Competing Chiral Orders in the Topological Haldane-Hubbard Model of Spin-1/2 Fermions and Bosons

1.8 MB·English

by  Ciarán Hickey

Competing chiral orders in the topological Haldane-Hubbard model of spin-1/2
fermions and bosons
C. Hickey1, P. Rath1,2, and A. Paramekanti1,3
1Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
2Department of Physics, Indian Institute of Technology, Kanpur 208106, India and
3Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
MotivatedbyexperimentsonultracoldatomswhichhaverealizedtheHaldanemodelforaChern
insulator, we consider its strongly correlated Mott limit with spin-1/2 fermions. We find that slave
5 rotor mean field theory yields gapped or gapless chiral spin liquid Mott insulators. To study com-
1 peting magnetic orders, we consider the strong coupling effective spin Hamiltonian which includes
0 chiralthree-spinexchange. Weobtainitsclassicalphasediagram,uncoveringvariouschiralmagnetic
2 orders including tetrahedral, cone, and noncoplanar spiral states which can compete with putative
chiralquantumspinliquids. Westudytheeffectofthermalfluctuationsonthesestates,identifying
r
p crossovers in the spin chirality, and phase transitions associated with lattice symmetry breaking.
A We also discuss analogous effective spin Hamiltonians for correlated spin-1/2 bosons. Finally, we
point out possible experimental implications of our results for cold atom experiments.

6
1
I. INTRODUCTION Hubbard model suggests the appearance of fractional-

] izedCherninsulators28 ortopologicalN´eelorderatmod-
s
a erate coupling with a staggered sublattice potential.29
Momentum space topology is key to our understand-
g An alternative slave rotor mean field theory of spin-1/2
- ingofphasessuchastopologicalinsulators1,2orquantum
nt anomalous Hall insulators.3,4 The interplay of momen- fcehramrgioends,eginrewehoifchfrHeeudbobma,rdlerdeptuolsaionmeloacnalfiizeelds odnelsycrtihpe-
a tum space topology and local real space interactions is
tionofagappedchiralspinliquid(CSL).30,31 Gutzwiller
u expectedtoleadtoarichvarietyofcorrelatedinsulating
projected wave function studies show that such a CSL
q phases.5–12 This has motivated an extensive investiga-
. wouldbeinthesamephaseastheν=1/2bosonicLaugh-
t tion of the time-reversal invariant Kane-Mele-Hubbard
a lin state.32,33 More generally, such CSLs have also been
model or more realistic variants,13–19 which provide the
m showntobeinducedbychiral3-spininteractions,34,35 or
simplestexamplesofinteractingquantumspinHallinsu-
- lators. Recently, cold atoms in a shaken optical lattice20 appear naturally for SU(N) fermions36 at large N.
d
n or laser-induced tunneling or transitions21,22 have been However, previous work has not considered the var-
o proposed for realizing topological states of matter, and ious types of magnetic orders which can compete with
c employed to realize the Haldane honeycomb model of a quantum spin liquid phases. Indeed, even the type of
[ Chern insulator (CI),23 and Chern bands in the square short-range spin correlations in the correlated Mott in-
4 lattice Hofstadter model with flux π/2 per plaquette.24 sulator, which could potentially be probed in cold atom
v Preliminary experiments suggest that interactions in the experiments, have not been explored. Finally, since the
4 Haldane model may not lead to excessive heating.23 Mo- topological order associated with such chiral spin liquids
0 tivated by this, we explore strong correlation effects in willnotpersistatnonzerotemperature,itisimportantto
3
the Haldane model for spin-1/2 fermions and bosons. gain an understanding of the thermal fluctuation effects
1
0 The Haldane model is defined on the two-dimensional on the Mott insulating state if we are to make connec-
. (2D) honeycomb lattice in Fig. 1(a), with a real near- tion with experiments. However, previous studies of this
1
est neighbor hopping amplitude t , and a complex model have all been restricted to the T =0 limit.
0 1
5 next-neighbor hopping t2eiφ which breaks time-reversal This motivates us to revisit the effect of strong corre-
1 symmetry.3 For t2 =0, the energy dispersion is identical lations on the Haldane-Hubbard model. Our key results
v: to graphene, supporting two inequivalent massless Dirac arethefollowing. (i)Weextendtheslaverotormeanfield
i fermion modes. For small t2(cid:54)=0, and φ(cid:54)=0,π, the Dirac theoryofthefermionicHaldane-Hubbardmodeltolarger
X fermions acquire a mass, leading to a quantized Hall ef- t . We show that this parton construction leads to CSLs
2
r fect σxy =±e2/h at half-filling. For spin-1/2 fermions, withgappedorgaplessbulkspinonexcitationswhichde-
a
witheachspeciesathalf-filling,σx↑y=σx↓y=±e2/h. What scend from CI or Chern metal (CM) phases. The gap-
is the fate of this quantum Hall insulator when interac- less spin liquids display multiple spinon Fermi pockets.
tions lead to Mott localization? Does the Mott insulator Both types of spin liquids are expected to support gap-
supportunusualmagnetismorspinliquidgroundstates? less spinon edge states. Realizing the Haldane-Hubbard
For spinless fermions, nearest-neighbor repulsion model with a broad range of t would thus allow one
2
induces a topologically trivial charge density wave to explore the physics of gapped as well as gapless chi-
insulator,25,26 while spinless bosons with a local Hub- ral spin liquid phases. (ii) The gapped or gapless spin
bard repulsion form a plaquette Mott insulator with liquids uncovered in the slave rotor parton description
loop currents.27 A slave-spin approach to the Haldane- may be unstable to magnetic ordering due to fluctua-
2
tion effects beyond mean field theory. To study mag-
neticinstabilitiesinthestronglycorrelatedlimit,wecon-
sider the strong coupling limit of the Haldane-Hubbard
model, and derive and study the effective spin Hamilto-
nian which has chiral 3-spin interactions. This leads us
to uncover a rich variety of competing chiral magnetic
states such as tetrahedral, cone, and noncoplanar spirals
which could compete with spin liquid phases. Quantum
melting such phases will typically lead to broken sym-
metry states. However, unlike other states, we find that
the tetrahedral state has spin correlations which respect
all lattice symmetries and a large uniform chirality. We
tentativelyidentifythisstateastheclassicalparentstate
FIG. 1: (a) Honeycomb lattice showing the two sublattices
of a chiral quantum spin liquid. Such tetrahedral states with hoppings t ,t and phase φ defining the noninteracting
1 2
have also been recently uncovered in a honeycomb lat- Haldanemodel,withlargeandsmalltrianglesenclosingfluxes
tice Hubbard model at intermediate correlations37. (iii) ±3φand−φrespectively. (b)Phasediagramathalf-fillingfor
Turning to physics at nonzero temperature, we discuss eachspinspeciesshowingChernbandmetalandChernband
thermal fluctuation effects and lattice symmetry break- insulator, and the Hall conductance per spin σxy in units of
ing thermal phase transitions associated with many of e2/h.
these states using a combination of analytic Landau the-
ory arguments supplemented by classical Monte Carlo
simulations. Suchthermaltransitionsmaybeobservable correlated Mott insulator once interactions exceed the
inexperiments. (iv)Finally,wehighlightsomeanalogous bandwidth. The critical repulsion U = 8η|K|, where
c
results for spin-1/2 bosons which can also be studied in K ≡K(t ,t ,φ) is the kinetic energy per site per spin
1 2
cold atom experiments, but which have not been theo- of the noninteracting bands at half-filling.
retically explored. Slave rotor mean field theory38,39, provides a concrete
realization of the Mott phase and the Mott transition;
it is a parton construction in which the spin and charge
II. FERMI HALDANE-HUBBARD MODEL
degrees of freedom are carried by two partons, a neutral
spin-1/2fermionandachargedspinlessrotor. Thelocal-
For fermions with spin-1/2, the Haldane Hubbard ization of the rotor angular momentum corresponds to
model is described by the Hamiltonian the transition into the Mott insulator phase. A single-
site slave rotor mean field theory38,39 yields η=1. How-
(cid:88) (cid:88)
HHH =−t1 (c†iσcjσ+h.c.)−t2 (eiνijφc†iσcjσ+h.c.) ever, a comparison with quantum Monte Carlo studies
(cid:104)ij(cid:105)σ (cid:104)(cid:104)ij(cid:105)(cid:105)σ of the honeycomb lattice Hubbard model40,41 at t2 = 0,
(cid:88) and variational numerical studies of the triangular lat-
+ U n n (1)
i↑ i↓ tice Hubbard model42 with t = φ = 0 show that the
1
i
single site slave rotor theory overestimates η; the Mott
where (cid:104).(cid:105) and (cid:104)(cid:104).(cid:105)(cid:105) denote, respectively, first and second transition occurs at a smaller renormalized η≈0.6-0.7.
neighbors, and ν = ±1, depending on whether we hop Fig. 2 shows the phase boundaries which delineate the
ij
along or opposite to the arrows shown in Fig. 1(a), and topological CM or CI from the correlated Mott insula-
U is the local Hubbard repulsion. tor (assuming U =8η|K| with η=0.65) at an interac-
c
For U = 0, diagonalizing the Hamiltonian leads to tion strength U/t = 10. Within slave rotor mean field
1
ChernbandswithChernnumbersC=±1. Dependingon theory, the spinon band structure in the Mott insulator
t /t and φ, these bands may overlap in energy, leading resembles the CI or CM band structure from which the
2 1
toaCMwithanon-quantizedHalleffect,orbewellsep- Mott insulator descends. Thus, depending on t ,φ, one
2
aratedinenergyleadingtoaCI(equivalently,aquantum could have gapped or gapless spinon excitations in the
Hallinsulator)withaHallconductivityσ =±e2/hper bulk, accompanied by gapless chiral edge modes; we in-
xy
spin. This phase diagram is shown in Fig. 1(b), along dicate these mean field states as ‘gapped CSL’ and ‘gap-
with σ which is quantized in the CI but non-quantized lessCSL’.Thespinonbandstructuremeanfieldtheoryis
xy
intheCM.(Notethatthet =0axisisaDiracsemimetal; simply inherited from the parent noninteracting fermion
2
on this singular line, the flux plays no role.) dispersion. Since the parent Chern metal phase exhibits
multiple Fermi pockets, as shown in Fig. 2, the gapless
CSL will inherit these as spinon Fermi pockets. Gapless
III. MEAN FIELD CHIRAL SPIN LIQUIDS chiral spin liquids have also been recently discussed in
kagome antiferromagnets.45
Upon increasing the Hubbard repulsion, this topolog- Goingbeyondmeanfieldtheory,weexpectthe‘gapped
ical band metal or band insulator will transition into a CSL’tobecomeagenuinespinliquidwithsemionexcita-
3
configurations. This approximation may be formally set
up by scaling the 2-spin exchange couplings by 1/(4S2)
and the 3-spin exchange couplings by 1/(8S3) and then
takingS →∞whichleadstoaclassicalspinmodelwith
quantum fluctuations suppressed by O(1/S). Below we
will use this classical approximation for S = 1/2 as an
uncontrolledapproximation,whichisneverthelessknown
toworkwellinmanycasesinpredictingthecorrectmag-
neticorderin2Dquantumspinmodels. Forplottingthe
groundstatespinconfigurationsandstructurefactors,we
view the honeycomb lattice as a square lattice with 1/4
deleted bonds (‘brickwall’ lattice), similar to the geome-
tryrealizedinopticallatticeexperiments.23Theresulting
FIG.2: SlaverotormeanfieldtheoryoftheHaldane-Hubbard phase diagram, shown in Fig. 3 for U/t = 10, contains
model showing the topological Chern metal (CM, green),
six phases: N´eel, Tetrahedral, Cantellated Tetrahedral,
Cherninsulator(CI,brown),andtopologicalspinliquidMott
Spiral, Cone-I and Cone-II, with a line indicating the
insulators(black)atU/t =10. ThetopologicalMottinsula-
1 regime below which this strong coupling description is
tor could have gapless or gapped spinon excitations depend-
appropriate.
ing on whether it descends from a CM or CI phase, leading
to gapless or gapped chiral spin liquids. An example of the Our main finding, atleast for t2/t1 (cid:46)1.5, as discussed
parent Fermi surface of the non-interacting model, which is below, is that the gapped spin liquid phase found in the
inherited by the spinons at mean-field level, is shown at a slaverotormeanfieldtheoryinSectionIIImaybepoten-
particular point. The slave rotor result has been rescaled by tially unstable to commensurate magnetic orders such as
η=0.65 (see text). NeelorTetrahedralorder. Ontheotherhand,theregime
we identify as a gapless spin liquid with Fermi pockets
in the slave rotor mean field theory appears to be domi-
tions and topological degeneracy.32,33 While these CSLs nantly unstable to various incommensurate spiral phases
are expected to be stable for weak gauge fluctuations, such as Spiral or Cone phases. Since the phase diagrams
strong gauge fluctuations might drive magnetic ordering obtainedinFig.2andFig.3involvedifferentapproxima-
instablilties, or, spinon pairing instabilities in the case tions,neitheroneisfullyreliable;furthernumericalwork
of the gapless CSL.43,44 Since slave rotor theory does is necessary to definitely identify whether the spin liquid
not capture competing magnetic ordering tendencies of phasesareindeedstableorgivewaytomagneticorother
the Mott insulator, we address such potential competing symmetry breaking orders. However, the true phase di-
phases using a strong coupling expansion.46 agram is expected to display phases shown in the phase
diagrams in Fig. 2 and Fig. 3.
We describe these various competing magnetic ground
IV. COMPETING MAGNETIC ORDERS states below. In the Appendix, we include “Common
origin” plots48 for these states for further insights.
At large U, a standard derivation46,47 leads to an ef- (i) N´eel: The N´eel state is collinear with spins pointing
fective spin Hamiltonian oppositetoeachotheronthetwosublatticesofthebrick-
wall lattice, with (cid:104)χˆ (cid:105) = 0 on all triangular plaquettes.
(cid:52)
Hspin=4Ut21 (cid:88)Si·Sj + 4Ut22 (cid:88)Si·Sj hItisbisttsruacBturraeggfapcteoarkSa(tqq)==(N1π,(cid:80)π)i.,j(cid:104)Si·Sj(cid:105)eiq·(ri−rj) ex-
(cid:104)ij(cid:105) (cid:104)(cid:104)ij(cid:105)(cid:105)
(ii) Tetrahedral: In this state, the spins point from the
−24t21t2 (cid:88) χˆ sinΦ − 24t32(cid:88) χˆ sinΦ (2) origin towards the four corners of a tetrahedron, and are
U2 (cid:52) (cid:52) U2 (cid:52) (cid:52) tiled on the lattice as shown in Fig. 3. This noncoplanar
small−(cid:52) big−(cid:52) √
state has a uniform chirality (cid:104)χˆ (cid:105) = −1/6 3 on each
(cid:52)
where χˆ ≡S ·S ×S is the scalar spin chirality oper- small-(cid:52). The tetrahedral state is a triple-q state, with
(cid:52) i j k
ator defined around triangular plaquettes, with the sites the structure factor S(q) exhibiting Bragg peaks at q=
{ijk} being labelled going anticlockwise around the tri- ±(π/2,±π/2) and q=(0,π).
angles. Φ being the flux enclosed by the correspond- (iii) Cantellated Tetrahedral: The Cantellated Tetrahe-
(cid:52)
ing triangle. As shown in Fig. 1(a), the set of triangles dral(CT)statedescendsfromaparenttetrahedralstate.
includes small triangles that enclose flux Φ = −φ, big It is reminiscent of the cuboctahedral state in frustrated
(cid:52)
triangleswithinahexagonwhichencloseafluxΦ =3φ, kagome antiferromagnets.49–51 In the parent tetrahedral
(cid:52)
and big triangles defined around a site which enclose a configuration, the spins marked A (or B,C,D) in Fig. 3
flux Φ =−3φ. form a honeycomb (or brickwall) lattice, with a larger
(cid:52)
Treating the spin as a classical vector, we have ob- unit cell. We can deform these ferromagnetically aligned
tained the magnetically ordered ground states of H spins by dividing the larger honeycomb lattice into two
spin √ √
usingnumericalsimulatedannealingandvariationalspin sublattices, and then allowing for a 3× 3 canting on
4
angle θ varies with t ,φ; at large t /t , the spins on
2 2 1
each large-(cid:52) tend to form an orthonormal triad, with
√
θ = tan−1(1/ 2). The Triad-I is a triple-q state; S(q)
exhibits Bragg peaks at q = (0,±2π/3), q = (π,±π/3),
and q = (0,0). Let us now consider an axis nˆ which is
s
perpendicular to the plane formed by nˆ and any one of
the spins. The Cone-I state is obtained by rotating all
the spins about this axis by an angle θ(r) = q·r where
q is an incommensurate spiral wavevector; moving along
q, each spin of the triad again traces out a cone, and the
Bragg peaks acquire a weak incommensuration.
(v) Cone-II: The Cone-II state occurs for φ > π/3,
where the sin3φ term in Eq. 2 changes sign. Here,
the parent state is a Triad-II state in which the spins
on sublattice-A are reordered, with SA = cosθnˆ +
j
sinθcos(4πj/3)nˆ +sinθsin(4πj/3)nˆ , whilethecone
⊥1 ⊥2
onsublattice-Bissimplyflipped,sothatSB =−cosθnˆ+
j
sinθcos(ϕ+2πj/3)nˆ +sinθsin(ϕ+2πj/3)nˆ ,withthe
⊥1 ⊥2
ground state energy being independent of ϕ. The Triad-
II state has a nonzero staggered magnetization, with
Bragg peaks in S(q) at q = (0,±2π/3), q = (π,±π/3),
and q = ±(π,π). The Cone-II state is obtained by ro-
tating all the spins of Triad-II about the nˆ =nˆ axis by
s
θ(r)=q·rwhereqisanincommensuratespiralwavevec-
FIG.3: Top:PhasediagramofH showingvariousclassical
spin
magnetically ordered ground states at U/t = 10: N´eel, Spi- tor, leading to weakly incommensurate Bragg peaks.
ral, Tetrahedral, Cantellated Tetrahedral (CT), Cone-I, and (vi) Noncoplanar Spiral: At φ = 0, the Hamiltonian
Cone-II. Below the dashed (white) line, the slave rotor re- H is the frustrated J -J Heisenberg antiferromagnet
spin 1 2
sult shows that we are in the Mott insulator, so the strong on the honeycomb lattice. The classical ground states
coupling expansion is expected to be valid. Stars indicate of this model consists of coplanar incommensurate spi-
points where we numerically study the effect of thermal fluc- rals for J > J /6.52 For φ (cid:54)= 0, chiral terms in H
2 1 spin
tuations. Bottom:SpinconfigurationsintheTetrahedraland
cause spins on the A and B sublattices to cant in oppo-
Cone states (viewing the honeycomb lattice as a ‘brickwall’
site directions away from the spiral plane, leading to a
for visual convenience). The indicated Triad states act as
noncoplanar spiral with a uniform χ (cid:54)=0 on all big-(cid:52)s.
parent states for Cone-I and Cone-II, which are obtained by (cid:52)
spiralling the Triads about the indicated nˆ axis. The CT
√ √ s
descendsfromtheTetrahedralbya 3× 3cantingpattern.
V. THERMAL FLUCTUATIONS
eachsublattice; thissplitseachvertexofthetetrahedron In 2D, the Mermin-Wagner theorem53 precludes spin
intosixvertices(formingasmallhexagon)leadingtothe SU(2) symmetry breaking at T > 0. However, discrete
CT state with a 24-site unit cell. Here, the CT state ex- ordersassociatedwithlatticesymmetrybreakingcansur-
ists over an extremely narrow window between the N´eel vive thermal fluctuations.
and Tetrahedral states. Third-neighbor Si · Sj terms, TheN´eelandTetrahedralordersbreakSU(2)symme-
which favor the cuboctahedral state on the kagome lat- try, but their SU(2) invariant spin correlations, such as
tice, could enhance the regime of CT order. (cid:104)S ·S (cid:105)and(cid:104)S ·S ×S (cid:105),respectalllatticesymmetries.
i j i j k
(iv) Cone-I: This state descends from a Triad-I state ThesestatesatT >0canthusbesmoothlyconnectedto
in which spins on the large-(cid:52) tend to form a triad on thehigh-T paramagneticstate. AsshowninFig.4(a),for
each sublattice, leading to a nonzero chirality. The or- the Tetrahedral state at U/t = 10, there is a crossover
1
ganization of the spins is shown in Fig. 3. In Cone-I, temperature ∼0.05t below which the spin chirality χ
1 √ (cid:52)
which occurs for φ < π/3, we start with triads on the on small triangles becomes large, saturating to 1/6 3 as
two sublattices that are lined up with a common axis T → 0. The T > 0 regime here can thus be viewed as a
nˆ, with spins making an angle θ with nˆ. Forming an or- classical CSL. By contrast, χ vanishes at T =0 in the
(cid:52)
thonormalbasis{nˆ,nˆ⊥1,nˆ⊥2},weparameterizethethree N´eelstate,butitexhibitsapeakatnonzerotemperature.
spins of the triad on sublattice-A, SA (j = 0,1,2), as Previous work has shown that at φ = 0, the spiral
j
SA = cosθnˆ+sinθcos(2πj/3)nˆ +sinθsin(2πj/3)nˆ , ground states of the frustrated J -J Heisenberg anti-
j ⊥1 ⊥2 1 2
while the triad on sublattice-B is rotated by ϕ about nˆ, ferromagnet have nematic order associated with broken
withSB =cosθnˆ+sinθcos(ϕ+2πj/3)nˆ +sinθsin(ϕ+ C honeycomb lattice symmetry which survives up to
j ⊥1 3
2πj/3)nˆ . This state has a nonzero net magnetization. a thermal Z -clock transition.52 The noncoplanar spiral
⊥2 3
The ground state energy is independent of ϕ. The triad doesnotinvolveanyadditionalsymmetrybreaking,soit
5
Here, we focus on the strong coupling limit
U/t ,U/t (cid:29) 1. The spin Hamiltonian in the result-
1 2
ing Mott insulator can be derived in a manner similar to
thefermioncase, althoughitismoretedioustocalculate
the terms to O(t3/U2). We find
(cid:88) (cid:88)
HBose=J (φ) S ·S +J (φ) S ·S
spin 1 i j 2 i j
(cid:104)ij(cid:105) (cid:104)(cid:104)ij(cid:105)(cid:105)
FIG.4: (a)Temperaturedependenceofthespinchiralityχ(cid:52) +24t21t2 (cid:88) χˆ sinΦ + 24t32 (cid:88) χˆ sinΦ(4)
on small triangles for U/t=10 and φ=π/3, showing its low U2 (cid:52) (cid:52) U2 (cid:52) (cid:52)
temperature saturation in the Tetrahedral phase (t /t =1) small−(cid:52) big−(cid:52)
2 1
and its vanishing in the T =0 N´eel phase (t /t =0.4). We
have plotted 5×χ(cid:52) in the N´eel state for c2lar1ity. (b) Log with J1(φ) = −4tU21 −24tU21t22 cosφ and J2(φ) = −4Ut22 −
plot of finite size scaling of critical susceptibility for the Z3- 6t21t2 cosφ−12t32 cos3φ. Similar to the fermion case, the
clock ordering transition in the Cone-II state at U/t =10, U2 U2
1 set of triangles includes small triangles that enclose flux
φ=9π/20, t /t =1.5 showing χ ∼Lγ/ν, with γ/ν=1.80(8).
2 1 3 Φ = −φ, big triangles within a hexagon which enclose
(cid:52)
a flux Φ = 3φ, and big triangles defined around a site
(cid:52)
which enclose a flux Φ = −3φ. As expected, the two-
undergoes a similar Z -clock transition at φ(cid:54)=0. (cid:52)
3 spin exchanges are ferromagnetic at φ=0, and the chiral
In the Cone states, the choice of the common axis
terms also change sign relative to the fermions.
nˆ represents a spontaneously broken SU(2) symmetry;
In order to localize bosons, it is well known that the
long wavelength fluctuations will disorder nˆ at arbitrar-
required repulsive interactions are much stronger. Thus,
ilysmallT >0,restoringSU(2)symmetry. UsingMonte
t/U is much smaller in the Mott insulating phase of
Carlosimulations,wefindthatthenearestneighborspin
spinor bosons, so that the chiral terms are much weaker
correlations (cid:104)S ·S (cid:105) are modulated in the Cone states,
i j inmagnitudecomparedtothefermioncase. Inaddition,
leadingtoenergymodulationsonthebondsofthelattice.
unlike the fermion case, both J and J are ferromag-
In the Cone-I state, these energy modulations resemble 1 2
√ √ netic,whichmeansthesetwo-spinexchangecouplingsdo
a 3× 3columnarvalence-bondsolidpattern,whilein
not frustrate each other. Due to both these reasons, we
the Cone-II state the energy modulations on the bonds
find that ferromagnetic order almost completely domi-
resemble a staggered valence-bond solid bond pattern.
nates the magnetic phase diagram in the Mott insulator
This leads to a Z symmetry breaking in both states.
3 of spinor bosons.
We thus expect the Cone states to undergo a thermal
In the ferromagnetic Mott insulator, the spin of the
Z -clock transition into the high temperature paramag-
3 bosons plays no role. Thus, our model should yield re-
netic state. In the Cone-II state at U/t =10, φ=9π/20,
1 sultsidenticaltothatforspinlessbosons. Indeed,assum-
t /t =1.5, the computed peak susceptibility χ (L) for
2 1 3 ing ferromagnetic order in our model, the chiral terms
thistransitionisplottedinFig.4(b);itsfinitesizescaling
vanish. Nevertheless, we find that the bond energies dis-
showsT ≈0.10t ,withχ (L)∼Lγ/ν,withγ/ν=1.80(8),
c 1 3 playsafluxdependencearisingfromtheflux-dependence
consistent with the exact Z -clock result54 for the expo-
3 ofthetwo-spinexchangecouplings. Thisleadstoaweak
nent ratio γ/ν =26/15.
nonzeroloopcurrents∼t3/U2 aroundtriangularplaque-
ttes, consistent with weak loop currents predicted in the
plaquette Mott insulator of spinless bosons using quite
VI. MOTT INSULATOR OF SPINOR BOSONS
differentapproaches.27 Thisresultforspinorbosonsisin
IN THE HALDANE HUBBARD MODEL
strikingcontrasttothefermioncase, wheretheonlyflux
dependencearisesfromthechiralterms,soloopcurrents
Cold atom experiments can also study interacting
to O(t3/U2) only arise in fermion Mott insulators which
bosonsintheHaldane-Hubbardmodel. Previousworkon
have non-coplanar spins with a nonzero scalar chirality.
spinlessbosonshasshowntheemergenceofchiralsuper-
fluidsandplaquetteMottinsulatorswithloopcurrents.27
Forpseudospin-1/2bosons, theHaldane-Hubbardmodel
VII. DISCUSSION
is
HBose =−t(cid:88)(b† b +h.c.)−t (cid:88)(eiνijφb† b +h.c.) We have discussed spin-1/2 fermions or bosons in the
HH 1 iσ jσ 2 iσ jσ strongly correlated Haldane-Hubbard model. Our main
(cid:104)ij(cid:105)σ (cid:104)(cid:104)ij(cid:105)(cid:105)σ
finding is the rich variety of competing chiral magnetic
1 (cid:88)
+ U n n (3) orders with varying flux and hopping. Such orders,
2 σσ(cid:48) iσ iσ(cid:48) even if they are short-ranged, could be probed via mag-
iσσ(cid:48)
netic Bragg scattering,55,56 which has been shown to
For atoms such as 87Rb, the background scattering be a useful probe in atomic gases. For fermions, since
lengths are nearly isotropic, so we set U =U. ∂H /∂φ∼χˆ , measuring the energy change upon an
σσ(cid:48) spin (cid:52)
6
adiabaticchangeoffluxisanindirectwaytomeasureχ . phaseofspinorbosonsisdominatedbytheferromagnetic
(cid:52)
This relation also ties the spin chirality to (weak) charge phase, lowering U might lead to superfluids with exotic
currents in the Mott insulator which could potentially magnetic orders; this will be explored elsewhere.
be detected using quantum quenches.57–59 More direct
routes to detect the spin chirality of atoms are desirable.
Doping these magnetically ordered states leads to non-
Acknowledgments
trivialChernbandsandanonzerochargeHalleffect,sim-
ilar to that in certain frustrated metallic magnets.60–62
Quantum melting the Tetrahedral state, which is uni- We acknowledge useful discussions with I. Bloch, D.
form with a large chirality, may lead to a quantum CSL Greif, Z. Papic, S. Parameswaran, and N. Perkins. This
ground state with short range magnetic correlations at research was funded by NSERC of Canada and Mi-
the same wavevectors. Further work is needed to clar- tacs through the Mitacs-Globalink program. AP thanks
ify the energetic competition between CSLs and chiral the Aspen Center for Physics (Grant No. NSF PHY-
magnetic orders. Finally, although the Mott insulating 1066293) where part of this work was completed.
1 M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 134010 (2013).
(2010). 23 G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat,
2 X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237
(2011). (2014).
3 F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). 24 M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T.
4 C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, Barreiro, S. Nascimbene, N. R. Cooper, I. Bloch, and
M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, et al., Sci- N. Goldman, Nat Phys advance online publication
ence 340, 167 (2013). (2014).
5 D. Pesin and L. Balents, Nat Phys 6, 376 (2010). 25 C. N. Varney, K. Sun, M. Rigol, and V. Galitski, Phys.
6 D. N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Nat Com- Rev. B 82, 115125 (2010).
mun 2, 389 (2011). 26 C. N. Varney, K. Sun, M. Rigol, and V. Galitski, Phys.
7 T. Neupert, L. Santos, C. Chamon, and C. Mudry, Phys. Rev. B 84, 241105 (2011).
Rev. Lett. 106, 236804 (2011). 27 I.Vasic,A.Petrescu,K.LeHur,andW.Hofstetter,ArXiv
8 N. Regnault and B. A. Bernevig, Phys. Rev. X 1, 021014 e-prints (2014), 1408.1411.
(2011). 28 J.MaciejkoandA.Ruegg,Phys.Rev.B88,241101(2013).
9 D. A. Abanin and D. A. Pesin, Phys. Rev. Lett. 109, 29 D. Prychynenko and S. D. Huber, ArXiv e-prints (2014),
066802 (2012). 1410.2001.
10 D.Cocks,P.P.Orth,S.Rachel,M.Buchhold,K.LeHur, 30 J.He,Y.-H.Zong,S.-P.Kou,Y.Liang,andS.Feng,Phys.
and W. Hofstetter, Phys. Rev. Lett. 109, 205303 (2012). Rev. B 84, 035127 (2011).
11 A. M. La¨uchli, Z. Liu, E. J. Bergholtz, and R. Moessner, 31 J.He,S.-P.Kou,Y.Liang,andS.Feng,Phys.Rev.B83,
Phys. Rev. Lett. 111, 126802 (2013). 205116 (2011).
12 W.Witczak-Krempa,G.Chen,Y.B.Kim,andL.Balents, 32 Y. Zhang, T. Grover, and A. Vishwanath, Phys. Rev. B
AnnualReviewofCondensedMatterPhysics5,57(2014). 84, 075128 (2011).
13 D. Zheng, G.-M. Zhang, and C. Wu, Phys. Rev. B 84, 33 Y. Zhang, T. Grover, A. Turner, M. Oshikawa, and
205121 (2011). A. Vishwanath, Phys. Rev. B 85, 235151 (2012).
14 M.Hohenadler,T.C.Lang,andF.F.Assaad,Phys.Rev. 34 M. Greiter, D. F. Schroeter, and R. Thomale, Phys. Rev.
Lett. 106, 100403 (2011). B 89, 165125 (2014).
15 C. Griset and C. Xu, Phys. Rev. B 85, 045123 (2012). 35 B. Bauer, L. Cincio, B. P. Keller, M. Dolfi, G. Vidal,
16 H.-H. Hung, L. Wang, Z.-C. Gu, and G. A. Fiete, Phys. S. Trebst, and A. W. W. Ludwig, Nat Commun 5 (2014).
Rev. B 87, 121113 (2013). 36 M. Hermele, V. Gurarie, and A. M. Rey, Phys. Rev. Lett.
17 M.Hohenadler,Z.Y.Meng,T.C.Lang,S.Wessel,A.Mu- 103, 135301 (2009).
ramatsu, and F. F. Assaad, Phys. Rev. B 85, 115132 37 K. Jiang, Y. Zhang, S. Zhou, and Z. Wang,
(2012). arXiv:1411.6019 (unpublished).
18 A. Ru¨egg and G. A. Fiete, Phys. Rev. Lett. 108, 046401 38 S.FlorensandA.Georges,Phys.Rev.B70,035114(2004).
(2012). 39 E. Zhao and A. Paramekanti, Phys. Rev. B 76, 195101
19 T.Liu,B.Douc¸ot,andK.LeHur,Phys.Rev.B88,245119 (2007).
(2013). 40 Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad, and
20 P.Hauke,O.Tieleman,A.Celi,C.O¨lschl¨ager,J.Simonet, A. Muramatsu, Nature 464, 847 (2010).
J. Struck, M. Weinberg, P. Windpassinger, K. Sengstock, 41 S. Sorella, Y. Otsuka, and S. Yunoki, Sci. Rep. 2 (2012).
M.Lewenstein,etal.,Phys.Rev.Lett.109,145301(2012). 42 H. Morita, S. Watanabe, and M. Imada, Journal of the
21 D. Jaksch and P. Zoller, New Journal of Physics 5, 56 Physical Society of Japan 71, 2109 (2002).
(2003). 43 R. B. Laughlin, Phys. Rev. Lett. 60, 2677 (1988).
22 N. Goldman, F. Gerbier, and M. Lewenstein, Journal of 44 W. Bishara and C. Nayak, Phys. Rev. Lett. 99, 066401
Physics B: Atomic, Molecular and Optical Physics 46, (2007).
7
FIG. 5: Common origin plots for the Neel, Tetrahedral, Cantellated Tetrahedral, Spiral, Cone-I and Cone-II states, with red
and black dots representing spins on the two sublattices of the honeycomb (or ‘brickwall’) lattice.
45 S. Bieri, L. Messio, B. Bernu, and C. Lhuillier, ArXiv e- 55 T. A. Corcovilos, S. K. Baur, J. M. Hitchcock, E. J.
prints (2014), 1411.1622. Mueller,andR.G.Hulet,Phys.Rev.A81,013415(2010).
46 A. H. MacDonald, S. M. Girvin, and D. Yoshioka, Phys. 56 R. A. Hart, P. M. Duarte, T.-L. Yang, X. Liu, T. Paiva,
Rev. B 37, 9753 (1988). E. Khatami, R. T. Scalettar, N. Trivedi, D. A. Huse, and
47 O. I. Motrunich, Phys. Rev. B 73, 155115 (2006). R. G. Hulet (2014), 1407.5932.
48 S. R. Sklan and C. L. Henley, Phys. Rev. B 88, 024407 57 M. Killi and A. Paramekanti, Phys. Rev. A 85, 061606
(2013). (2012).
49 L.Messio,C.Lhuillier,andG.Misguich,Phys.Rev.B83, 58 M. Killi, S. Trotzky, and A. Paramekanti, Phys. Rev. A
184401 (2011). 86, 063632 (2012).
50 S. Ghosh, P. O’ Brien, M. J. Lawler, and C. L. Henley, 59 A.Dhar,T.Mishra,M.Maji,R.V.Pai,S.Mukerjee,and
ArXiv e-prints (2014), 1407.5354. A. Paramekanti, Phys. Rev. B 87, 174501 (2013).
51 S.-S. Gong, W. Zhu, L. Balents, and D. N. Sheng, ArXiv 60 I.MartinandC.D.Batista,Phys.Rev.Lett.101,156402
e-prints (2014), 1412.1571. (2008).
52 A. Mulder, R. Ganesh, L. Capriotti, and A. Paramekanti, 61 Y. Kato, I. Martin, and C. D. Batista, Phys. Rev. Lett.
Phys. Rev. B 81, 214419 (2010). 105, 266405 (2010).
53 N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 62 J. W. F. Venderbos, M. Daghofer, J. van den Brink, and
(1966). S. Kumar, Phys. Rev. Lett. 109, 166405 (2012).
54 F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982).
Appendix A: Common origin plots for the various magnetic orders
Forvisualizingspinconfigurations, weemploycommonoriginplots48 whereallspinsonthelatticeareplottedwith
their tails at the center of a unit sphere and heads marked as dots on the surface of the sphere. We mark the spins
on the two sublattices of the honeycomb (i.e., brickwall) lattice using different colors, red and black. Doing this for
the Tetrahedral state, for instance, we see that all spins point along one of four directions, towards the vertices of
a tetrahedron. For the Cantellated Tetrahedral, the vertices split into six points, three on each sublattice. For the
Spiralstate,wefindthatthespinsonthetwosublatticeswhichsupportincommensuratespiralscantoutoftheplane
of the coplanar spiral in opposite directions. The Cone-I state, which is obtained by spinning a triad state about a
specific axis leads to one great circle and two small circles, while Cone-II leads to two parallel small circles.