-
Topological Invariant and Quantum Spin Models from Magnetic �
Fluxesin Correlated Topological Insulators
F. F. Assaad, M. Bercx, and M. Hohenadler
Institut für Theoretische Physik und Astrophysik, Universität
Würzburg, Am Hubland, 97074 Würzburg, Germany(Received 23 April
2012; revised manuscript received 5 October 2012; published 26
February 2013)
The adiabatic insertion of a � flux into a quantum spin Hall
insulator gives rise to localized spin and
charge fluxon states. We demonstrate that � fluxes can be used
in exact quantum Monte Carlo simulations
to identify a correlated Z2 topological insulator using the
example of the Kane-Mele-Hubbard model.
In the presence of repulsive interactions, a � flux gives rise
to a Kramers doublet of spin-fluxon states with
a Curie-law signature in the magnetic susceptibility. Electronic
correlations also provide a bosonic mode
of magnetic excitons with tunable energy that act as exchange
particles and mediate a dynamical
interaction of adjustable range and strength between spin
fluxons. � fluxes can therefore be used to
build models of interacting spins. This idea is applied to a
three-spin ring and to one-dimensional spin
chains. Because of the freedom to create almost arbitrary spin
lattices, correlated topological insulators
with � fluxes represent a novel kind of quantum simulator,
potentially useful for numerical simulations
and experiments.
DOI: 10.1103/PhysRevX.3.011015 Subject Areas: Strongly
Correlated Materials, Topological Insulators
I. INTRODUCTION
A topological insulator represents a novel state of
mattercharacterized by a special band structure that can
result,e.g., from strong spin-orbit interaction [1,2]. In two
dimen-sions, this state is called a quantum spin Hall insulator
andhas deep connections with the quantum Hall effect, includ-ing
the coexistence of a bulk band gap and metallic edgestates, the
absence of symmetry breaking, and the possi-bility of a
mathematical classification [3,4]. Importantly,because of the
absence of a magnetic field, the quantumspin Hall insulator
preserves time-reversal symmetry,which provides protection against
interactions and disorder[5–7]. The quantum spin Hall insulator has
been realized inHgTe quantum wells [8,9].
Correlated topological insulators with strong electron-electron
interactions are the focus of current research [10].Intriguing
concepts include electron fractionalization inthe presence of
time-reversal symmetry, [11–14] spinliquids [14–16], and
topological Mott insulators [17,18].Remarkably, some of the
theoretical models can be studiedusing exact numerical methods. A
central problem in thiscontext is the question of how to detect a
topologicalstate directly from bulk properties, for example, in
caseswhere the bulk-boundary correspondence breaks
down.Experimentally, this issue also arises in the absence ofsharp
edges in proposed cold-atom realizations as a resultof the trapping
potential [19,20]. The classification interms of a Z2 Chern-Simons
index relies on Bloch wave
functions and is therefore only valid for noninteractingsystems.
Generalizations involve twisted boundary condi-tions [21] or Green
functions [22–27] and are challengingto use in experiments or exact
simulations. Indirect signa-tures such as the closing of gaps [16]
or the crossing ofenergy levels [28] require, among other
difficulties, experi-mental tuning of microscopic
parameters.Topological insulators show a unique response to
topological defects such as dislocations [29,30] or �
fluxes[12,30,31]. Upon adiabatic insertion of a � flux,
Faraday’slaw, together with the quantized transverse
conductivity,gives rise to midgap charge and spin-fluxon states
[12,31].These states are exponentially localized around the
flux[12,31]. The existence of these states is ensured, even in
thepresence of interactions or disorder, by time-reversal
sym-metry, and has been suggested as a bulk probe of the Z2index
[12,31]. The concept of fluxons can also be general-ized to
situations where spin is not conserved, such as inthe presence of
Rashba coupling. In three dimensions, amagnetic flux gives rise to
the wormhole effect [32].Electron-electron repulsion lifts the
degeneracy of chargeand spin fluxons, but the two degenerate
spin-fluxonstates constitute a localized spin with Sz ¼ �1=2
[12].Dynamical� fluxes emerge in the context of
fractionalizedtopological insulators [12,13].Previous work on �
fluxes in noninteracting quantum
spin Hall insulators [12,30,31] was based on
square-latticemodels such as that for HgTe quantum wells [8]. Here,
weconsider the half-filled Kane-Mele model on the honey-comb
lattice [3] (historically the first model with a Z2topological
phase), which has close connections to gra-phene [3], the integer
quantum Hall effect [33], and, whenincluding interactions, to
correlated Dirac fermions [15].Topological phases of interacting
fermions on honeycomblattices may be realized in transition metal
oxides [34],
Published by the American Physical Society under the terms ofthe
Creative Commons Attribution 3.0 License. Further distri-bution of
this work must maintain attribution to the author(s) andthe
published article’s title, journal citation, and DOI.
PHYSICAL REVIEW X 3, 011015 (2013)
2160-3308=13=3(1)=011015(12) 011015-1 Published by the American
Physical Society
http://dx.doi.org/10.1103/PhysRevX.3.011015http://creativecommons.org/licenses/by/3.0/
-
semiconductor structures [35], graphene [36], or coldatoms [37]
(see also Ref. [10]).
Here, we use � fluxes in combination with exact quan-tum Monte
Carlo simulations and show that they can beused efficiently to
probe the topological invariant of corre-lated topological
insulators. In particular, this method doesnot rely on an adiabatic
connection to a noninteractingstate, and it may also be used for
fractional states. Inaddition, we demonstrate that� fluxes permit
the construc-tion of quantum spin models of almost arbitrary
geometryand with tunable, dynamical interactions. The spins
corre-spond to the spin fluxons created by inserting � fluxes,
andthe interaction is mediated by magnetic excitons corre-sponding
to collective magnetic fluctuations of the topo-logical insulator.
These spin models can be studiedtheoretically with the quantum
Monte Carlo method, orexperimentally. As examples, we show that a
ring of threespins has a ground state with magnetization 1=2 and
that aone-dimensional chain of fluxons undergoes a Mott tran-sition
and is described at low energies by an XXZ model.
The article is organized as follows. In Sec. II, we in-troduce
the Kane-Mele and Kane-Mele-Hubbard models.Section III provides
details about the methods. The use of� fluxes as a probe for
topological states is discussed inSec. IV, whereas the construction
of quantum spin modelsis the topic of Sec. V. Conclusions are given
in Sec. VI, andwe provide three appendixes.
II. MODEL
The half-filled Kane-Mele model with additionalelectron-electron
interactions can be studied with powerfulquantum Monte Carlo
methods [16,18]. Using the spinor
notation ĉyi ¼ ðĉyi"; ĉyi#Þ, where ĉyi� is a creation
operator foran electron in a Wannier state at site i with spin �,
theHamiltonian reads
HKM ¼ �tXhi;ji
�ijĉyi ĉj þ i�
Xhhi;jii
�ijĉyi ð�ij � �Þĉj
þ i�Xhi;ji
�ijĉyi ðs� d̂ijÞ � ẑĉj: (1)
The notations hi; ji and hhi; jii indicate that the sites i and
jare nearest neighbors and next-nearest neighbors, respec-tively,
and implicitly include the Hermitian conjugateterms.
The first term describes the hopping of electrons be-tween
neighboring lattice sites. The second term representsthe spin-orbit
coupling which reduces the SUð2Þ spinrotation symmetry to a Uð1Þ
symmetry. The third term isan additional Rashba spin-orbit coupling
[38]. The addi-tional factors �ij ¼ �1 take into account any �
fluxespresent, whereas the original Kane-Mele model (without�
fluxes) is recovered from Eq. (1) by setting �ij ¼ 1.
The spin-orbit term corresponds to a next-nearest-neighbor
hopping with a complex amplitude i� and has
been derived from the spin-orbit coupling in graphene [3].This
hopping acquires a sign �1, depending on its direc-tion, the
sublattice, and the electron spin. This sign isencoded in (�ij �
�), where
� ij ¼dik � dkjjdik � dkjj : (2)
dik is the vector connecting sites i and k, and k is
theintermediate lattice site involved in the hopping processfrom i
to j. For a coordinate-independent representation,the vectors d��
are defined in three-dimensional space,
although the z component vanishes. The vector � is de-fined by �
¼ ð�x; �y; �zÞ, with the Pauli matrices ��.The last term in Eq. (1)
is the Rashba spin-orbit inter-
action [3,5]. It is defined in terms of the spin vector
s ¼ �=2 and the unit vector d̂ij, which can be expressedin terms
of the nearest-neighbor vectors �1, �2, �3 [39].The Rashba coupling
breaks the z � �z inversionsymmetry and has to be taken into
account, for example,in the presence of a substrate. Because this
term includesspin-flip terms, spin is no longer conserved. The
Rashbaterm has been included in the results for the
noninteractingmodel (1), but cannot be included in quantum Monte
Carlosimulations of the interacting model (3) due to a minus-sign
problem.The model (1) can be solved exactly [3,5,40]. In the
absence of Rashba coupling, � ¼ 0, the Kane-Mele modeldescribes
a Z2 quantum spin Hall insulator for any � > 0.
This state is characterized by a bulk band gap�sp ¼
3ffiffiffi3
p�,
a spin gap �s ¼ 2�sp, and a quantized spin Hall conduc-tivity
�sxy ¼ e22� . The topological state survives for Rashbainteractions
�< 2
ffiffiffi3
p� (for chemical potential� ¼ 0) and
has protected, helical edge states for geometries with
openboundaries [3,5,40]. We use t as the unit of energy (@ ¼
1),take �=t ¼ 0:2, and consider periodic lattices with L� L0unit
cells.To investigate the effect of electron-electron repulsion,
we consider the paradigmatic Hubbard interaction [41] andarrive
at the Kane-Mele-Hubbard model [42],
HKMH ¼ HKM þHU; HU ¼ 12UXi
ðĉyi ĉi � 1Þ2: (3)
Hamiltonian (3) without Rashba coupling has been
studiedintensely [16,42–48]. In particular, its symmetries
permitthe application of exact quantum Monte Carlo methodswithout a
sign problem [16,43,48].On a lattice with periodic boundaries, �
fluxes can only
be inserted in pairs, as illustrated for the minimal numberof
two fluxes in Fig. 1. The flux pair is connected by abranch cut (or
string), and every hopping process crossingthe cut acquires a phase
ei� ¼ �1, as encoded by �ij inEq. (1). Different choices of the
branch cut for fixed fluxpositions are related by a gauge
transformation.
F. F. ASSAAD, M. BERCX, AND M. HOHENADLER PHYS. REV. X 3, 011015
(2013)
011015-2
-
III. METHOD
We have used the auxiliary-field quantum MonteCarlo method [49],
which was previously applied to theHubbard model on the honeycomb
lattice [15], and theKane-Mele-Hubbard model [16,43,48]. The
central idea ofthis stochastic method is to use a path integral
representationof the interacting model (3). By means of a
Hubbard-Stratonovich transformation, the Hubbard term is
decoupled,leading to a problem of noninteracting fermions in
anexternal, space-dependent and imaginary-time-dependentfield. The
sampling is over different configurations of theseauxiliary fields
in terms of local updates. For a givenconfiguration of fields,
Wick’s theorem can be used tocalculate arbitrary correlation
functions from the single-particle Green function. We refer to a
review [50] andprevious work [15,16,48] for technical details such
as thecalculation of energy gaps.
Here, we have used a projective formulation (withprojection
parameter �t ¼ 40) to obtain ground-stateresults, starting from a
trial wave function (the ground stateof the U ¼ 0 case) [48] and a
finite-temperature formula-tion to calculate thermodynamic
properties. Both variantsrely on a Trotter discretization of
imaginary time (we used�� ¼ �=L ¼ �=L ¼ 0:1), but the associated
systematicerror is smaller than the statistical errors. At half
filling,time-reversal invariance ensures that simulations can
becarried out without a minus-sign problem, even in thepresence of
� fluxes.
IV. USING � FLUXES TO PROBE CORRELATEDTOPOLOGICAL STATES
A. Thermodynamic signature of � fluxes
In the topological phase of the model (1), each � fluxgives rise
to four fluxon states which are exponentiallylocalized (due to the
bulk energy gap �sp) near the corre-
sponding flux-threaded hexagons [12,31] (see Fig. 1). Thestates
correspond to the spin fluxons j "i, j #i (with energyE"#), forming
a Kramers pair related by time reversal, andthe charge fluxons jþi,
j�i (with energies Eþ, E�), relatedby particle-hole symmetry. As we
show in Fig. 2, thefluxon states lie inside the bulk band gap, and
for non-interacting electrons, E"# ¼ Eþ ¼ E�.
The fluxons leave a characteristic signature in the staticspin
and charge susceptibilities,
s ¼ �ðhM̂2zi � hM̂zi2Þ; c ¼ �ðhN̂2i � hN̂i2Þ; (4)
which are defined in terms of the operators of total spin
in the z direction (M̂z ¼P
iĉyi �
zĉi) and of the total
charge (N̂ ¼ Piĉyi ĉi); the inverse temperature is given by� ¼
1kBT . At low temperatures, kBT � �sp, we can restrictthe Hilbert
space to fj "i; j #i; jþi; j�ig. If the spin fluxonsare
independent, and for � ¼ 0, we expect a Curie laws ¼ c ¼ 12kBT per
� flux, and hence s ¼ c ¼ 1kBT fortwo independent � fluxes (see
Appendix A). The prefactorof the Curie law follows from the
quantized spin Hallconductance in the absence of Rashba coupling
[3].Similarly, a Curie law was also predicted for
topologicalexcitations in polyacetylene [51].
FIG. 1. For a lattice with periodic boundaries, � fluxes can
beinserted in pairs. Each flux threads a hexagon (highlighted
inblue) of the honeycomb lattice, and the pair is connected by
abranch cut (blue line). Hopping processes crossing the branchcut
acquire a phase ei� ¼ �1.
1
10
100
1000
0.001 0.01 0.1 1 10
FIG. 3. Spin susceptibility of the Kane-Mele model(�=t ¼ 0:2)
with two � fluxes at the maximal distance on an18� 18 lattice, for
different Rashba couplings �. At tempera-tures kBT & 0:1t, each
� flux contributes
12kBT
to the suscepti-
bility, leading to s � 1kBT . Also shown is the spin-gap
energyscale 0:2�s for T ¼ 0, � ¼ 0. For �> 0, the chemical
potentialis adjusted to retain a half-filled band.
FIG. 2. In a quantum spin Hall insulator, a � flux gives riseto
four states (with charge q and spin Sz) localized near theflux,
which lie inside the bulk energy gap between the valenceand
conduction bands (labeled ‘‘VB’’ and ‘‘CB’’ in the
figure,respectively) [12,31]. The states correspond to a Kramers
dou-blet of spin fluxons j "i, j #i with energy E"# and a doublet
ofcharge fluxons jþi, j�i with energies Eþ, E�.
TOPOLOGICAL INVARIANT AND QUANTUM SPIN MODELS . . . PHYS. REV. X
3, 011015 (2013)
011015-3
-
Figure 3 shows results for s as a function of tempera-ture for
the Kane-Mele model with two � fluxes located atthe largest
possible distance. At temperatures kBT � �s,s is dominated by bulk
effects. For kBT & 0:1t, we ob-serve the expected Curie law.
The latter is robust withrespect to Rashba coupling, which is
crucial for possibleexperimental realizations.
B. Probing correlated topological insulators
Figure 3 establishes the existence and thermodynamicsignature of
degenerate spin and charge fluxons in a quan-tum spin Hall
insulator threaded by a pair of � fluxes. Wenow consider the effect
of electron-electron interactions inthe framework of the
Kane-Mele-Hubbard model (3). For� > 0, the phase diagram of the
latter includes a correlatedquantum spin Hall insulating phase that
is adiabaticallyconnected to that of the Kane-Mele model (i.e., U ¼
0),and a Mott insulating phase with long-range antiferromag-netic
order [42,48]. Figure 4(a) shows the quantum phasetransition
between these two phases as a function of U=t at�=t ¼ 0:2. At the
transition, the spin gap �s—as obtainedfrom finite-size scaling
(see Ref. [48] for details)—closes,corresponding to the
condensation of magnetic excitons[47,48]. The magnetic order is of
the easy-plane type, andthe transition has 3D XY universality
corresponding tothe ordering of local moments [47,48]. For U �
Uc,
time-reversal symmetry is spontaneously broken, and
thesingle-particle gap �sp remains open across the transition
[48] [see Fig. 4(a)].The location of the critical point can be
estimated from
the scaling behavior of the real-space spin-spin
correlationfunction
SxxðrÞ ¼ hSxAðrÞSxAð0Þi (5)at the largest distance r ¼ L=2.
Here, we consider corre-lations between spins on the A sublattice,
but results are thesame for the B sublattice. The limit
limL!1SxxðL=2Þ isidentical to m2, with m being the magnetization
per site.This critical value can be obtained by considering the
3DXY scaling behavior at the transition. Following Ref. [48],
we plot L2�=SxxðL=2Þ as a function of U for differentsystem
sizes by using the critical exponents z ¼ 1, ¼0:6717ð1Þ, and � ¼
0:3486ð1Þ [52]. Figure 4(b) reveals theexpected intersection of
curves at the critical point andgives Uc=t ¼ 5:70ð3Þ.The
well-understood magnetic transition of the model
(3) provides a test case for the use of � fluxes to probea
correlated quantum spin Hall state, as well as to trackthe
interaction-driven transition to a topologically trivialstate. We
solve the interacting model with two � fluxes byusing exact quantum
Monte Carlo simulations. Spin flux-ons can be detected by
calculating the lattice-site-resolved,dynamical spin-structure
factor at T ¼ 0, defined as
Sði; !Þ ¼ �Xn
jhnjĉyi �zĉij0ij2�ðEn � E0 �!Þ: (6)
Here, HKMHjni ¼ Enjni, and j0i denotes the ground state.Sði; !Þ
corresponds to the spectrum of spin excitations atlattice site i. A
real-space map of the spin-fluxon states j "i,j #i is obtained by
integrating Sði; !Þ up to an energy scale�=t ¼ 0:2, well within the
charge gap �c � 2�sp, givingS�ðiÞ ¼
R�0 d!Sði; !Þ. For U=t ¼ 4, corresponding to the
quantum spin Hall phase [see Fig. 4(a)], we see in Fig. 5(a)very
sharply defined spin fluxons localized at the two flux-threaded
hexagons. The value of S�ðiÞ is about 3 orders ofmagnitude smaller
at lattice sites that are further awayfrom a flux so that the spin
fluxons can easily be detectednumerically. In Fig. 5(b), we show
results for the magneticinsulating phase at U=t ¼ 6. As expected
for this topologi-cally trivial state, no well-defined spin fluxons
exist.The dependence of S�ðiÞ on U=t across the magnetic
quantum phase transition is shown in Fig. 5(c). A clearsignal is
found deep in the topological-insulator phase,whereas a strong drop
is observed on approaching thecritical point at Uc=t ¼ 5:70ð3Þ.
Hence, the spin-fluxonsignal can be used in quantum Monte Carlo
simulationsto distinguish topological and nontopological phases.As
for the noninteracting case (Fig. 3), the spin fluxons
created by the fluxes give rise to a characteristic Curie lawin
the spin susceptibility. Figure 6(a) shows quantumMonte Carlo
results for the spin and charge susceptibilities
0.0
0.5
1.0
1.5
2.0
3 4 5 6 7 8
(a)
Topologicalinsulator
AntiferromagneticMott insulator
0.0
0.2
0.4
0.6
0.8
1.0
1.2
5.3 5.4 5.5 5.6 5.7 5.8 5.9 6
(b)
FIG. 4. (a) Spin gap �sðq ¼ 0Þ and single-particle gap�spðq ¼ KÞ
in the thermodynamic limit as a function of theHubbard repulsion U,
at T ¼ 0 (�=t ¼ 0:2, � ¼ 0). (b) Scalingof SxxðL=2Þ using the
critical exponents of the 3D XY model,z ¼ 1, ¼ 0:6717ð1Þ, and � ¼
0:3486ð1Þ [52]. The intersectiongives the critical point Uc=t ¼
5:70ð3Þ. The lattice size is L� L.Error bars are smaller than the
symbols.
F. F. ASSAAD, M. BERCX, AND M. HOHENADLER PHYS. REV. X 3, 011015
(2013)
011015-4
-
defined in Eq. (4) in the topological phase (U=t ¼ 4).We again
consider two fluxes at the maximal separation.At low temperatures,
kBT � �s, s is well described bys ¼ 2kBT , or 1kBT per � flux. The
additional factor of 2compared to the case U ¼ 0 comes from the
splitting ofspin and charge states, which only leaves the
Kramers
doublet fj "i; j #ig at low energies (see Appendix A). TheCurie
law holds down to the lowest temperatures consid-ered in Fig. 6(a).
Finally, the charge susceptibility isstrongly suppressed at low
temperatures and reveals theabsence of low-energy charge fluxons as
a result of theHubbard repulsion.
C. Interaction between spin fluxons
So far, we have exploited the thermodynamic andspectral
signatures of independent spin-fluxon excitations(i.e., free
spins). On periodic lattices, spin fluxons can onlybe created in
pairs, and it is therefore interesting to con-sider their mutual
interactions. Such interactions will playa key role in Sec. V,
where we consider quantum spin Hallinsulators with multiple �
fluxes to create and simulatesystems of interacting
spins.Interaction effects due to a coupling between two spin
fluxons in a lattice with one pair of� fluxes become visiblefor
larger U=t, i.e., closer to the magnetic transition.Figures 6(b)
and 6(c) show a deviation from the Curielaw below a temperature
scale determined by the interac-tion between spin fluxons. In the
model (3), this interactionis mediated by the exchange of
collective spin excitations(magnetic excitons), which are the
lowest-lying excitationsof the correlated topological insulator,
and they evolve intothe gapless Goldstone mode of the magnetic
state. Sincemagnetic order is of the easy-plane type, the
dominantcontribution of the resulting interaction is expected
tohave the general form
Sint ¼ �g2Xr�r0
ZZ �0d�d�0½Sþr ð�ÞDðr� r0; �� �0Þ
� S�r0 ð�0Þ þ H:c:�; (7)where S�r ð�Þ are spin-flip operators
acting on a spin fluxonat position r at time �, Dðr� r0; �� �0Þ is
the Fourier
(a) Topological insulator
(b) Antiferromagnet
0.00
0.01
0.02
0.03
0.04
0.05
3 4 5 6 7 8
(c)Topologicalinsulator
AntiferromagneticMott insulator
FIG. 5. Integrated dynamical spin-structure factor S�ðiÞ atT ¼ 0
on a 9� 9 lattice. (a) Localized spin fluxons created inthe
topological-insulator phase at U=t ¼ 4. (b) Absence ofspin fluxons
in the magnetic phase at U=t ¼ 6. (c) Maximumof S�ðiÞ, as a
function of U=t. Here �=t ¼ 0:2, � ¼ 0, and�=t ¼ 0:2.
FIG. 6. (a) Spin (s) and charge (c) susceptibilities of the
Kane-Mele-Hubbard model (�=t ¼ 0:2, � ¼ 0) at U=t ¼ 4. We
considerL� L lattices with one pair of � fluxes placed at the
maximal distance. At low temperatures, the spin susceptibility
reveals a Curie laws ¼ 2kBT , whereas the charge susceptibility is
suppressed by the charge gap. (b), (c) Spin susceptibility as a
function of temperature fordifferent values of U=t (�=t ¼ 0:2, � ¼
0). (a)–(c) show that with increasing U=t, the range of the
interaction between spin fluxonsincreases, leading to deviations
from the Curie law s ¼ 2kBT at low temperatures. (d) For U >Uc ¼
5:70ð3Þt, s reflects the presenceof long-range magnetic order in
the bulk. Error bars are smaller than the symbol size. The arrows
indicate the energy scale associatedwith the spin gap.
TOPOLOGICAL INVARIANT AND QUANTUM SPIN MODELS . . . PHYS. REV. X
3, 011015 (2013)
011015-5
-
transform of the exciton propagator Dðq; i�mÞ(q: momen-tum; �m ¼
2n�=�: bosonic Matsubara frequency), and gis a coupling constant.
At long wavelengths, the dispersionrelation of the collective spin
mode can be written as
!ðqÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2jq�Qj2
þ�2sp , where v is the spin velocity,�s is the spin gap, and Q is
the magnetic-orderingwave vector. The minimal exciton energy is
given by!ðq ¼ QÞ ¼ �s. Fourier transformation of the propagator(see
Appendix C) gives, in the limit of low energies andlong
wavelengths,
Dðr; �Þ expðiQ � rÞ expð��s�Þ exp��jrj
2�s2v2�
�: (8)
The first term determines the sign of the interaction. Thedecay
at large imaginary time � is governed by the spin gap�s. The fast
decay as a function of jrj underlies the clearCurie law seen, e.g.,
in Fig. 6. The interaction rangeand strength can be tuned via the
spin gap and hence[cf. Fig. 4(a)] by varying U=t.
From Eq. (8), we expect the interaction range to in-crease with
increasing U=t due to the decrease of �s[see Fig. 4(a)]. Indeed,
Figs. 6(b) and 6(c) reveal anenhanced effect of the spin-fluxon
separation at low tem-peratures with increasing U=t. In particular,
for U=t ¼ 5:5(close to the magnetic transition), Fig. 6(c) shows a
Curielaw corresponding to two free spin fluxons, only for
thelargest system sizes (L ¼ 18). As U ! Uc, the interactionrange
diverges, and free spin fluxons can no longer exist.For U >Uc,
time-reversal invariance is broken, and �fluxes do not create spin
fluxons. Instead, the spin suscep-tibility in Fig. 6(d) is that of
an antiferromagnet; the finitevalue of s=L
2 at T ¼ 0 reflects the density of spin-waveexcitations.
To illustrate the dependence of the interaction strengthon �s,
we consider two fluxes at a fixed, small distance, asillustrated in
the inset of Fig. 7. We show the spin suscep-tibility for different
values of U=t in Fig. 7. For U=t ¼ 3, aCurie law s � 2kBT may be
inferred at temperatures kBT �0:1t. Increasing U=t, the interaction
between the spinfluxons becomes too large to observe free spin
fluxonsbelow the temperature range set by the bulk spin gap �s.The
downturn of s occurs at higher and higher tempera-tures with
increasing U=t, and it reflects a tunable,correlation-induced
energy scale for the interactionbetween spin fluxons that is absent
in Fig. 3.
V. �-FLUX QUANTUM SPIN MODELS
The possibility of inserting � fluxes to create localizedspin
fluxons with a tunable interaction mediated by mag-netic excitons
provides a toolbox to engineer interactingspin models in correlated
topological insulators. The com-putational effort for quantum Monte
Carlo simulationsdoes not depend on the number of � fluxes, and the
lattercan be arranged in almost arbitrary geometries on
thehoneycomb lattice.
A. Three-spin system
As a first extension of the two-spin cases considered sofar, we
consider four spin fluxons emerging from two pairsof � fluxes. The
fluxes are arranged so that three spinfluxons form a ring, and the
fourth spin fluxon is locatedat the largest distance from the
center of the ring. For largeenough lattices, the separated spin
fluxon will not couple tothe other three, and the physical problem
is similar toexperiments on coupled quantum dots [53] or flux
qubits[54] in the context of quantum computation. The three
spinfluxons experience a transverse interaction of the form (7)
1
10
100
1000
0.01 0.1 1
FIG. 7. Spin susceptibility as a function of temperature for
two� fluxes arranged as shown in the inset (�=t ¼ 0:2, � ¼ 0, 9�
9lattice). With increasing U=t, the strength of the
interactionbetween spin fluxons increases, as revealed by the shift
of thetemperature below which deviations from a 2kBT
Curie law occur.
Statistical errors are smaller than the symbol size. Inset:
S�ðiÞfor U=t ¼ 4, using the same color coding as in Fig. 5(a).
1
10
100
1000
0.01 0.1 1
FIG. 8. Spin susceptibility as a function of temperature for
four� fluxes arranged as shown in the inset (�=t ¼ 0:2, � ¼ 0,U=t ¼
4) on L� L lattices. The data reveal a Curie law 4kBT
atintermediate temperatures and 2kBT
at low temperatures.
Statistical errors are smaller than the symbol size. Inset:
S�ðiÞfor L ¼ 15, using the same color coding as in Fig. 5(a).
F. F. ASSAAD, M. BERCX, AND M. HOHENADLER PHYS. REV. X 3, 011015
(2013)
011015-6
-
and behave as an effective spin with Sz ¼ �1=2 atlow
temperatures (see Appendix B). The spin susceptibilityfor U=t ¼ 4
shown in Fig. 8 reveals that, at low tempera-tures, the two
independent spins indeed give rise to theexpected Curie law s ¼
2kBT . At higher temperatureskBT � 0:1t, we find s ¼ 4kBT ,
corresponding to four in-dependent spin fluxons. In the regime
where s ¼ 2kBT , thesign of the interaction between the spin
fluxons determinesthe ground-state degeneracy of the three-spin
cluster. A netferromagnetic interaction results in a spin-1=2
doublet,whereas an antiferromagnetic coupling gives rise to a
four-fold degenerate, chiral ground state (see Appendix B).
Inprinciple, the sign of the exchange coupling can be deter-mined
from entropy measurements. Since Q ¼ 0 for themodel (3), Eq. (8)
suggests that the interaction isferromagnetic.
B. Simulation of one-dimensional fluxon chains
Whereas the study of systems with a small number ofspins is
relevant for applications such as quantum comput-ing, many
questions in condensed matter physics are re-lated to periodic spin
lattices. In this section, we thereforeconsider one-dimensional
chains of � fluxes in the honey-comb lattice with periodic boundary
conditions.
We begin with the noninteracting Kane-Mele modelwith a periodic
flux chain. The fluxon excitations arevisible in Fig. 9, which
shows the integrated local densityof states, A�ðiÞ ¼
R�0 d!Aði; !Þ; the single-particle spec-
tral function Aði; !Þ is defined as usual in terms of
thesingle-particle Green function, Aði; !Þ ¼ �ImGði; !Þ.Whereas the
fluxons are well localized in the directionnormal to the chain, the
overlap of neighboring fluxons inthe chain gives rise to a
tight-binding band inside thetopological band gap, which can be
seen in the spectrumshown in Fig. 10. The specific form of the band
structurecan be attributed to the fact that the smallest unit cell
forthe fluxon chain contains two flux-threaded hexagons
(and is four hexagons wide) (see Fig. 9). Exploitingthe fact
that the four possible fluxon states per hexagon,fj "i; j #i; jþi;
j�ig, can formally be written in terms of thefermion Fock states fj
"i; j #i; j0i; j "#ig, and assumingnearest-neighbor hopping, a
suitable Hamiltonian isgiven by
H ¼ �~tXi�
ð�yi�c i� � c yi��iþa�� þ H:c:Þ; (9)
where �, c refer to the two flux-threaded hexagonsin the unit
cell, and i numbers the unit cells. The resultingband dispersion
ðkÞ ¼ �2~t sinð2kaÞ matches the low-energy bands in the spectrum
(Fig. 10). The form of theeffective low-energy Hamiltonian, and
especially thegapless nature of the spectrum, stems from the fact
thatthe unit cell is a gauge choice; a translation by halfa lattice
vector, a�=2, corresponds to a gauge transfor-mation. This symmetry
allows the intercell and intracellhopping integrals to differ only
by a phase ei�. Imposingtime-reversal symmetry pins the phase
factor to � ¼ 0and � ¼ �, thus leading to the dispersion
relations�2~t cos½ðkþ �=a�Þa�=2�. The choice � ¼ � producesthe
above-mentioned dispersion relation, and the choice� ¼ 0 merely
corresponds to translating the reciprocallattice by half a
reciprocal lattice unit vector.In contrast to the helical edge
states of a quantum spin
Hall insulator, each of the two fluxon bands is spin
degen-erate. As a result, and because the system is half filled,
weexpect a Mott transition of charge fluxons for any
nonzeroelectron-electron repulsion. Figure 11 shows the spin
andcharge susceptibilities of the Kane-Mele-Hubbard modelon L� 12
lattices with L=2 � fluxes and U=t ¼ 4. TheHubbard U causes an
exponential suppression of thecharge susceptibility at low
temperatures [see Fig. 11(b)and inset], whereas low-energy
spin-fluxon excitations
FIG. 9. Integrated local density of states A�ðiÞ (� ¼ 0:2t;
seetext) at T ¼ 0 for the Kane-Mele model (�=t ¼ 0:2, � ¼ 0) witha
periodic chain of � fluxes. We show a part of the 72� 12lattice
used and the size of the magnetic unit cell containing two� fluxes.
The latter has width a� ¼ 4a, where a 1 corre-sponds to the norm of
the lattice vectors of the underlyinghoneycomb lattice.
FIG. 10. Spectrum of eigenvalues of the Kane-Mele modelwith a
periodic chain of � fluxes (cf. Fig. 9). Here, �=t ¼ 0:2,� ¼ 0, and
the honeycomb lattice has dimensions 72� 12.Points correspond to
eigenvalues and lines to the band structure
ðkÞ ¼ �2~t sinð2kaÞ, with ~t � 0:126t and a 1.
TOPOLOGICAL INVARIANT AND QUANTUM SPIN MODELS . . . PHYS. REV. X
3, 011015 (2013)
011015-7
-
remain [Fig. 11(a)]. Hence, similar to the
one-dimensionalHubbard model, the fluxon chain undergoes a
Motttransition to a state with a nonzero charge gap but gaplessspin
excitations.
In the Mott phase of the fluxon chain, the low-energyphysics is
expected to be described by spin fluctuationsand hence by an
effective spin model with spins corre-sponding to Kramers doublets
of localized spin fluxons.Because the interaction range depends
exponentially onthe spin gap, we expect nearest-neighbor
interactionsJxy, Jzz between spin fluxons to dominate, except for
theclose vicinity of the magnetic transition. As argued be-fore,
the magnetic exciton is of predominantly easy-planetype, and we
therefore expect anisotropic interactions,jJxyj � jJzzj. The
minimal model for the spin chain isthe one-dimensional XXZ
Hamiltonian,
H ¼ JzzXi
Szi Sziþ1 þ Jxy
Xi
ðSþi S�iþ1 þ S�i Sþiþ1Þ: (10)
Using the ALPS 1.3 implementation [55], we can simulatethis
model in the stochastic-series-expansion representa-tion to
calculate the spin susceptibility as a function oftemperature.
There is one free parameter, Jxy=Jzz, which
is varied to obtain the best fit to the
low-temperaturesusceptibility (at high temperatures, bulk states of
thetopological insulator begin to play a role) of the
Kane-Mele-Hubbard model. For example, considering sixspins, a
rather good match between the spin-fluxon dataand the XXZ model is
obtained for Jzz=jJxyj ¼ �0:1(the sign of Jxy is irrelevant) (see
Fig. 12). Importantly,taking the same parameters, and simulating
ten spins withboth spin fluxons and the XXZ model, equally
goodagreement is found in Fig. 12. These results demonstratethat
the spin fluxons form a one-dimensional spin systemwith
well-defined interactions and that a quantum spinHall insulator
with � fluxes can indeed be used as aquantum simulator.
VI. CONCLUSIONS
In this work, we have presented quantum Monte Carloresults for a
correlated quantum spin Hall insulator withtopological defects in
the form of � fluxes. Such fluxesrepresent a universal probe for
the topological index thatcan be used in the presence of electronic
correlations anddoes not rely on spin conservation or an adiabatic
connec-tion to a noninteracting topological insulator. Our
resultsdemonstrate that � fluxes can be combined with
exactnumerical simulations and lead to clear signatures
ofnontrivial topological properties in spectral and thermody-namic
properties. As a concrete example, we have studiedthe magnetic
quantum phase transition of the Kane-Mele-Hubbard model at which
time-reversal symmetry is spon-taneously broken. In principle, �
fluxes can also be used inconnection with fractional quantum spin
Hall states.More generally, � fluxes in correlated topological
insulators allow one to construct and simulate quantumspin
models and hence lead to a novel class of quantum
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
(a)
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
(b)
0 5 10 15 20 25
(no fluxes)
FIG. 11. (a) Spin and (b) charge susceptibility of
theKane-Mele-Hubbard model (�=t ¼ 0:2, � ¼ 0) at U=t ¼ 4.We
consider L� 12 lattices with L=2 � fluxes arranged in aperiodic
one-dimensional chain. The inset in (b) shows thecharge
susceptibility as a function of inverse temperature on alogarithmic
scale. The key in (a) applies to all panels.
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.02 0.04 0.06 0.08 0.1
XXZ model-flux model-flux model
FIG. 12. Spin susceptibility as a function of
temperature.Symbols correspond to quantum Monte Carlo results for
theKane-Mele-Hubbard model (�=t ¼ 0:2, � ¼ 0, U=t ¼ 4) onL� 12
lattices with L=2 � fluxes arranged in a periodic one-dimensional
chain. Lines are quantum Monte Carlo results forthe one-dimensional
XXZ model with Jzz=jJxyj ¼ �0:1 and L=2lattice sites (spins).
F. F. ASSAAD, M. BERCX, AND M. HOHENADLER PHYS. REV. X 3, 011015
(2013)
011015-8
-
simulators. This finding is not restricted to the
Kane-Mele-Hubbard model considered here. In particular,
magnetismdriven by electronic correlations—the origin of
theinteraction between spin fluxons—is a common phenome-non. The
physics described here relies on the coexistenceof magnetic
correlations and time-reversal symmetry andcannot be captured by
static mean-field descriptions. Thespin models share the
topological protection of their hostagainst, for example, disorder.
In general, they are charac-terized by a dynamical, time-dependent
interaction remi-niscent of spin-boson problems. The detailed form
and signof the interaction, whose strength and range can be
tunedvia the electronic interactions, depend on the
electronicHamiltonian and the lattice geometry of the
underlyingtopological insulator. Because of the spin-orbit
interaction,the spin symmetry is Uð1Þ, and—similar to
cold-atomrealizations of the quantum Ising model [56]—the spin-spin
interaction is generically anisotropic. We have pro-vided explicit
evidence for the feasibility of our idea interms of simulations of
two and four spins, as well as ofone-dimensional spin chains.
Additional Rashba termslead to spin models with a discrete Z2 Ising
symmetry.Although spin-fluxon states are still well defined
[12,31], itis a priori not clear which operators have to be
measured inthe numerical simulations. Finally, the concept of
fluxonsoriginating from � fluxes carries over to
three-dimensionaltopological insulators [12,32].
An open question of central importance is whether theuse of �
fluxes will enable us to study quantum spinsystems that are
currently not accessible to numericalmethods, for example, due to a
sign problem in the pres-ence of frustrated interactions. Whereas
we have providedevidence for the possibility to simulate arrays and
chains ofquantum spins, and to tune the interaction strength
andrange, entropy measurements are required to determine thesign of
the interactions. However, the latter are extremelydemanding to
carry out using discrete-time quantumMonte Carlo methods. A
systematic effort to study spinfluxon chain and ladder geometries
is currently in progress.
Our idea may potentially also be used in experiments.A strongly
correlated topological insulator on the honey-comb lattice may be
realized with Na2IrO3 [34] or withmolecular graphene [57]. It has
been suggested that �fluxes can be created in a quantum spin Hall
insulator bymeans of an adjacent superconductor and a magnetic
field[31]. This idea can be generalized to arrays of � fluxesusing
Abrikosov lattices. Alternatively, � fluxes may berealized using
SQUIDs. A potential problem is that thediameter of the � fluxes
will not be of the order of thelattice constant. Other exciting
recent proposals that arerelevant for the realization of our idea
include artificialsemiconductor honeycomb structures [35], cold
atoms inoptical lattices [19], and cold atoms on chips [58]. In
solid-state setups, � fluxes can also be created by
dislocations[29,30] or wedge disclinations [59].
ACKNOWLEDGMENTS
We thank N. Cooper, T. L. Hughes, L. Molenkamp,J. Moore, J.
Oostinga, X.-L. Qi, C. Xu, and S. C. Zhangfor insightful
conversations, and acknowledge supportfrom DFG Grants No. FOR1162
and No. As 120/4-3, aswell as generous computer time at the
LRZMunich and theJSC. We made use of ALPS 1.3 [55].
APPENDIX A: SPIN SUSCEPTIBILITYFOR TWO � FLUXES
A single � flux in a topological insulator gives rise tofour
states, j "i, j #i, jþi, j�i. In the absence of correla-tions,
these states are degenerate. At low temperatures, thespin
susceptibility, defined in Eq. (4), can be calculatedusing the
Hilbert space formed by only these states.Defining an effective
Hamiltonian H� ¼
PcE
c jc ihc jwith c 2 fþ;�; "; #g and Eþ ¼ E� ¼ E" ¼ E# ¼ E"#,
weobtain
s¼�ðhM̂2zi�hM̂zi2Þ¼ 1kBTP
c hc jM̂2ze��H� jc iPc hc je��H� jc i
¼ 12kBT
:
(A1)
For U � kBT, the spin fluxons j "i, j #i are the only low-energy
excitations, and s can be calculated by restrictingc to f"; #g.
Since E" ¼ E# ¼ E"# due to time-reversal sym-metry, we get
s ¼ 1kBT : (A2)
For the case of two independent � fluxes, the aboveresults imply
s ¼ 1kBT at U ¼ 0 and s ¼ 2kBT for U > 0.These results agree
with the numerical results shown inFig. 3 for U ¼ 0 and in Figs. 6
and 7 for U > 0.Our derivation is only valid in the absence of
the Rashba
spin-orbit coupling �. However, the numerical results inFig. 3
show that the low-temperature Curie law in s is thesame also for �
� 0.
APPENDIX B: SPIN SUSCEPTIBILITYAND GROUND-STATE DEGENERACY
FOR FOUR � FLUXES
The results for the Kane-Mele-Hubbard model with four� fluxes
shown in Fig. 8 reveal a 2kBT Curie law at low
temperatures and a 4kBTCurie law at higher temperatures.
This finding can be understood as corresponding to eithertwo or
four noninteracting spins. The latter case corre-sponds to the
spatially separate spin fluxon and an effectivespin-1=2Kramers
doublet (formed by the three nearby spinfluxons) in the regime
where s � 2kBT and to four non-interacting spin fluxons in the
regime where s � 4kBT .The cluster formed by the three nearby spin
fluxons has
the possible configurations
TOPOLOGICAL INVARIANT AND QUANTUM SPIN MODELS . . . PHYS. REV. X
3, 011015 (2013)
011015-9
-
jMzj ¼ 32: fj """i; j ###ig;
jMzj ¼ 12: fj "##i; j #"#i; j ##"i; j #""i; j "#"i; j ""#ig;
(B1)
whereMz denotes the total spin in the z direction. Since
theexciton-mediated interaction in the Kane-Mele-Hubbardmodel has
the form given in Eq. (7), and hence promotesspin flips, the ground
state can be expected to have jMzj ¼1=2. The above-mentioned
effective spin-1=2 doublet thencorresponds to the two possible
values Mz ¼ �1=2.
The degeneracy of the ground state depends on the sign ofthe
interaction. ConsideringMz¼1=2, we have the allowedstates j #""i, j
"#"i, and j""#i. The spin-flip terms that con-nect these states are
of the form JðSþiþ1S�i þSþi S�iþ1Þ, withperiodic boundary
conditions. An equivalent representa-tion is given by the
Hamiltonian
H ¼ JXj
ðjjþ 1ihjj þ jjihjþ 1jÞ; (B2)
which describes the hopping of a particle (the spin-down)on a
three-site ring, with j1i ¼ j #""i, etc. The eigenstatesare
obtained by Fourier transformation and have the form
jki ¼ 1ffiffiffi3
p X3j¼1
eikjjji; k ¼ 0;� 2�3
: (B3)
The eigenvalues are given by
EðkÞ ¼ 2J cosk: (B4)For J < 0, the ground state has k ¼ 0 and
energy equal to2J. For J > 0, the ground state is chiral, with k
¼ �2�=3and energy equal to �J. Taking into account the sectorMz ¼
�1=2, we find a total ground-state degeneracy oftwo in the
ferromagnetic case (J < 0) and four in theantiferromagnetic case
(J > 0).
APPENDIX C: FOURIER TRANSFORM OF THEEXCITON PROPAGATOR
The exciton propagator in Eq. (7) takes the form
Dðq; �Þ ¼ e�!ðqÞ
e�!ðqÞ � 1�e��!ðqÞ
e��!ðqÞ � 1 ; (C1)
with the exciton energy
!ðqÞ
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2jq�Qj2
þ �2s
q: (C2)
In the low-temperature limit � ! 1, the propagator be-comes Dðq;
�Þ ¼ e��!ðqÞ. Setting q0 ¼ q�Q, the Fouriertransform is given
by
Dðr; �Þ ¼ eiQ�rZ
d2q0eiq0�re�!ðq0þQÞ�: (C3)
Assuming �s � v, we can expand to obtain
!ðq0 þQÞ � �s�1þ v
2jq0j22�2s
�: (C4)
Taking the continuum limit, the Fourier transform involvesthe
product of two Gaussian integrals, and the result isgiven by Eq.
(8).
[1] J. E. Moore, The Birth of Topological Insulators,
Nature(London) 464, 194 (2010).
[2] M. Z. Hasan and C. L. Kane, Colloquium:
TopologicalInsulators, Rev. Mod. Phys. 82, 3045 (2010).
[3] C. L. Kane and E. J. Mele, Z2 Topological Order and
theQuantum Spin Hall Effect, Phys. Rev. Lett. 95, 146802(2005).
[4] Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, andAndreas
W.W. Ludwig, Classification of TopologicalInsulators and
Superconductors in Three SpatialDimensions, Phys. Rev. B 78, 195125
(2008).
[5] C. L. Kane and E. J. Mele, Quantum Spin Hall Effect
inGraphene, Phys. Rev. Lett. 95, 226801 (2005).
[6] Congjun Wu, B. Andrei Bernevig, and Shou-ChengZhang, Helical
Liquid and the Edge of Quantum SpinHall Systems, Phys. Rev. Lett.
96, 106401 (2006).
[7] Cenke Xu and J. E. Moore, Stability of the Quantum SpinHall
Effect: Effects of Interactions, Disorder, andZ2Topology, Phys.
Rev. B 73, 045322 (2006).
[8] B. Andrei Bernevig, Taylor L. Hughes, and Shou-ChengZhang,
Quantum Spin Hall Effect and Topological PhaseTransition in HgTe
Quantum Wells, Science 314, 1757(2006).
[9] Markus König, Steffen Wiedmann, Christoph Brüne,Andreas
Roth, Hartmut Buhmann, Laurens W.Molenkamp, Xiao-Liang Qi, and
Shou-Cheng Zhang,Quantum Spin Hall Insulator State in HgTe
QuantumWells, Science 318, 766 (2007).
[10] M. Hohenadler and F. F. Assaad, Correlation Effects
inTwo-Dimensional Topological Insulators [J. Phys.Condens. Matter
(to be published)].
[11] Chang-Yu Hou, Claudio Chamon, and Christopher
Mudry,Electron Fractionalization in Two-DimensionalGraphenelike
Structures, Phys. Rev. Lett. 98, 186809(2007).
[12] Ying Ran, Ashvin Vishwanath, and Dung-Hai Lee, Spin-Charge
Separated Solitons in a Topological BandInsulator, Phys. Rev. Lett.
101, 086801 (2008).
[13] Andreas Ruegg and Gregory A. Fiete, Topological Orderand
Semions in a Strongly Correlated Quantum Spin HallInsulator, Phys.
Rev. Lett. 108, 046401 (2012).
[14] G. A. Fiete, V. Chua, X. Hu, M. Kargarian, R. Lundgren,A.
Ruegg, J. Wen, and V. Zyuzin, Topological Insulatorsand Quantum
Spin Liquids, Physica (Amsterdam) 44, 845(2012).
[15] Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad, and
A.Muramatsu, Quantum Spin-Liquid Emerging in Two-Dimensional
Correlated Dirac Fermions, Nature(London) 464, 847 (2010).
F. F. ASSAAD, M. BERCX, AND M. HOHENADLER PHYS. REV. X 3, 011015
(2013)
011015-10
http://dx.doi.org/10.1038/nature08916http://dx.doi.org/10.1038/nature08916http://dx.doi.org/10.1103/RevModPhys.82.3045http://dx.doi.org/10.1103/PhysRevLett.95.146802http://dx.doi.org/10.1103/PhysRevLett.95.146802http://dx.doi.org/10.1103/PhysRevB.78.195125http://dx.doi.org/10.1103/PhysRevLett.95.226801http://dx.doi.org/10.1103/PhysRevLett.96.106401http://dx.doi.org/10.1103/PhysRevB.73.045322http://dx.doi.org/10.1126/science.1133734http://dx.doi.org/10.1126/science.1133734http://dx.doi.org/10.1126/science.1148047http://dx.doi.org/10.1103/PhysRevLett.98.186809http://dx.doi.org/10.1103/PhysRevLett.98.186809http://dx.doi.org/10.1103/PhysRevLett.101.086801http://dx.doi.org/10.1103/PhysRevLett.108.046401http://dx.doi.org/10.1016/j.physe.2011.11.011http://dx.doi.org/10.1016/j.physe.2011.11.011http://dx.doi.org/10.1038/nature08942http://dx.doi.org/10.1038/nature08942
-
[16] M. Hohenadler, T. C. Lang, and F. F. Assaad,
CorrelationEffects in Quantum Spin-Hall Insulators: A QuantumMonte
Carlo Study, Phys. Rev. Lett. 106, 100403 (2011).
[17] S. Raghu, Xiao-Liang Qi, C. Honerkamp, and Shou-Cheng
Zhang, Topological Mott Insulators, Phys. Rev.Lett. 100, 156401
(2008).
[18] Dmytro Pesin and Leon Balents, Mott Physics and
BandTopology in Materials with Strong Spin-Orbit Interaction,Nat.
Phys. 6, 376 (2010).
[19] B. Beri and N. R. Cooper, Z2 Topological Insulators
inUltracold Atomic Gases, Phys. Rev. Lett. 107, 145301(2011).
[20] K. Sun, W.V. Liu, A. Hemmerich, and S. Das
Sarma,Topological Semimetal in a Fermionic Optical Lattice,Nat.
Phys. 8, 67 (2011).
[21] Qian Niu, D. J. Thouless, and Yong-Shi Wu, QuantizedHall
Conductance as a Topological Invariant, Phys. Rev.B 31, 3372
(1985).
[22] Zhong Wang, Xiao-Liang Qi, and Shou-Cheng Zhang,Topological
Order Parameters for InteractingTopological Insulators, Phys. Rev.
Lett. 105, 256803(2010).
[23] V. Gurarie, Single-Particle Green’s Functions
andInteracting Topological Insulators, Phys. Rev. B 83,085426
(2011).
[24] Zhong Wang, Xiao-Liang Qi, and Shou-Cheng Zhang,Topological
Invariants for Interacting TopologicalInsulators with Inversion
Symmetry, Phys. Rev. B 85,165126 (2012).
[25] Zhong Wang and Shou-Cheng Zhang, SimplfiedTopological
Invariants for Interacting Insulators, Phys.Rev. X 2, 031008
(2012).
[26] Tsuneya Yoshida, Satoshi Fujimoto, and NorioKawakami,
Correlation Effects on a TopologicalInsulator at Finite
Temperatures, Phys. Rev. B 85,125113 (2012).
[27] Jan Carl Budich, Ronny Thomale, Gang Li, ManuelLaubach, and
Shou-Cheng Zhang, Fluctuation-InducedTopological Quantum Phase
Transitions in QuantumSpin-Hall and Anomalous-Hall Insulators,
Phys. Rev. B86, 201407 (2012).
[28] Christopher N. Varney, Kai Sun, Marcos Rigol, andVictor
Galitski, Topological Phase Transitions forInteracting Finite
Systems, Phys. Rev. B 84, 241105(2011).
[29] Y. Ran, Y. Zang, and A. Vishwanath,
One-DimensionalTopologically Protected Modes in Topological
Insulatorswith Lattice Dislocations, Nat. Phys. 5, 298 (2009).
[30] V. Juricic, A. Mesaros, R.-J. Slager, and J.
Zaanen,Universal Probes of Two-Dimensional TopologicalInsulators:
Dislocation and � Flux, Phys. Rev. Lett.108, 106403 (2012).
[31] Xiao-Liang Qi and Shou-Cheng Zhang, Spin-ChargeSeparation
in the Quantum Spin Hall State, Phys. Rev.Lett. 101, 086802
(2008).
[32] G. Rosenberg, H.-M. Guo, and M. Franz,Wormhole Effectin a
Strong Topological Insulator, Phys. Rev. B 82,041104 (2010).
[33] F. D.M. Haldane, Model for a Quantum Hall Effect with-out
Landau Levels: Condensed-Matter Realization of theParity Anomaly,
Phys. Rev. Lett. 61, 2015 (1988).
[34] A. Shitade, H. Katsura, J. Kunes, X.-L. Qi, S.-C. Zhang,and
N. Nagaosa, Quantum Spin Hall Effect in a TransitionMetal Oxide
Na2IrO3, Phys. Rev. Lett. 102, 256403(2009).
[35] A. Singha, M. Gibertini, B. Karmakar, S. Yuan, M. Polini,G.
Vignale, M. I. Katsnelson, A. Pinczuk, L. N. Pfeier,K.W. West, and
V. Pellegrini, Two-Dimensional Mott-Hubbard Electrons in an
Artificial Honeycomb Lattice,Science 332, 1176 (2011).
[36] Conan Weeks, Jun Hu, Jason Alicea, Marcel Franz, andRuqian
Wu, Engineering a Robust Quantum Spin HallState in Graphene via
Adatom Deposition, Phys. Rev. X 1,021001 (2011).
[37] L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and
T.Esslinger, Creating, Moving and Merging Dirac Pointswith a Fermi
Gas in a Tunable Honeycomb Lattice, Nature(London) 483, 302
(2012).
[38] Y. A. Bychkov and E. I. Rashba Oscillatory Effects and
theMagnetic Susceptibility of Carriers in Inversion Layers,J. Phys.
C 17, 6039 (1984).
[39] A. H. Castro Neto, F. Guinea, N.M.R. Peres, K. S.Novoselov,
and A.K. Geim, The Electronic Propertiesof Graphene, Rev. Mod.
Phys. 81, 109 (2009).
[40] D. N. Sheng, Z. Y. Weng, L. Sheng, and F. D.M.
Haldane,Quantum Spin-Hall Effect and Topologically InvariantChern
Numbers, Phys. Rev. Lett. 97, 036808 (2006).
[41] J. Hubbard, Electron Correlations in Narrow EnergyBands,
Proc. R. Soc. A 276, 238 (1963).
[42] Stephan Rachel and Karyn Le Hur, Topological Insulatorsand
Mott Physics from the Hubbard Interaction, Phys.Rev. B 82, 075106
(2010).
[43] Dong Zheng, Guang-Ming Zhang, and Congjun Wu,Particle-Hole
Symmetry and Interaction Effects in theKane-Mele-Hubbard Model,
Phys. Rev. B 84, 205121(2011).
[44] C. Griset and C. Xu, Phase Diagram of the Kane-Mele-Hubbard
Model, Phys. Rev. B 85, 045123 (2012).
[45] S.-L. Yu, X. C. Xie, and J.-X. Li, Mott Physics
andTopological Phase Transition in Correlated DiracFermions, Phys.
Rev. Lett. 107, 010401 (2011).
[46] Wei Wu, Stephan Rachel, Wu-Ming Liu, and Karyn LeHur,
Quantum Spin Hall Insulators with Interactions andLattice
Anisotropy, Phys. Rev. B 85, 205102 (2012).
[47] Dung-Hai Lee, Effects of Interaction on Quantum SpinHall
Insulators, Phys. Rev. Lett. 107, 166806 (2011).
[48] M. Hohenadler, Z. Y. Meng, T. C. Lang, S. Wessel,
A.Muramatsu, and F. F. Assaad, Quantum Phase Transitionsin the
Kane-Mele-Hubbard Model, Phys. Rev. B 85,115132 (2012).
[49] J. E. Hirsch, D. J. Scalapino, R. L. Sugar, and
R.Blankenbecler, Monte Carlo Simulations of One-Dimensional Fermion
Systems, Phys. Rev. B 26, 5033(1982).
[50] F. F. Assaad and H.G. Evertz, in Computational ManyParticle
Physics, edited by H. Fehske, R. Schneider, andA. Weie, Lecture
Notes in Physics Vol. 739 (SpringerVerlag, Berlin, 2008), p.
277.
[51] W. P. Su, J. R. Schrieer, and A. J. Heeger, Solitons
inPolyacetylene, Phys. Rev. Lett. 42, 1698 (1979).
[52] Massimo Campostrini, Martin Hasenbusch, AndreaPelissetto,
Paolo Rossi, and Ettore Vicari, Critical
TOPOLOGICAL INVARIANT AND QUANTUM SPIN MODELS . . . PHYS. REV. X
3, 011015 (2013)
011015-11
http://dx.doi.org/10.1103/PhysRevLett.106.100403http://dx.doi.org/10.1103/PhysRevLett.100.156401http://dx.doi.org/10.1103/PhysRevLett.100.156401http://dx.doi.org/10.1038/nphys1606http://dx.doi.org/10.1103/PhysRevLett.107.145301http://dx.doi.org/10.1103/PhysRevLett.107.145301http://dx.doi.org/10.1038/nphys2134http://dx.doi.org/10.1103/PhysRevB.31.3372http://dx.doi.org/10.1103/PhysRevB.31.3372http://dx.doi.org/10.1103/PhysRevLett.105.256803http://dx.doi.org/10.1103/PhysRevLett.105.256803http://dx.doi.org/10.1103/PhysRevB.83.085426http://dx.doi.org/10.1103/PhysRevB.83.085426http://dx.doi.org/10.1103/PhysRevB.85.165126http://dx.doi.org/10.1103/PhysRevB.85.165126http://dx.doi.org/10.1103/PhysRevX.2.031008http://dx.doi.org/10.1103/PhysRevX.2.031008http://dx.doi.org/10.1103/PhysRevB.85.125113http://dx.doi.org/10.1103/PhysRevB.85.125113http://dx.doi.org/10.1103/PhysRevB.86.201407http://dx.doi.org/10.1103/PhysRevB.86.201407http://dx.doi.org/10.1103/PhysRevB.84.241105http://dx.doi.org/10.1103/PhysRevB.84.241105http://dx.doi.org/10.1038/nphys1220http://dx.doi.org/10.1103/PhysRevLett.108.106403http://dx.doi.org/10.1103/PhysRevLett.108.106403http://dx.doi.org/10.1103/PhysRevLett.101.086802http://dx.doi.org/10.1103/PhysRevLett.101.086802http://dx.doi.org/10.1103/PhysRevB.82.041104http://dx.doi.org/10.1103/PhysRevB.82.041104http://dx.doi.org/10.1103/PhysRevLett.61.2015http://dx.doi.org/10.1103/PhysRevLett.102.256403http://dx.doi.org/10.1103/PhysRevLett.102.256403http://dx.doi.org/10.1126/science.1204333http://dx.doi.org/10.1103/PhysRevX.1.021001http://dx.doi.org/10.1103/PhysRevX.1.021001http://dx.doi.org/10.1038/nature10871http://dx.doi.org/10.1038/nature10871http://dx.doi.org/10.1088/0022-3719/17/33/015http://dx.doi.org/10.1103/RevModPhys.81.109http://dx.doi.org/10.1103/PhysRevLett.97.036808http://dx.doi.org/10.1098/rspa.1963.0204http://dx.doi.org/10.1103/PhysRevB.82.075106http://dx.doi.org/10.1103/PhysRevB.82.075106http://dx.doi.org/10.1103/PhysRevB.84.205121http://dx.doi.org/10.1103/PhysRevB.84.205121http://dx.doi.org/10.1103/PhysRevB.85.045123http://dx.doi.org/10.1103/PhysRevLett.107.010401http://dx.doi.org/10.1103/PhysRevB.85.205102http://dx.doi.org/10.1103/PhysRevLett.107.166806http://dx.doi.org/10.1103/PhysRevB.85.115132http://dx.doi.org/10.1103/PhysRevB.85.115132http://dx.doi.org/10.1103/PhysRevB.26.5033http://dx.doi.org/10.1103/PhysRevB.26.5033http://dx.doi.org/10.1103/PhysRevLett.42.1698
-
Behavior of the Three-Dimensional XY Universality Class,Phys.
Rev. B 63, 214503 (2001).
[53] L. Gaudreau, G. Granger, A. Kam, G. C. Aers, S.
A.Studenikin, P. Zawadzki, M. Pioro-Ladriere, Z. R.Wasilewski, and
A. S. Sachrajda, Coherent Control ofThree-Spin States in a Triple
Quantum Dot, Nat. Phys.8, 54 (2011).
[54] A. Izmalkov, M. Grajcar, S. H.W. van der Ploeg, U.Hubner,
E. Il’ichev, H.-G. Meyer, and A.M. Zagoskin,Measurement of the
Ground-State Flux Diagram of ThreeCoupled Qubits as a First Step
Towards theDemonstration of Adiabatic Quantum Computation,Europhys.
Lett. 76, 533 (2006).
[55] A. F. Albuquerque, F. Alet, P. Corboz, P. Dayal, A.Feiguin,
S. Fuchs, L. Gamper, E. Gull, S. Gurtler, A.Honecker, R. Igarashi,
M. Korner, A. Kozhevnikov,A. Lauchli, S. R. Manmana, M. Matsumoto,
I. P.McCullochc, F. Michel, R.M. Noack, G. Pawlowski,L. Pollet, T.
Pruschke, U. Schollwock, S. Todo, S.
Trebst, M. Troyer, P. Werner, and S. Wessel for theALPS
Collaboration, The ALPS Project Release 1.3:Open-Source Software
for Strongly Correlated Systems,J. Magn. Magn. Mater. 310, 1187
(2007).
[56] M. J. Bhaseen, A.O. Silver, M. Hohenadler, and B.D.Simons,
Feshbach Resonance in Optical Lattices and theQuantum Ising Model,
Phys. Rev. Lett. 103, 265302(2009).
[57] K. K. Gomes, W. Mar, W. Ko, F. Guinea, and H. C.Manoharan,
Designer Dirac Fermions and TopologicalPhases in Molecular
Graphene, Nature (London) 483,306 (2012).
[58] N. Goldman, I. Satija, P. Nikolic, A. Bermudez,
M.A.Martin-Delgado, M. Lewenstein, and I. B. Spielman,Realistic
Time-Reversal Invariant Topological Insulatorswith Neutral Atoms,
Phys. Rev. Lett. 105, 255302 (2010).
[59] A. Rüegg and C. Lin, Bound States of ConicalSingularities
in Graphene-Based Topological Insulators,Phys. Rev. Lett. 110,
046401 (2013).
F. F. ASSAAD, M. BERCX, AND M. HOHENADLER PHYS. REV. X 3, 011015
(2013)
011015-12
http://dx.doi.org/10.1103/PhysRevB.63.214503http://dx.doi.org/10.1038/nphys2149http://dx.doi.org/10.1038/nphys2149http://dx.doi.org/10.1209/epl/i2006-10291-5http://dx.doi.org/10.1016/j.jmmm.2006.10.304http://dx.doi.org/10.1103/PhysRevLett.103.265302http://dx.doi.org/10.1103/PhysRevLett.103.265302http://dx.doi.org/10.1038/nature10941http://dx.doi.org/10.1038/nature10941http://dx.doi.org/10.1103/PhysRevLett.105.255302http://dx.doi.org/10.1103/PhysRevLett.110.046401