turkey.
(Dated:January7,2016)
Westudythemean-fieldBCS-BECevolutionofauniformFermigasonasingle-bandtriangularlattice,and
constructitsground-statephasediagrams,showingawealthoftopologicalquantumphasetransitionsbetween
gappedandgaplesssuperfluidsthatareinducedbytheinterplayofanout-of-planeZeemanfieldandageneric
6
non-Abeliangaugefield.
1
0
PACSnumbers: 03.75.Ss,03.75.Hh,67.85.Lm
2

n
a I. INTRODUCTION required SF transition temperature. Stimulated by these ex-
J periments, there has been a fruitful activity on many aspects

6 of spin-orbit coupled Fermi gases, but the majority of them
The intriguing possibility of superfluid (SF) phase transi- are focused on continuum systems with a lack of interest in
] tionsfromtopologyinmomentum(k)spacehaslongbeenof
s lattice ones [16]. For instance, even though topological SFs
interestnotonlytothecondensed-matterbutalsotothecold
a haverecentlybeencharecterizedforasquarelatticewithnon-
g atom and molecular physics communities in the broad con-
Abeliangaugefields[17,18],andtunablehoneycomblattices
- textsofnodalsuperconductors(d-wavesymmetryforhigh-T
t c (made of two triangular sublattices) are of both ongoing ex-
n materials),nodalSFs(p-wavesymmetriesforliquid3Heand
perimentalandtheoreticalinterest[19–23],thetriangularlat-
a single-component Fermi gases), and population-imbalanced
u ticesthemselvesarealmostentirelyoverlookedinthiscontext.
SFs (s-wave symmetry for two-component Fermi gases) [1].
q Here, we study the BCS-BEC evolution of a spin-1/2
AnalogoustotheLifshitztransitioninmetals[2],thesetopo-
. Fermi gas on a single-band triangular lattice, and construct
t logicalphasetransitionsaresolelyassociatedwiththeappear-
a its ground-state phase diagrams. Our primary objective is to
m anceordisappearanceofk-spaceregionswithzeroexcitation
establish that the interplay of an out-of-plane Zeeman field
energies, where the symmetry of the SF order parameter re-
- and a generic non-Abelian gauge field gives rise to a wealth
d mainsunchangedinsharpcontrasttotheLandau’sclassifica-
of topological phase transitions between gapped and gapless
n tionofordinaryphasetransitions. Notonlysuchchangesnat-
SFs that are accessible in atomic optical lattices. The rest
o urallycauseadramaticrearrangementofparticlesinkspace,
c of the paper is organized as follows. After we introduce the
andtherefore,arereadilyseenintheirk-resolveddistribution
[ model Hamiltonian in Sec. II, first we discuss the effects of
and/orspectralfunction,buttheyalsoleavenon-analyticsig-
agenericnon-Abeliangaugefieldonthesingle-particleprob-
1 naturesinthethermodynamicpropertiesofthesystem[3].
lem,andthenbrieflysummarizethemean-fieldformalismthat
v
5 Toobserveandstudytopologicalphasetransitions,onere- is used for tackling the many-body problem. In Sec. III, we
8 quires to have a reliable knob over either the density of par- thoroughly analyze the conditions under which the quasipar-
1 ticles or both the strength and symmetry of the inter-particle ticle/quasiholeexcitationspectrumoftheSFphasemayvan-
1 interactions[4]. Sincesuchcontrolsareeitherverylimitedor ish,andevaluatethecorrespondingchangesintheunderlying
0 not yet possible in condensed-matter systems, the cold-atom Chern number. These conditions are numerically solved in
1. systems initially thought to offer an ideal platform for real- Sec. IV together with the self-consistency equations, where
0 izing these transitions, thanks in particular to their precise- weconstructtheground-statephasediagramsasafunctionof
6 tuning capabilities over a wide range of laser parameters. particlefillingandSOCforawiderangeofpolarizationsand
1 However,despiteallthepastandongoingattemptswithFermi interactions. The paper ends with a briery summary of our
:
v gasesacrossp-waveFeshbachresonances[5–8],theshortlife- conclusionsandanoutlookgiveninSec.V.
i times of the resultant p-wave molecules have so far been the
X
biggest drawback in this line of research, which the experi-
r mentalistsyettoovercome.
a II. THEORETICALMODEL
On the other hand, given the recent progress in creating
artificial gauge fields [9, 10], there is a growing consensus
In this paper, we consider a spin-1/2 Fermi gas on a tri-
that one of the most promising ways to realize a topologi-
angular lattice, and study the effects of the following non-
calphasetransitionistoincorporates-waveFermigaseswith
Abelian gauge field A = (ασ ,−βσ ) on the ground state
spin-orbitcouplings(SOC)[11]. Forinstance, dependingon y x
SF phases, where {α,β} ≥ 0 characterize the SOC, and σ
theinter-particleinteraction,polarization,dimension,geome- x
andσ arethePauli-spinmatrices[17,18].Wetakethegauge
try,andsymmetryandstrengthofSOC,itispossibletocreate y
fieldintoaccountviathePeierlssubstitution,underwhichthe
azooofnodalSFswithpoint,lineorsurfacenodesinkspace
tunneling of atoms from site i to j is described by the hop-
in various numbers. Since several groups have already suc- pingHamiltonianH =−(cid:80) c† tσ(cid:48)σc .Here,weonly
ceededin creatingsuch setupsathigh temperatures[12–15], 0 σσ(cid:48)ij σ(cid:48)j ji σi
there is arguably no doubt that these new systems will soon allow nearest-neighbor hoppings with tσji(cid:48)σ = te−i(cid:82)rrijA·dr,
offerunforeseenpossibilitiesoncetheyarecooledbelowthe where t ≥ 0 is its amplitude, and r is the position of site
i
2
i. The lattice spacing a is set to unity in this paper. Us-
ing the Fo√urier series expansion of the annihilation operator
cσi = (1/ M)(cid:80)keik·ricσk and its Hermitian conjugate in
momentumk = (k ,k )space,whereM → ∞isthenum-
x y
beroflatticesitesinthesystem,thehoppingHamiltoniancan
bewrittenasH =(cid:80) ψ†((cid:15) σ +S ·(cid:126)σ)ψ .Here,thespinor
0 k k k 0 k k
ψ† = (c† ,c† )denotesthecreationoperators, (cid:15) istheen-
k ↑k ↓k k
ergy dispersion, σ is the identity matrix, S = (Sx,Sy,0)
0 k k k
is the SOC, and (cid:126)σ = (σ ,σ ,σ ) is a vector of spin matri-
x y z
ces. Foratriangularcrystall√atticewithprimitiveunitvectors
a =(1,0)anda =(1/2, 3/2),wefind
1 2
√
k 3k
(cid:15) =−2tcosαcosk −4tcosγcos x cos y, (1)
k x 2 2
√
√ sinγ k 3k
Sx =−2 3tβ cos x sin y, (2)
k γ 2 2
√
sinγ k 3k
Sy =2tsinαsink +2tα sin x cos y, (3)
k x γ 2 2
(cid:112)
where γ = α2+3β2/2. Note that the reciprocal of a tri-
angular lattice is a hexagonal latt√ice in k space with pr√imi-
tiveunitvectorsb =(2π,−2π/ 3)andb =(0,4π/ 3),
1 2 √
and therefore, the first BZ is bounded by |k | = 2π/ 3 for
√ √ y
|k | ≤ 2π/3, k = ±( 3k −4π/ 3) for 2π/3 ≤ k ≤
x y √ x √ x
4π/3 and k = ±( 3k + 4π/ 3) for −4π/3 ≤ k ≤
y x x√
−2π/3. Inaddition,sincetheareaofthefirstBZis8π2/ 3,
(cid:80)
we evaluate the sums by converting them into integrals
√ k
[M 3/(8π2)](cid:82) d2kinournumerics.
BZ
The total single-particle density-of-states (DoS) for the
hopping Hamiltonian can be written as D(ω) = D (ω) +
+
(cid:80)
D (ω), where D (ω) = (1/M) δ(ω − (cid:15) ∓ |S |)
− ± k k k
with δ(x) the Dirac-delta function. In Fig. 1, we show col-
ored maps of tD(ω) as a function of ω/t and (α,β), where
πδ(x) → η/(x2 +η2) is implemented in our numerics with
a small broadening η = 0.01t. First of all, after summing
over k, since the SOC can be gauged away from the single-
particle problem in the limits of α → 0 or β → 0, D(ω)
approachtotheno-SOCvalueinallfigures,showingasharp
peak at ω = 2t. This is also the reason behind the some-
whatfeaturelessstructureofFig.1(b)whereβ issmall. Fur-
thermore, for symmetric SOCs with α = β, it can analyt-
ically be shown that the DoS has D(ω,α) = D(ω,−α),
D(ω,π+α) = D(ω,π−α)andD(ω,α) = D(−ω,π−α)
symmetries for any α, leading to D(ω = 0,α = iπ/2) = 0
foranyintegeri. Thesesymmetriesandtheresultantgapsare
clearlyillustratedinournumericsshowninFig.1(a),andthey
play important roles in understanding the resultant ground- FIG.1. (Coloronline)ThetotalDoStD(ω)isshownasafunction
of energy ω/t and SOC (α,β) for (a) α = β, (b) β = π/4, (c)
stateSFphasesasdiscussedbelowinSec.IV.Thus,insharp
α=π/4,and(d)α=β(squarelattice).
contrasttothecontinuumsystemswherethelow-energyDoS
increases with increasing SOC, we show that the DoS has a
much richer dependence on energy and SOC on a triangular
lattice. For completeness, typical DoS data are illustrated in Since our primary objective in this paper is to character-
Figs. 1(b) and 1(c) for asymmetric SOCs with α (cid:54)= β (cid:54)= 0, izedistinctSFphasesof↑and↓fermionsinthepresenceof
showingnoparticularsymmetryingeneral. Incomparisonto on-siteattractiveinteractionsinbetween,weintroduceacom-
Fig.1(a),wealsoshowtheanalogousDoSdependencefora plexparameter∆ = g(cid:104)c c (cid:105),whichdescribesthelocalSF
i ↑i ↓i
nearest-neighbor square lattice in Fig. 1(d), manifesting the orderwithintheBCSmean-fielddescription,whereg ≥ 0is
particle-holesymmetryaroundω =0foranyα. thestrengthoftheinteraction,and(cid:104)···(cid:105)isathermalaverage.
3
Finally,includingapossibleout-of-planeZeemanfieldh,the andleftedgesoftheBZaddingintotaltotwofullpoints,the
etHorata=tolmrMΨea|†k∆n-=|fi2e/(lgcd†↑+Hk(cid:80),acm†↓kkilξ,tkoc↑+n,i−a(1nk/,c2ca↓)n,(cid:80)−bkek)cΨdoem†knHpoatkecΨstlytkh,wewrcihrteeteranetitaohsne[a1on8pd]- tacwnodso-kpfi5xonia=nltlyseαtthkes4ifno=γur(c-kop4xror,ien0st)posisentsduskct5hott=hwaot(chkoa5xsl,fk-±2p4xo2=iπn/t−s√os3innα)tαhweshi2ntγeoγrpe,
annihilationoperatorscollectively,thematrix 2 sinα 2γ
andtwohalf-pointsonthebottomedgesaddingintotaltotwo
Hk =ξkSk0⊥−∗h ξ−kS∆+k⊥∗h −ξ−k0∆+h S∆k0⊥∗  (4) ifktnou4ltlthh=peeo(sfii±inxrts4sctπ.oB/rNn3Ze,or0ftseo)roatfhαnatdht=ekk5B5xβZ==,ai(dn±kd4wxi2nπh±gi/ci3n2h,πt±coact2saaπeln/tto√ohnet3wl)ycoocbmofeurbrlsleinasptepoidosifinnsetdesdst.
∆∗ 0 S⊥ −ξ −h
k k Therefore,eitherthesetk ork butnotbothisrelevantwhen
4 5
is the Hamiltonian density, ξ = (cid:15) − µ with µ the α (cid:54)= β as illustrated in Fig. 2. The corresponding energy
k k
tchheemSiFcalorpdoetrenptiaarla,mSekt⊥er =∆ S=kx −g(cid:80)iSkky(cid:104)ci↑skcth↓,e−kS(cid:105)OCis, uannid- d2icsopserγs)iofnosrathtethfiersltocsaetti,o(cid:15)nk2of=ze−ro2st(acroes(cid:15)αk1−=2c−os2γt()cofosrαth+e
form in k space. The eigenvalues Eλk of the Hamilto- seconds(cid:16)et,(cid:15)k3 = 2tcosα(cid:17)forthethirdset,and(cid:15)k4 = (cid:15)k5 =
nian matrix with λ = {1,2,3,4} are simply given by 2tcosα 1− α2 sin2γ + t α sin(2γ) for the remaining
(cid:113) sin2α 2γ2 sinα γ
E = s ξ2 +h2+|∆|2+|S⊥|2+2p A , correspond- sets.
λk λ k k λ k
ing to the quasiparticle (s1,3 = p3,4 = +1) and quasihole k k k k k k
(p1,2 = s2,4 = −1)excitationenergiesofthesystem, where X5 X2 X5 5X X2 X 5
(cid:113)
A = (ξ2 +|∆|2)h2+|S⊥|2ξ2.IntermsofE ,theself- k k k k
k k k k λk 3X X 3 3X X 3
consistencyequationscanbewrittenas[18]
k k
kX kX kX X 4 kX 4X
−M|∆| = 1(cid:88)∂Eλkf(E ), (5) 4 1 4 1
g 4 ∂|∆| λk k k k k
λk 3X X 3 3X X 3
(cid:20) (cid:21)
N +N = 1(cid:88) 1−2∂Eλkf(E ) , (6) kX kX kX kX kX Xk
↑ ↓ 4 ∂µ λk 5 2 5 5 2 5
λk
N −N = 1(cid:88)∂Eλkf(E ), (7) α = β α = β
↓ ↑ 2 ∂h λk
λk FIG. 2. (Color online) The gapless k-space points are illustrated
withinthefirstBZ.Notethateitherthesetk ork butnotbothis
where f(x) = 1/[ex/(kBT) + 1] is the Fermi func- 4 5
relevantwhenα(cid:54)=β(seethetext).
tion with k the Boltzmann constant and T the temper-
B
ature. Here, the derivatives are ∂E /∂|∆| = (1 +
λk
p h2/A )|∆|/E for the order parameter, ∂E /∂µ = It is well-known that opening or closing of a gap in k
λ k λk λk
−[1+p (h2+|S⊥|2)/A ]ξ /E forthechemicalpotential, space gives rise to a topological phase transition between
λ k k k λk
and∂E /∂h=−[1+p (ξ2+|∆|2)/A ]h/E fortheZee- SFs with distinct k-space topologies. In our case, these
λk λ k k λk
manfield.WhileµdeterminesthetotalnumberN =N +N transitions are further signalled by changes in the topolog-
↑ ↓
of atoms where N = (cid:80) n with the local fermion filling ical invariant of the system [24], which by definition may
σ i σi
0 ≤ n = (cid:104)c† c (cid:105) ≤ 1,h ≥ 0determinesthepolarization only change due to a change in system’s underlying topol-
σi σi σi ogy. For instance, it can be shown that the change in CN at
P = (N −N )/N ≥ 0 of the system which is assumed to
↑ ↓ h = h is simply given by a sum over the Berry indices
bepositivewithoutloosinggenerality. Next,weanalyzeEλk atalltoku0chingpoints[17,18]∆CN(h ) = (cid:80) sign(Y )
fko-rspgaacpepteodp/goalopgleyssofsothlueitrioenxscittoatdioisntsin.guish SF phases by the whereYk0 = (cid:104)∂∂Skxkx(cid:16)∂∂Skyky − ∂∂Skxky ∂∂(cid:17)Skykx(cid:105)k0k.0Inparticku0lar,wefikn0d
thatY = 6t2αβ sinα + sinγ sinγ isalwayspositiveand
k1 α 2γ γ
III. TOPOLOGICALSUPERFLUIDS (cid:16) (cid:17)
∆CN(h ) = +1,Y = −6t2αβ sinα − sinγ sinγ isal-
k1 k2 α 2γ γ
It is clear that E1k and E2k may become gapless in k ways negative and ∆CN(hk2) = −1, Yk3 = −3t2αβsinγ22γ
space, i.e., E are precisely 0 at some special k points is always negative and ∆CN(h ) = −2, and Y =
1(2)k0 0 (cid:16) k3(cid:17) k4
(cid:112)satisfying the condition |Sk⊥0| = 0 when h = hk0 = Yk5 = 3t2αβ sinγ22γ − sinα22αsi4nγ44γ is always positive and
((cid:15) −µ)2+|∆|2. There are five sets of k points satis-
fyingk0|Sk⊥0| = 0: inadditiontothecenterk1 =0 (0,0)of√the ∆inCCNN(ahdkd4s)u=pt∆oC0Nfo(rhakl5l)pa=ram+e2t.erNsoatseathfuantcthtieontootaflinchcraenagse-
hexagon-shaped BZ, the two-point set k = (0,±2π/ 3) ingh. ThisisbecausesincetheSFphaseistopologicallytriv-
2
corresponds to the midpoints of the top and bottom edges of ialintheh→0limit,thenormalphasemustalsobetopolog-
the BZ adding i√n total to one full point, the four-point set icallytrivialintheh (cid:29) taswell. Inparticular,whenα = β,
k =(±π,±π/ 3)correspondstothemidpointsoftheright these expressions reduce to Y = −3Y = −3Y =
3 k1 k2 k3
4
4Y = 4Y = 9t2sin2α, and the corresponding energy consistency Eqs. (5), (6) and (7) at T = 0 as a function of
k4 k5
dispersionsaregivenby(cid:15) = −3(cid:15) = −3(cid:15) = −2(cid:15) = totalparticlefillingF =(N +N )/M andα=β. Wecon-
k1 k2 k3 k4 ↑ ↓
−2(cid:15) = −6tcosα. Thus, sincethe combinedset {k ,k } struct the ground-state phase diagrams for a number of po-
k5 4 5
correspondstothesixcornersofthefirstBZasillustratedin larization P = (N −N )/(N +N ) and g values. Note
↑ ↓ ↑ ↓
Fig. 2, and h = h and h = h are two-fold degen- that since 0 ≤ N /M ≤ 1 within the single-band approxi-
k2 k3 k4 k5 σ
erate, we find ∆CN(h ) = +1, ∆CN(h ) = −3 and mation,themaximumpossiblevalueofF dependsonP,i.e.,
k1 k2,3
∆CN(h )=+2. Basedonthisclassificationscheme,next F = 2/(1+|P|) and the majority (minority) component
k4,5 max
we explore the phase diagrams of the system for SF phases isabandinsulator(normal)forF ≥ F . Inaddition,since
max
withdistinctk-spacetopologies. thephasediagramsaresymmetricaroundπwithα→2π−α
symmetry, which is caused by the symmetry of the DoS as
shown in Fig. 1(a), we present the resultant phase diagrams
IV. SELF-CONSISTENTRESULTS onlyfortheinterval0≤α≤π.
For simplicity, we restrict our numerical analysis to the
ground state with symmetric SOCs, and solve the self-
For instance, in Fig. 3, we show colored maps of |∆|/t toenhancedpairing,oneofthemostnotablefindingsinFig.4
and the corresponding boundaries for the topological quan- isthatnotonlythequalitativebutalsothequantitativestruc-
tumphasetransitionswheng =4tandP ={0,0.2,0.4,0.6}. tureofthephasediagramsarequiterobustagainstincreasing
Here,thenormalregionischaracterizedby|∆| (cid:46) 10−3t. In g. Thus, weconclude thattopological phase transitionswith
the trivial case of zero polarization shown in Fig. 3(a), the ∆CN = {±1,±2,±3} are generally accessible on a trian-
entire phase diagram is a SF and even though some of the gular lattice. Having achieved our primary objective of con-
low-energyfeaturesaresmearedoutbyfiniteg, |∆|haspre- structing the ground-state phase diagrams, next we end this
cisely the symmetry of the DoS shown in Fig. 1(a), where paper with a briery summary of our conclusions and an out-
ω = 0correspondstoµ = 0,i.e.,thehalf-fillingF = 1. As look.
increasing P progressively weakens |∆| due to the Zeeman-
inducedpairingmismatchbetween↑and↓fermions,notonly
thenormalregionexpandsbutalsomorefootprintsofthelow-
V. CONCLUSIONS
energyDoSbecomegraduallysalientin|∆|,includingthegap
atω = 0whenα = π/2. Inparticular, wefindreentrantSF
phase transitions that are interfered by the normal phase in Insummary,todescribetheBCS-BECevolutionofaspin-
the neighbourhood of this gap in Figs. 3(c) and 3(d). Fur- 1/2 Fermi gas that is loaded on a uniform triangular optical
thermore, whiletheSFphaseistriviallygappedintheentire lattice, here we considered a single-band lattice Hamiltonian
phasediagramshowninFig.3(a),havingafiniteP givesrise within the mean-field approximation for on-site pairing. In
to the emergence of two phase-transition branches per each particular,weexploredtopologicalphasetransitionsbetween
k ,satisfyingthecriticalconditionh = h . Thesebranches gapped and gapless SF phases that are induced by the in-
ariise from the particle- and hole-pairingksiectors, and all of terplay of an out-of-plane Zeeman field and a non-Abelian
themeventuallymeetatF = 1andα = π/2withincreasing gauge field. These transitions are signalled by changes in
P. Since µ = 0 for all parameters as long as F = 1, and the underlying Chern number, and we found that ∆CN =
(cid:15) = 0 for all k points at α = β = π/2, all of the transi- {±1,±2,±3} are generally accessible on a triangular lat-
tikoin boundaries biecome degenerate precisely when h = |∆| tice. By constructing a number of ground-state phase dia-
issimultaneouslysatisfied. Therefore,thecriticalvalueofP grams self-consistently for a wide range of parameter space,
for such a crossing clearly increases with g, and it happens we also found reentrant SF phase transitions that are inter-
aroundP ≈0.2wheng =4tandP ≈0.4wheng =10t. fered by the normal phase, and traced their imprints to the
c c
DoS of the non-interacting problem. In sharp contrast to the
Similarly,inFig.4,weshowcoloredmapsof|∆|/tandthe continuumsystemswherethelow-energyDoSincreaseswith
corresponding boundaries for the topological quantum phase increasing SOC, we showed that the DoS has a much richer
transitionswhenP = 0.5andg = {3,4,5,7}. Theintricate dependence on energy and SOC on a triangular lattice, lead-
dependenceof|∆|onF andα,andtheresultantreentrantSF ing in return to an intricate dependence of the SF order pa-
phase transitions can again be traced back to the low-energy rameteronparticlefilling,SOCandinter-particleinteraction.
featuresofDoS,asthey playthemostimportantrolesinthe Sincethelow-energyDoSplaysthemostimportantroleinthe
weakly-interactinglimitwherethereentrantbehaviorismost weakly-interacting limit, the reentrant behavior is most emi-
eminentforanyP.Apartfromtheshrinkageofthenormalre- nentthereforanypolarization,anditgraduallydiminishesas
gionandgradualdisappearanceofthereentrantbehaviordue theinteractiongetsstronger.
5
FIG.3. (Coloronline)TheSForderparameter|∆|/t(left)andtheextractedboundariesforthetopologicalquantumphasetransitions(right)
areshownasafunctionoftotalparticlefillingF andSOCα=βfor(a)P =0,(b)0.2,(c)0.4and(d)0.6,whereg=4t.
6
Even though the topological phase transitions discussed hexagonallattices,higher-dimensionallattices,beyondmean-
in this paper occur in momentum space, and therefore, are field effects, etc. Theoretical understanding of these exten-
mostevidentinthemomentumdistributionsofatomsintime- sions in greater depth will surely have dramatic impacts for
of-flight measurements, it is well-established in the context notonlytothecold-atomandcondensed-mattercommunities
of unconventional (e.g., p- or d-wave) nodal-SFs that such but also to the others, where the interplay of SF pairing and
changesalsoleavenon-analytictracesinthethermodynamic SOCarecontemporaryconceptsofferingfuturistictechnolog-
properties of the system including the compressibility, spin icalapplications.
susceptibility, specificheat, etc. Thus, asanoutlook, webe-
lieve it is fruitful to extend this line of research towards all
sorts of directions, including finite center-of-mass pairings, VI. ACKNOWLEDGMENTS
finite temperatures, multi-band lattices, longer-ranged hop-
pings/interactions, Abelian gauge fields, confined systems, WegratefullyacknowledgefundingfromTU¨BI˙TAKGrant
No. 1001-114F232.
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7
FIG.4. (Coloronline)TheSForderparameter|∆|/t(left)andtheextractedboundariesforthetopologicalquantumphasetransitions(right)
areshownasafunctionoftotalparticlefillingF andSOCα=βfor(a)g=3t,(b)4t,(c)5tand(d)7t,whereP =0.5.