Table Of ContentChiral spin liquid from magnetic Wannier states
I. Panfilov,1 A. Patri,1 Kun Yang,2 and A.A. Burkov1,3
1Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
2Department of Physics and National High Magnetic Field Laboratory,
Florida State University, Tallahassee, Florida 32310, USA
3ITMO University, Saint Petersburg 197101, Russia
(Dated: March 28, 2016)
Wepresentamappingofatwo-dimensionalsystemofinteractingbosonsinastrongperpendicular
magnetic field to an equivalent system of interacting bosons on the square lattice in the absence of
the field. The mapping utilizes a magnetic Bloch and the corresponding magnetic Wannier single-
6 particlebasisinthelowestLandaulevel. Byconstruction,thegroundstatesoftheresultingmodel
1 of interacting bosons on the square lattice are gapped fractionalized liquids or gapless Bose metal
0 states with broken time reversal symmetry at specific rational filling fractions.
2
r
a I. INTRODUCTION absenceofanexternalmagneticfield(suchahypothetical
M material may be called a fractional Chern insulator).
Followingtheremarkablediscoveryoftopologicalinsu- Therehasbynowbeenasignificantamountofworkon
4
lators(TI),1,2 electronicstructure topologyhasbeenun- fractional Chern insulators, see Refs. 8–29 for an incom-
2
derstoodtobeapreviouslylargelyoverlooked,butessen- plete list. The purpose of this article is to derive, some-
] tialingredientinourunderstandingofthephasesofcon- what in the spirit of the Kalmeyer and Laughlin paper,4
el densed matter.3 One of the reasons nontrivial electronic a mapping between a model of interacting bosons in the
- structure topology is of significant importance is that it lowestLandaulevel(LLL),andalatticemodelofbosons
r
t is a purely quantum-mechanical phenomenon, with no in the absence of an external magnetic field (but with
s
classicalanalogs,yet,inmanycases,hasobservableman- broken time reversal symmetry), with the lattice filling
.
t ifestations on macroscopic scales. This makes such phe- identical to the LLL filling factor. By construction, the
a
m nomena not only interesting from the purely scientific ground states of this lattice model are equivalent to the
viewpoint, but also potentially useful technologically. ground states of interacting bosons in the LLL, i.e. may
-
d Quantum mechanical nature of the electrons in solids be fractionalized liquids or Bose metals30,31 with broken
n may also manifest on macroscopic scales through the timereversalsymmetryatspecificrationalfillingfactors.
o
electron-electron interactions, well-known examples be- While most of the calculations, presented below, may be
c
[ ingthe phenomenaofmagnetism andsuperconductivity. carried out for a model of interacting electrons in the
Perhaps particularly remarkable is the fractional quan- LLLjustaswell,wechooseinteractingbosons,havingin
3 tum Hall effect (FQHE), where electrons effectively frac- mindpotentialrealizationsinmagneticsystems,6 orcold
v tionalizeandthelow-energyquasiparticlesarecharacter- atoms in optical lattices.32 Some work along these lines,
8
6 ized by fractional quantum numbers and non-fermionic but valid only at high boson filling factors, has already
9 statistics. This amazing behavior is made possible by beendonebyoneofus.33 Inthispaperwepresentamore
0 the interplay of the strong electron-electron interactions complete analysis, containing points related to the LLL
0 (kinetic energy being completely quenched by the mag- topology, overlooked in Ref. 33, but crucially important
1. netic field), and the nontrivial topology of the individual atlowbosonfillingfactors,atwhichfractionalizedliquid
0 Landau levels. states are realized.
6
Animportantquestioniswhethersuchphenomenaare The rest of the paper is organized as follows. In sec-
1
unique to the system of two-dimensional electrons in a tion II we introduce our model of interacting bosons in
:
v strong perpendicular magnetic field, or they are more two dimensions (2D) in the presence of a strong perpen-
i
X general and may be found in other systems where both dicularmagneticfield,suchthattheLLLprojectionmay
interactions and nontrivial electronic structure topology be used. We introduce a magnetic Bloch and magnetic
r
a are present. This question was first raised in the semi- Wannier single particle basis in the LLL, following the
nal paper of Kalmeyer and Laughlin,4 who pointed out procedure, first described by Rashba et al.34 The advan-
strong similarities between the physics of FQHE and the tage of this particular realization of the magnetic Wan-
resonating valence bond theory5 of spin-liquid states in nier states is that they have the fastest possible decay
Mott insulators.6 The interest in this issue was reinvig- rate for such states, 1/r2, in all directions, and have the
orated recently, after the discovery of TI, which demon- full symmetry of the Bravais lattice.35 In section III we
strated that nontrivial electronic structure topology is derive a representation of the density operator in the
quite common among heavy-element compounds with magnetic Bloch basis and point out some of its most
strongspin-orbitinteractions.7 Thisgivesonesomehope important properties. Focusing on long-wavelength den-
that analogs of FQHE may be found in crystalline mate- sity modes, we perform a gradient expansion of the den-
rials with nontrivial electronic structure topology in the sity operator and derive a simplified expression, valid in
2
√
the long-wavelength limit. In section IV, using the re- ofthesquarelatticewiththelatticeconstanta= 2π(cid:96)2,
sults obtained in the previous sections, we rewrite the and m = (m ,m ) is a vector with integer components,
x y
HamiltonianofinteractingbosonsintheLLLinthemag- labeling the lattice sites.
netic Wannier basis, using the long-wavelength expres- The set of functions c (r) is overcomplete by exactly
m
sions for the density operators. We show that in this one state, which is a consequence of the Perelomov iden-
long-wavelength limit the Hamiltonian has a very simple tity36
form,consistingonlyofafewdistinct(long-range)terms.
(cid:88)
This provides a mapping between the Hamiltonian of in- (−1)mx+mycm(r)=0. (4)
teracting bosons in the LLL and a lattice boson Hamil- m
tonian, withnoexplicitlypresentexternalmagneticfield
This property plays an important role in what follows.
(brokentimereversalsymmetryisstillexplicit, however,
ThemagneticBlochstatesmaynowbeconstructedas
sincesomeofthetermsintheHamiltonianarecomplex).
linear combinations of the LLL orbitals c (r) as
m
WeconcludeinsectionVwithadiscussionofourresults
and a brief summary. Ψk(r)= (cid:112) 1 (cid:88)cm(r)eik·rm
Nν(k)
m
(cid:115)
II. MAGNETIC BLOCH AND MAGNETIC = 2 e−2aπ2r2e−a2π2[ky+aπ2(x−iy)]2
WANNIER BASES IN THE LLL a2Nν(k)
(cid:18) (cid:19) (cid:18) (cid:19)
k a k a iπ
We start from a model of bosons of charge −e, inter- ×θ + ,e−π θ − − (x−iy),e−π . (5)
3 2 3 2 a
actingvia sometwo-body interactionpotentialin 2D(to
be specified in more detail below), in the presence of a where N = L L /2π(cid:96)2 is the number of the magnetic
x y
perpendicular magnetic field B = Bzˆ. We will assume flux quanta, contained in the sample area L L , θ (z,q)
x y 3
that the magnetic field is sufficiently strong, such that areJacobithetafunctionsand ν(k)isneeded tonormal-
only states in the LLL are important. ize the Bloch function to unity in the sample volume.
WewillusetheLLLbasisofmagneticWannierstates, Explicitly, the normalization factor is given by
firstintroducedinRef.34. Theadvantageofthisparticu-
larrealizationofthemagneticWannierstatesisthatthey ν(k)=(cid:88)(−1)mxmye−4r2m(cid:96)2eik·rm
have the fastest decay rate at long distances, compatible
m
walistohatrheehniognhtlyrivsiyaml LmLeLtritcopaonldogayl,lowwh,icahs iwsi1ll/br2e.3d5eTmhoeny- =√2e−ky22πa2θ3(cid:18)k+2a,e−π(cid:19)θ3(cid:18)k−2a,e−π(cid:19). (6)
strated below, for the construction of Wannier Hamilto-
nians with the full symmetry of any 2D Bravais lattice. The probability density, corresponding to a magnetic
Toconstructthisbasis,weadoptthesymmetricgauge Blochstate,|Ψ (r)|2,hastheformofasquareAbrikosov
k
A= 1B×r,andstartfromthezero-angular-momentum vortex lattice, as shown in Fig. 1.
2
symmetric gauge orbital in the LLL The function ν(k) is non-negative everywhere in the
first Brillouin zone (BZ). As immediately follows from
c0(r)= √ 1 e−4r(cid:96)22, (1) the Perelomov identity, Eq. (4), ν(k) vanishes at the BZ
2π(cid:96)2 corner k = (π/a,π/a), ν(k ) = 0. Near k , ν(k) be-
0 0 0
haves as
(cid:112)
where(cid:96)= c/eB isthemagneticlengthandwillusethe γ
¯h=1 units throughout. We then construct an overcom- ν(k0+k)≈ 2k2a2, (7)
plete basis of the LLL orbitals by translating the c (r)
0
orbital, localized at the origin, to sites of any 2D Bra- where
vaislatticewithunitcellarea2π(cid:96)2. Wewillfocusonthe 1 (cid:88)
simplest case of the square lattice henceforth, as the lat- γ =−2a (−1)mx+mycm(0)r2m, (8)
est geometry is unimportant here, see discussion of this m
point below. We obtain is a positive constant of order unity. We will use the
above results extensively later.
c (r)=T T c (r)
m mxax myay 0 The quantum geometry of the magnetic Bloch states,
= (−√1)mxmye−(r−4r(cid:96)m2)2+2(cid:96)i2zˆ·(r×rm). (2) defined by Eq. (5), turns out to be closely connected to
2π(cid:96)2 the properties of the function ν(k). Defining a “periodic
part” of the Bloch function in the standard way as (it is
Here
periodic, but with respect to the magnetic translations,
TR =e−iR·(p−ecA), (3) nthoetBoredrrinyacroynonneecst)iounkA(r()k=)=e−−iki(cid:104)·ruΨk|∇(r),|uan(cid:105)d, weevaolubatatiinng
k k k
is the magnetic translation operator in the symmetric 1
gauge,ax,y =axˆ,ayˆaretheprimitivetranslationvectors A(k)= 2(zˆ×∇k)ln[ν(k)]. (9)
3
which is equivalent to Eq. (12).
The magnetic Wannier states are related to the Bloch
states in the standard way
1 (cid:88)
Φ (r)= √ Ψ (r)e−ik·rm. (14)
m k
N
k
Itisstraightforwardtoshow34 thatthedivergenceofthe
normalizationfactoroftheBlochwavefunctionattheBZ
corner
1 1
∼ , (15)
(cid:112)
ν(k +k) k
0
leads to power-law 1/r2 tail in the long-distance decay
of the Wannier states Φ (r). Nonetheless, the functions
m
Φ (r) form a complete orthonormal set of states, since
m
the Bloch function Ψ (r) is still well-defined, the sin-
k0
FIG.1. (Coloronline). Plotof|Ψ (r)|2 fork=0. Ingeneral gularity, in the form of a momentum-space vortex, only
k
|Ψ (r)|2 has the form of a square Abrikosov vortex lattice, existing in its phase
k
shiftedwithrespecttothelattice,showninthefigure,bythe
vector (cid:96)2zˆ×k. ieiφk (cid:88)
Ψ (r)= √ (−1)mx+my(m −im )c (r),
k0+k 2Nγ x y m
m
(16)
This may be particularly easily evaluated near the BZ
where k→0 and φ is the azimuthal angle of the vector
corners. In this case, using Eq. (7), we obtain k
k. This phase singularity is again a consequence of the
zˆ×k Dirac string, as in Eq. (10). Since the functions Φm(r)
A(k0+k)≈ k2 . (10) form a complete orthonormal set of states, the question
of the LLL Hamiltonian in the magnetic Wannier basis
This expression is singular when k→0, which expresses is also well-defined.
the impossibility of choosing a smooth gauge for the
Bloch functions in the LLL, due to the nonzero Chern
number, to be calculated below. The BZ corner is where III. DENSITY OPERATOR IN THE MAGNETIC
a “Dirac string” must enter the first BZ, to ensure that BLOCH AND WANNIER BASES
the circulation of the Berry connection around the BZ
boundary is equal to 2π. Evaluating the z-component of In this section we will construct the density opera-
the Berry curvature, we obtain tor in the Bloch and Wannier bases, introduced in the
previous section. As is well-known, the peculiar algebra
Ω (k)=∇ ×A(k)= 1∇2ln[ν(k)]=−a2, (11) (Girvin-MacDonald-Platzman,orGMPalgebra)37 ofthe
z k 2 k 2π LLL-projecteddensityoperatorplaysacrucialroleinthe
appearance of the fractional quantum Hall liquid states
which follows immediately from Eq. (6). The integral
in the LLL. It is thus important to understand how this
of Ω (k) over the BZ then gives the nontrivial Chern
z algebra is realized when the density operator is written
number of the LLL, as it should
in the magnetic Bloch and Wannier bases.
1 (cid:90) π/a EvaluatingtheFouriertransformoftheLLL-projected
C = dk dk Ω (k)=−1. (12) density operator, one obtains
2π x y z
−π/a
(cid:90)
(cid:37)(q)= d2rΨ†(r)Ψ(r)e−iq·r
This result may also be obtained using the expression
EIafrqow.ue(n1de0v)tahfleourafittrehseBthBZeebrcroiyrucnucdolaantrniyoe,ncittiooifnstcnhleeeaarrBttehhrraeyt,BcdZounecnoterocntetirohsne. =e−q24πa2 (cid:88)ν(cid:0)(cid:112)kν+(kq2+−q2i)zνˆ(×k)q(cid:1)b†kbk+q. (17)
k
periodicity of the function ν(k) in the first BZ, only the
This expression may be simplified further either using
singularpointsattheBZcornerswillactuallycontribute
Jacobi theta function identities or invoking properties of
to the circulation, see Fig. 2. Using the Eq. (10), one
theBlochfunctionsΨ (r). Wewilltakethesecondroute
obtains k
as it is more transparent.
(cid:73)
ThepropertyoftheBlochfunctionswewilluseisthat
A(k)·dk=−2π, (13)
they are fully determined, up to a k-dependent phase
∂BZ
4
factor, by their zeroes, which form a square Abrikosov
vortex lattice with the lattice constant a, as shown in
Fig.1. Mathematically,thisstatementmaybeexpressed
in the form of the following relation33
Ψk+q(r)=eiγ(k+q,k)e2iq·rΨk(r−(cid:96)2zˆ×q), (18)
where eiq·r/2 is an Aharonov-Bohm phase factor and
(cid:90)
eiγ(k+q,k) = d2rΨk+q(r)Ψ∗k(r−(cid:96)2zˆ×q)e−2iq·r
=e−q28πa2 ν(cid:0)(cid:112)k+ q2 − 2izˆ×q(cid:1). (19)
ν(k+q)ν(k)
This immediately gives
(cid:18) (cid:19) FIG. 2. (Color online). First Brillouin zone with corner
q i
γ(k+q,k)=Imlnν k+ − zˆ×q , (20) patchesshownbyshadedcirclesofradius1/ξ. Circulationof
2 2
theBerryconnectionvectoraroundtheBZboundary,exclud-
ingthecorners,asshownbyarrows,givestheChernnumber
or, equivalently
C =−1.
ν(cid:0)(cid:112)kν+(kq2+−q2i)zνˆ(×k)q(cid:1) =eq28πa2eiImlnν(k+q2−2izˆ×q). (21) Taking qa (cid:28) 1, the Berry phase given by Eq. (20) is
an analytic function of q everywhere, except in a circu-
The physical meaning of the phase γ(k + q,k) is the lar patch of radius 1/ξ (cid:28)1/a around the BZ corner, see
momentum-space Berry phase, accumulated upon adia- Fig. 2. This is again a consequence of the Dirac string,
baticevolutionoftheBlochstatefromktok+q(strictly enteringtheBZatthecorner. Letusanalyzethebehav-
speaking, a path in the first BZ needs to be specified for ior of γ(k+q,k) near the BZ corner k = k in detail.
0
this identification to be precise, but this will not be nec- Using Eq. (7), one obtains
essary for our purposes).
(cid:18) (cid:19)
q i (zˆ×k)·q
Thus we finally obtain the following expression for the Imlnν k +k+ − zˆ×q ≈atan .
density operator 0 2 2 k2+k·q
(24)
(cid:37)(q)=e−q28πa2 (cid:88)eiγ(k+q,k)b†b ≡e−q28πa2(cid:37)¯(q). (22) The meaning of Eq. (24) is simply the azimuthal angle
k k+q between the directions of the vector k+q and vector k.
k
This may be viewed as a direct consequence of Eq. (16).
Using Eqs. (11) and (18) it is straightforward to show Outside of the BZ corner patch, where q <k, this gives
that the density operators (cid:37)¯(q) satisfy the GMP algebra (cid:18) (cid:19)
q i
Imlnν k +k+ − zˆ×q
(cid:20)a2 (cid:21) 0 2 2
[(cid:37)¯(q),(cid:37)¯(q(cid:48))]=−2isin zˆ·(q×q(cid:48)) (cid:37)¯(q+q(cid:48)), (23) zˆ×k
4π ≈ ·q≈A(k +k)·q, (25)
k2 0
as they should.
The above discussion makes it clear that, in general,
To make further progress we will assume that qa may
(cid:37)¯(q)isanonanalyticfunctionofqinthevicinityofq=0
be taken to be small, i.e. only the long-wavelength den-
and thus may not be expanded in Taylor series with re-
sity modes are of interest to us. This might, perhaps, be
spect to q. However, as will be seen below, the nonana-
justifiedusingrenormalization-group-typearguments,al-
lyticityappearsexplicitlyonlywhenonegoesbeyondthe
though it is not easy in the present case, as we are inter-
first order in q, or, in other words, ∇(cid:37)¯(q → 0) is finite,
ested in gapped fractionalized liquid phases with short
even though all the higher gradients are not. Thus, the
correlation length. We will thus take a more simple-
gradient expansion of (cid:37)¯(q) does exist, if it is restricted
minded approach here and assume the interparticle in-
to terms of up to first order in qa, or a/ξ. Expanding to
teraction potential has a long, but finite, range ξ (cid:29) a.
only this order, we thus obtain
Contribution of the density modes with q > 1/ξ is then
(cid:88) (cid:88)
suppressednaturally, withoutrenormalization. Thisalso (cid:37)¯(q)≈ eiA(k)·qb†b ≈ [1+iA(k)·q]b†b .
k k+q k k+q
gives us a natural small parameter a/ξ, which we will
k k
usetocontrolourtheory. Extensiveearlierstudiesofthe (26)
FQHE in finite-width quantum well systems38 indicate This is an expression for the LLL-projected density op-
that ξ may be safely taken to be as long as 10 magnetic erator to leading nontrivial order in the small parameter
lengths. a/ξ.
5
IV. LLL HAMILTONIAN IN THE MAGNETIC and
WANNIER BASIS H =iaU1 (cid:88) zˆ· aξ(rm−rm(cid:48))×(rm(cid:48) −rn)
1 ξ |r −r |2|r −r |2
m(cid:48) n m m(cid:48)
WenowrewriteEq.(26)inthemagneticWannierbasis mm(cid:48)n
using ×J2(|rm(cid:48) −rn|/ξ)b†mbm(cid:48)b†nbn, (33)
where J are Bessel functions of the corresponding or-
1 (cid:88) 1,2
b† = √ b† eik·rm. (27) der, U = U/4π, and U = U/4π2. Eqs. (32) and (33)
k N m 0 1
m constitute the main result of our paper.
It may be useful, especially for possible future numer-
One obtains
ical studies of this model, to extend it by introducing an
(cid:88)
(cid:37)¯(q)= e−iq·rmb† b ordinary kinetic energy term (hopping) for the bosons.
m m
Physically this may be achieved by adding an external
m
i (cid:88) (cid:88) potential,withexactlythesameperiodicityasthesquare
+ N A(k)·qeik·(rm−rm(cid:48))e−iq·rm(cid:48)b†mbm(cid:48).(28) lattice, formed by the magnetic Wannier state centers.
mm(cid:48) k This potential would introduce a boson hopping term of
EventhoughtheBerryconnectionA(k)issingularinthe the form
liinmteigtrkal→ovker0kduientEoqt.h(e2p8)ressteinllcceoonfvtehrgeeDsi(rbauctstdriivnegr,gtehnet H2 =−t (cid:88) eik·(rm−rm(cid:48))b†mbm(cid:48). (34)
(cid:104)mm(cid:48)(cid:105)
terms will appear if expansion to higher orders in q is
attempted). However,duetothedivergenceoftheBerry Here hopping is assumed to be restricted to the nearest-
connection, the main contribution to the integral over k neighbor pairs of sites, t > 0 and the phase factor
at long distances, i.e. when |r −r |(cid:29)a, comes from eik·(rm−rm(cid:48)) depends on the location of the Wannier or-
m m(cid:48)
the vicinity of the BZ corner. This also makes sense bitalcenterwithintheunitcellofthephysicalsquarelat-
physically since, as discussed at the end of Section II, tice. The ground states of H0 correspond to the bosons
it is the contribution of vicinity of the BZ corner that condensing into one of the Bloch states Ψk(r) and form-
leads to the 1/r2 tail of the Wannier function Φ (r). In ing an Abrikosov vortex lattice state. The value of k de-
m
this case A(k) may be approximated by Eq. (10) and terminesthelocationofthevortexcoresoftheAbrikosov
the integral over k in Eq. (28) is then easily evaluated lattice relative to the Wannier orbital centers.
analytically. We obtain ThefullHamiltonianH =H0+H1+H2 willthencon-
tainbothanordinarysuperfluidphase,whentheH term
2
(cid:37)¯(q)=(cid:88)e−iq·rmb† b − a2 (cid:88) q· zˆ×(rm−rm(cid:48)) is dominant, and the fractionalized chiral liquid phases
m m 2π (r −r )2 when H + H is dominant. This may be generalized
m mm(cid:48) m m(cid:48) 0 1
even further by allowing the coupling constants U and
×eik0·(rm−rm(cid:48))e−iq·rm(cid:48)b†mbm(cid:48). (29) U1 tobeindependent, whichwillalsointroduceord0inary
Mott insulator phases with broken translational symme-
The oscillating phase factor eik0·(rm−rm(cid:48)) may be elim-
try when U (cid:29)t,U .
inated by a gauge transformation of the boson creation 0 1
and annihilation operators
b e−ik0·rm →b , (30) V. DISCUSSION AND CONCLUSIONS
m m
and we will ignore this factor henceforth. In the previous sections we have derived, using a con-
The interacting-boson Hamiltonian, projected to the trolledexpansioninthesmallparametera/ξ,amagnetic
LLL, is given by Wannierstaterepresentationofinteractingbosonsinthe
LLL. This Hamiltonian describes interacting bosons on
H = 1 (cid:88)U(q)e−q42πa2(cid:37)¯(q)(cid:37)¯(−q), (31) a square lattice (the lattice geometry does not play a
2LxLy role here, as discussed below). The magnetic field does
q
not enter explicitly in this Hamiltonian, but time rever-
where U(q) is the Fourier transform of the interparticle sal symmetry is still explicitly broken in the H part of
1
interaction potential. In accordance with the discussion the Hamiltonian. The form of the H term is a direct
1
above we take U(q) to be negligible when q > 1/ξ and consequence of Eq. (26) and in this sense it may be re-
equal to a constant U(q) = Uξ2 when q < 1/ξ, where garded as a realization of the GMP algebra, satisfied by
U has dimensions of energy. In this case the integral the LLL-projected density operators.
over q in Eq. (31) is easily done analytically. Restricting Byconstruction,thegroundstatesofthelatticeHamil-
ourselves to only the terms of zeroth and first order in tonian must be the same as the ground states of inter-
thesmallparametera/ξ,weobtainH =H0+H1,where actingbosonsintheLLL,whichincludegappedfraction-
alized quantum Hall liquids at some rational filling frac-
H =U (cid:88)J1(|rm−rn|/ξ)b† b b†b , (32) tions, e.g. at filling factor 1/2. In this sense, the Hamil-
0 0 |rm−rn|/ξ m m n n toniangivenbyEqs.(32),(33),(34)mayberegardedasa
mn
6
parent Hamiltonian of chiral spin liquids (although time strange since the lattice geometry naturally plays a cru-
reversal is broken explicitly here). Chiral spin liquids cial role within the prevailing paradigm in the search for
have attracted considerable attention,4,39–43 particularly quantum spin liquid physics, that of geometrically frus-
duetorecentworkdemonstratingtheymayberealizedin tratedmagnets. Inourmodel, however, latticegeometry
spin-1/2 antiferromagnets on the kagome lattice.40,44–50 plays no role at all due to the long range nature of the
While in our model time reversal symmetry is broken interactions. The choice of the square lattice was thus
explicitly, it may still be a useful starting point for con- dictated only by its simplicity. Our construction, how-
structing models in which it is broken spontaneously. ever, may be repeated on any 2D Bravais lattice with
TheexpressionforthekineticpartoftheHamiltonian, identical results.
Eq. (33), reveals several features, which are presumably Inconclusion,wehaveprovidedaderivationofalattice
important to achieve a chiral spin liquid. First, H has Hamiltonian which, by construction, will have gapped
1
theformofacorrelatedhoppingHamiltonian, wherethe fractionalizedliquidgroundstatesatcertainrationalfill-
boson hopping amplitude from site m to m(cid:48) depends on ing fractional filling fractions, such as 1/2. The con-
the boson density at site n. This reminds one, not acci- struction employs magnetic Wannier states, which are
dentally, of flux attachment.51 Second, both H and H highly symmetric (the Hamiltonian has the full symme-
0 1
terms are long-range, exhibiting power-law decay. This try of the Bravais lattice used) and have the fastest pos-
is a direct consequence of the long-range 1/r2 tail of the sibledecayrate, compatiblewiththenontrivialtopology
Wannier functions Φ (r), which in turn is rooted in the of the LLL, i.e. 1/r2. It would be interesting to con-
m
nontrivialtopologyoftheLLL.Thisfeatureisalsoshared firm our construction by an explicit numerical solution
with previous constructions of parent Hamiltonians for of Eqs. (32),(33),(34).
chiral spin liquids,41,43 employing mapping to the LLL
in some form. The power law decay in Eqs. (32), (33)
is the fastest possible for this type of construction, since ACKNOWLEDGMENTS
weareusingtheWannierstateswiththefastestpossible
decay rate in all directions. WeacknowledgeausefuldiscussionwithD.Hawthorn.
In our construction of the magnetic Wannier basis we Financial support was provided by Natural Sciences and
havechosentoplacetheWannierorbitalsonthesitesofa Engineering Research Council (NSERC) of Canada (IP,
squarelattice. Thischoiceisofcoursearbitrary: any2D AP and AAB) and by the National Science Foundation
Bravaislatticewouldworkjustaswell. Thismayappear through grants DMR-1157490 and DMR-1442366 (KY).
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