Parent Hamiltonian for the Chiral Spin Liquid
Ronny Thomale, Eliot Kapit, Darrell F. Schroeter, Martin Greiter
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Parent Hamiltonian for the Chiral Spin Liquid
Ronny Thomale, Eliot Kapit, Darrell F. Schroeter, and Martin Greiter Institut f¨ur Theorie der Kondensierten Materie,Universit¨at Karlsruhe, 76128 Karlsruhe, Germany Department of Physics, Cornell University, Ithaca, NY 14853 Department of Physics, Reed College, Portland, OR 97202 (Dated: October 29, 2018)We present a method for constructing parent Hamiltonians for the chiral spin liquid. We findtwo distinct Hamiltonians for which the chiral spin liquid on a square lattice is an exact zero-energyground state. We diagonalize both Hamiltonians numerically for 16-site lattices, and find that thechiral spin liquid, modulo its two-fold topological degeneracy, is indeed the unique ground state forone Hamiltonian, while it is not unique for the other.
PACS numbers: 75.10.Jm, 05.30.Pr, 71.10.Hf
I. INTRODUCTION
The first notion of fractional excitations in condensedmatter physics goes back to the appearance of solitonmid-gap modes in polyacetylene , where the effective netcharge of one kink excitation is e/ i.e. , one half of theelectron charge. At a similar time, the field of fractionalstatistics, founded in the work by Leinaas and Myrheim ,attained broad attention due to the work by Wilczek in1982 . In strongly-correlated many-body systems, thephenomenon of fractionalization, where the elementaryexcitations of the system carry only a fraction of thequantum numbers of the constituents, has become knownto occur in a variety of cases.The first physical system in which fractional excita-tions and the associated fractional statistics have beendiscussed on a unified footing is the fractional quantumHall effect (FQHE). There, the quantum statistics ofthe anyonic quasiparticles can be understood in terms ofa generalized Berry’s phase , which is acquired by thewave function as quasiparticles wind around each other.This is a sensible concept in two dimensions only whereone can uniquely define a winding number for the braid-ing. In the FQHE, the fractional statistics is known tooccur in the presence of a magnetic field violating par-ity (P) and time-reversal (T) symmetry. In recent years,there have been tremendous efforts to study the frac-tional excitations of the FQHE experimentally in orderto confirm the prediction from theory and to validatefractional statistics as a concept being realized in nature.This, however, has remained inconclusive in certain as-pects and thus is still a subject of current discussion andwork .Later, the concept of fractional statistics has beenfound to occur in one-dimensional spin-1/2 antiferro-magnets, where it can be defined in terms of a gener-alized Pauli principle obeyed by the excitations and,as shown recently, by a phase the wave function acquireswhen two spinons move through each other . The frac-tional charge of the quasiparticles in the FQHE corre-sponds to the spin 1 / . In particular, various propertiesof fractional excitations in spin chains have been observedexperimentally .In general, it appears to be that P and T violation isintimately related to the occurrence of excitations obey-ing fractional statistics in two dimensions, which bothapplies to quasiparticles in the FQHE and spinons in aquantum antiferromagnet. These symmetries may be ex-plicitly broken as in the FQHE or generated by sponta-neous symmetry breaking. For two-dimensional antifer-romagnets, the concept of fractional excitations is lessestablished than for the one-dimensional case . In par-ticular, finding solvable theoretical models in which thephenomenon occurs has been one predominant area ofresearch in the field. Significant progress has been ac-complished for dimer models .In addition to important questions with regard tothe general principle underlying fractional statistics,two-dimensional spin liquids are of special interestwith regard to investigation of the hypothesized linkbetween fractionalization and high- T c superconductiv-ity . Moreover, in many systems where fractional-ization occurs, there is the ambition to use the topolog-ical degeneracy contained in these systems for quantumcomputing, where topological information can serve asa quantum bit with negligibly small local decoherencerates .The paradigmatic state for a S = 1 / , which is constructed to spontaneously vio-late the symmetries P and T, and can be defined on anyregular lattice including both bipartite and non-bipartitelattices. The universality class of chiral spin liquid states,and in particular the order parameter and the topologicaldegeneracy , was defined by Wen, Wilczek, and Zee .A CSL state has also been constructed by Yao and Kivel-son in the Kitaev model on a Fisher lattice, i.e. , ahoneycomb lattice of triangles. Recently, a family of non-belian CSL states has been proposed for general spin S , whose wave functions correspond to the bosonic Read-Rezayi series of FQH states . The non-Abelian statisticsof the spinons has even been conjectured to be a generalproperty of spin S antiferromagnets .As in the one-dimensional case mentioned above, thespinons in the CSL exhibit quantum-number fractional-ization and carry only half the spin of the bosonic spin ex-citations in conventional magnetically-ordered systems,which carry spin 1. Whereas the spinon appears tobe the fundamental field describing excitations in two-dimensional S = 1 / ; bothHamiltonians contain 6-body interactions. One of thekey issues we address here is whether the CSL is the only ground state of these Hamiltonians. To answer this ques-tion, we perform exact diagonalization studies of bothmodels for a 16-site square lattice. In particular, weintroduce an adapted Kernel sweeping method, whichallows for an efficient numerical implementation of thecomplex and technically cumbersome Hamiltonians weinvestigate. We find that the model we introduced pre-viously has indeed the CSL as its (modulo the two-foldtopological degeneracy) unique ground state. For theother Hamiltonian we present, however, we find that theCSL is not the unique ground state. Hence only the for-mer model is useful for further analysis of e.g. the spinonspectrum.The paper is organized as follows. In Section II, wereview the chiral spin liquid ground state and its ba-sic properties. After outlining the general constructionscheme for the Hamiltonians in Section III, we formu-late a destruction operator for the CSL state in Sec-tion IV and exploit the spin rotational invariance of theCSL state to decompose the destruction operator into itsspherical tensor components, which annihilate the CSLstate individually. The proof that the destruction oper-ator annihilates the CSL ground state is given in Sec-tion V. In Section VI, we introduce a Kernel sweepingmethod to compute the CSL Hamiltonians. We present LL FIG. 1: The model is defined on a square lattice length L ona side such that the total number of sites is given by N = L . The image shows the lattice for N = 16. The shadedcircles (including the origin) indicate those lattice sites forwhich G ( z ) = − G ( z ) = +1. The sites on which G ( z ) = − the method in detail and emphasize its applicability to ef-ficiently compute n -body interactions for finite-size exactdiagonalization studies. The numerical results obtainedwith this method are discussed in Section VII. We con-clude this work with a summary in Section VIII. II. CHIRAL SPIN LIQUID
The CSL was originally conceived by D.H. Lee as a spinliquid constructed by condensing the bosonic spin flipoperators on a two-dimensional lattice into a FQH liquidat Landau level filling factor ν = 1 /
2. The ground statewave function for a circular droplet with open boundaryconditions, on a square lattice with lattice constant oflength one, is given by h z · · · z M | ψ i = M Y j 2. The z ’s in the above expression arethe complex positions of the up-spins on the lattice: z = x + iy , with x and y integer. G ( z ) = ( − ( x +1)( y +1) is agauge factor, which ensures that (1) is a spin singlet (seeFig. 1). Lattice sites not occupied by z ’s correspond todown-spins.For our purposes, it is propitious to choose periodicboundary conditions (PBCs) with equal periods L = L = L , L even, and with N = L sites. FollowingHaldane and Rezayi , the wave function for the CSLthen takes the form h z · · · z M | ψ i = Y ν =1 ϑ (cid:16) πL [ Z − Z ν ] (cid:17) M Y j In order to construct a parent Hamiltonian for the chi-ral spin liquid, one first derives the destruction operatorsfor the ground state. In our formulation, the destruc-tion operators are constructed from a set of operators ω j where j = 1 , . . . , N indexes the lattice sites. The oper-ators ω j , to be introduced in Section IV below, are notthemselves destruction operators, but have the propertythat, acting on the ground state, they produce a resultindependent of the site index j : ω i | ψ i = ω j | ψ i . There-fore, once the above result is established in Section V, itfollows that the difference of any two of the operators is adestruction operator for the ground state: d ij = ω i − ω j .In order to construct a sensible parent Hamiltonian,one must minimally demand that it be a translationally-invariant scalar operator. In order to put the Hamilto-nian in this form, it is shown in Appendix A that theoperators may be written as ω j = Ω j + ✵ j where Ω j and ✵ j are vector and third-rank spherical tensor oper-ators respectively and where the 0 superscript indicatesthe component in spherical notation. The operators Ω j and ✵ j are given explicitly in terms of spin operators inSections IV A and IV B.As is discussed in detail in Section IV, the Wigner-Eckhart theorem guarantees that all components of theoperators D ij = Ω i − Ω j as well as D ij = ✵ i − ✵ j aredestruction operators for the chiral spin liquid groundstate so long as the reducible tensor operator d ij is. Onecan then construct Hamiltonians based on either set ofoperators: H = X h i j i D † ij · D ij (3)for the vector destruction operators or H = X h i j i X ν = − (cid:0) D νij (cid:1) † D νij (4) for the rank-3 spherical tensor operators. Either Hamil-tonian is a scalar and is translationally invariant, bothof these properties guaranteed by the construction. Ad-ditionally, since the Hamiltonians are positive semi-definite, the chiral spin liquid is a ground state of themodel. It should be noted that these models are notthemselves unique as one could include any coefficients J ij into the sums of Eqs. 3 and 4 and remove the restric-tion that only nearest-neighbor sites are summed over.These two models do, however, represent the simplestmodels from each class.In Section VI, a numerical method is developed forperforming the exact diagonalization of these Hamilto-nians that can handle the large number of interactionsefficiently. This method is used in Section VII to showthat the model given by Eq. 3 has exactly two groundstates, as expected due to the topological degeneracy ofthe chiral spin liquid on a torus, and that these statesare precisely the chiral spin liquid ground states givenin Section II above. Adopting the same procedure, theHamiltonian given in Eq. 4 is shown to have a largerground-state manifold which is not exhausted by the chi-ral spin liquid ground states. IV. ANNIHILATION OPERATOR FOR THECHIRAL SPIN LIQUID The Hamiltonian which stabilizes the chiral spin liquidis generated by first finding a set of operators ω i , where i is a site index. These operators are not themselvesdestruction operators, but the bond operators ω i − ω j ,where i and j are any two distinct sites, will be shownto destroy the CSL ground state. The operators may bewritten as ω j = ω + j − ω − j where ω + j = T j + V j and T j = 12 ′ X i k = j K ijk S + j S − k (cid:18) 12 + S zi (cid:19) (5) V j = X i = j U ij (cid:18) 12 + S zi (cid:19) (cid:18) 12 + S zj (cid:19) . (6)The two sets of coefficients U ij and K ijk are defined inSection IV C below and the prime on the sum indicatesthat one must exclude the coincidences of i and k .The operator ω − j is related to ω + j by a π/ x -axis that maps S z and S y into − S z and − S y .This means that the entire operator ω j is given by ω j = ′ X i k = j K ijk (cid:20) i ( S j × S k ) z +( S j · S k ) S zi − S zi S zj S zk (cid:21) + X i = j U ij S zi . (7)In writing down Eq. 7, the fact that P i = j U ij = 0, has been employed. This will be demonstrated in Sec-3ion IV C below. While the operators ω i are not them-selves destruction operators for the CSL ground state, itwill be shown in Section V that d ij = ω i − ω j is a de-struction operator for the ground state for any choice of i and j .The operators ω j are reducible and can be decomposedinto irreducible tensor operators, in this case of ranks 1and 3. From Eq. 7 it is clear that every term except forthe S zi S zj S zk term is the 0 (or z ) component of a rank-1(vector) operator. This final term can be decomposedinto rank-3 and vector components.It is straightforward to show that if an operator d is adestruction operator for the CSL ground state, then eachof its irreducible components are as well. This is becausethe Wigner-Eckhart theorem tells us that acting with anoperator T jm on a state | n q m q i with angular momentum q and z -component m q gives T jm | n q m q i = X j ′ m ′ C mj m q q m ′ j ′ | n ′ j ′ m ′ i , (8)where n and n ′ are any quantum numbers other thanangular momentum. Since the CSL is a spin singlet: q = m q = 0, it follows that there is only a single non-zero term in the above sum corresponding to j ′ = j and m ′ = m . This means that by decomposing the destruc-tion operator for the ground state d into its tensor com- ponents, which may be written d = P j a j T j , acting onthe ground state to obtain0 = d | ψ i = X j a j | n ′ j i , (9)and noting that states with different values of j are neces-sarily orthogonal, it immediately follows that each of thestates in the sum are themselves zero and hence the op-erators T j are destruction operators for the ground state.In Sections IV A and IV B we give two classes of opera-tors that are obtained from the reducible tensor operator ω j in Eq. 7. A. Vector destruction operator As shown in Appendix A, the operator S zi S zj S zk maybe written as the sum of the 0-components of a vectorand a third-rank tensor. The vector component is givenby 15 (cid:2) ( S i · S j ) S zk +( S j · S k ) S zi +( S k · S i ) S zj (cid:3) (10)and, working from Eq. 7, the vector operator Ω j is givenby Ω j = ′ X i,k = j K ijk (cid:20) i ( S j × S k ) + 45 ( S j · S k ) S i − 15 ( S k · S i ) S j − 15 ( S i · S j ) S k (cid:21) + X i = j U ij S i . (11)Since Ω i − Ω j is a destruction operator for the groundstate, it immediately follows that one may construct aHamiltonian for which the chiral spin liquid is the exactground state as H = X h i j i ( Ω i − Ω j ) † · ( Ω i − Ω j ) , (12)where the sum runs over all nearest-neighbors on the lat-tice. By construction, the Hamiltonian is a scalar opera-tor and translationally invariant.However, note that there is nothing restricting possiblemodels to run only over next-nearest neighbors. Rather,one can consider any combination of bond-operators (in-cluding arbitrary coefficients so long as one maintains positive semi-definiteness in H ) in constructing a parentHamiltonian for the CSL. B. Tensor destruction operator It is also possible to create a set of third-rank tensordestruction operators. As shown in Appendix A, theoperator S zi S zj S zk may be fully decomposed into the 0-components of a vector operator (given in Eq. 10) and athird-rank tensor operator, which is necessarily just thedifference between S zi S zj S zk and the operator in Eq. 10.This gives a destruction operator whose 0-component is ✵ j = − √ ′ X i,k = j K ijk (cid:2) ( S i · S j ) S zk +( S j · S k ) S zi +( S k · S i ) S zj − S zi S zj S zk (cid:3) . (13)4he other components are straightforward to obtain (seeAppendix A) and one may again use these operators toform a Hamiltonian for the chiral spin liquid accordingto H = X h i j i X ν = − (cid:0) ✵ νi − ✵ νj (cid:1) † (cid:0) ✵ νi − ✵ νj (cid:1) . (14)The Hamiltonian in Eq. 14 has two significant advantagesover the model in Eq. 12: it depends only on one set ofcoefficients ( K ijk but not U ij ) and, because the operatorin the sum in Eq. 13 is symmetric under interchange of i and k , one may replace K ijk by A ijk = ( K ijk + K kji ) / C. Coefficients The coefficients appearing in Eq. 7 are functions ofthe distance between the sites of the form K ijk = K ( z k − z j , z i − z j ) where K ( x, y ) = 1 N/ − R →∞ X ≤ z ≤ R P ( x − z , y ) x − z , (15)and the sum over z is a sum over all lattice translations: z = ( m + i n ) L for m and n integer. This sum guar-antees that the function K ( x, y ) is periodic in its firstargument.The coefficients U ij = π U ( π [ z j − z i ] /L ) /L are givenby πL U (cid:16) πL z (cid:17) = πL W (cid:16) πL z (cid:17) + 1 N − ddx P ( x, − z ) (cid:12)(cid:12)(cid:12)(cid:12) + lim R → X < | z |≤ R P ( z , − z ) z , (16)where W ( z ) is the periodic extension of 1 /z to the torus and also related to the logarithmic derivatives of the thetafunctions: πL W (cid:16) πL z (cid:17) = ddz ln ϑ (cid:16) πL z (cid:17) + πL z − z ∗ L . (17)The function P ( x, y ) is given by P ( x, y ) = lim R →∞ X ≤| z − y |≤ R Co (cid:0) π L [ z − y ] (cid:1) Co (cid:0) π L [ x − ( y − z )] (cid:1) e − πL | z − y | n ( y ) , (18)where Co( x ) = cos x +cosh x and where n ( y ) is a normal-ization factor chosen such that P (0 , y ) = 1 which entailsthe choice n ( z ) = ϑ (cid:16) πL Re [ z ] (cid:12)(cid:12)(cid:12) i (cid:17) ϑ (cid:16) πL Im[ z ] (cid:12)(cid:12)(cid:12) i (cid:17) . (19)While the form of the coefficients as given by Eqs. 15–17 are essential for forming a Hamiltonian that stabilizesthe CSL, there is significant freedom in how one choosesthe function P ( x, y ). The only requirements are thatit be a periodic function of y , fall off faster than 1 /x with increasing x , and be analytic apart from first-orderpoles that occur at the coincidence of the two arguments: x = y . It is straightforward to show that U ( z ) is an oddfunction; this in turn guarantees that P i U ij = 0 andlets this sum be dropped, as was done in writing downEq. 7. V. PROOF OF SOLUTION In order to prove that either of the Hamiltonians givenin Eqs. 3 and 4 are true parent Hamiltonians for the chiralspin liquid, we must demonstrate that ω j | ψ i = ω i | ψ i which we will demonstrate by first showing that h z · · · z M | ω j | ψ i = f ( Z ) h z · · · z M | ψ i , (20)where f ( Z ) is a function only of the center of mass: Z = P Mi =1 z i . This identity in turn follows from the fact that h z · · · z M | ω + j | ψ ih z · · · z M | ψ i = (cid:26) f ( Z ) z j ∈ { z · · · z M } , (21)and the result that the function f ( Z ) is both odd andperiodic. To see this, recall that one can write ω − j = U ω + j U † where U performs the π/ x -axis as discussed in Section IV above. The CSL ground5tate is invariant under such a rotation so that h z · · · z M | ω + j | ψ i = h z · · · z M | U † ω − j U | ψ i = h w · · · w M | ω − j | ψ i , (22)where { w i } , the locations of the down spins on the lattice,is the complement of { z i } . It then follows from Eq. 21that h z · · · z M | ω − j | ψ ih z · · · z M | ψ i = (cid:26) z j ∈ { z · · · z M } f ( W ) otherwise . (23)Assuming that the origin of the lattice is chosen suchthat the sites occupy positions z i = ( ℓ + i m ) for ℓ and m integer, it is straightforward to show that Z + W = L ( L − i ) L, (24) and since L is even it follows that the sum of Z and W isequivalent to a translation of the lattice z . Because thefunction f ( Z ) is periodic and odd, both properties willbe shown below, it immediately follows that f ( W ) = f ( z − Z ) = − f ( Z ). Combining this fact with Eq. 23completes the proof that Eq. 21 entails Eq. 20. A. Action of T j In order to prove Eq. 21, we first consider the off-diagonal terms in the operator ω + j which come from T j defined in Eq. 5. We consider a general element of thevector T j | ψ i : h z · · · z M | T j | ψ i = 12 ′ X i,k = j K ijk (cid:28) z · · · z M (cid:12)(cid:12)(cid:12)(cid:12) S + j S − k (cid:18) 12 + S zi (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:29) . (25)The element is clearly zero unless z j ∈ { z · · · z M } . When this is satisfied, acting onto the bra on the right-hand sideof the equation with the spin operators wipes out the matrix element unless z i ∈ { z · · · z M } and replaces z j with z k : h z · · · z M | T j | ψ i = 12 M X i = j N X k = j K ijk h z · · · z j − z k z j +1 · · · z M | ψ i . (26)The upper limit of M = N/ N ) on the sum on i indicates that z i must be a member of the up-spins.Rewriting K ijk = K ( z k − z j , z i − z j ) and defining z = z k − z j , this may be rewritten as h z · · · z M | T j | ψ i = 12 M X i = j X z =0 K ( z, z i − z j ) h z · · · z j + z · · · z M | ψ i . (27)Using the definition of the coefficient K from Eq. 15, this can be rewritten as h z · · · z M | T j | ψ i = 1 N − M X i = j X z =0 lim R →∞ X ≤ z 1) andthis gives, renaming ℓ as i , h z · · · z M | T j | ψ ih z · · · z M | ψ i = f ( Z ) − M X i = j πL W (cid:16) πL [ z j − z i ] (cid:17) − N − M X i = j "X z P ( z , z i − z j ) z + ddx P ( x, z i − z j ) (cid:12)(cid:12)(cid:12)(cid:12) , (34)where f ( Z ) = − X ν =1 πL W (cid:16) πL [ Z − Z ν ] (cid:17) . (35)The fact that f ( Z ) is both odd and periodic, requiredfor the proof of Eq. 20 above, follows from these sameproperties of the W function. Comparison with Eq. 16shows that h z · · · z M | T j | ψ ih z · · · z M | ψ i = f ( Z ) − M X i = j U ij (36)if z j is an element of the up-spins and zero otherwise. B. Action of V j The action of the operator V j on the CSL ground stateis straightforward to compute. Proceeding in an analo-gous manner, we have h z · · · z M | V j | ψ i = N X i = j U ij h z · · · z M | (cid:18) 12 + S zi (cid:19) (cid:18) 12 + S zj (cid:19) | ψ i . (37) The matrix element vanishes unless both z i and z j areelements of { z · · · z M } . Therefore, the diagonal contri-bution to the operator ω j gives h z · · · z M | V j | ψ ih z · · · z M | ψ i = M X i = j U ij (38)if z j ∈ { z · · · z M } and 0 otherwise. Combining Eqs. 38and 36 yields Eq. 21 and therefore proves that the chi-ral spin liquid is an exact ground state of either of theHamiltonians in Eqs. 12 or 14. VI. KERNEL SWEEPING METHOD To implement the Hamiltonians given in Eq. 12 andEq. 14, one has to take into account that 6-body termsappear in the Hamiltonians. For microscopic models con-taining many-body interactions, one must be very effi-cient if one hopes to write down the Hamiltonian in areasonable amount of time. For our Hamiltonians, thisis because there are, even for a lattice with only N = 16sites, literally thousands of terms in the Hamiltonian cor-responding to all the different ways to choose six sites outof sixteen. In contrast, a model with only two-site inter-actions on the same lattice would only have 15 terms to7ompute after taking into account translational symme-try, even if the model had infinite range. In this section,we describe an algorithm for calculating the Hamiltonianvery efficiently, called the kernel sweeping method.As an example to illustrate the kernel sweep-ing method, we will consider the computation of aHeisenberg-type Hamiltonian such as H = X ij J ij S i · S j . (39)We work in an S z basis and label the states by a binarynumber where up-spins are treated as 1’s and down-spinsare treated as 0’s. We first note that since this is a two-site interaction, in order to implement this model all wereally need to know is how the operator S i · S j acts onthe four-dimensional basis | s i s j i . This action may besummarized as h↓↓| h↓↑| h↑↓| h↑↑| / |↓↓i − / / |↓↑i / − / |↑↓i / |↑↑i (40)where the table format shows the order of the basis vec-tors. It is only necessary to compute this matrix onceat the beginning of running the code. One stores thismatrix as a set of rules R = { [ { s } m , { s } n ] → Ω mn } (41)where the { s } m and the { s } n are a binary shorthand forthe states in this two-dimensional basis and Ω mn are theelements in the matrix. In this example we would have R = (cid:26) [00 , → , [01 , → − , [01 , → , [10 , → − , [11 , → (cid:27) (42) where, since we are dealing with a Hermitian operator, weonly need to include the upper triangle. The extensionof this array to a p -site operator is straightforward; inthat case one must consider the action of the operatoron a 2 p -dimensional basis. Therefore, the correspondingoperator for the chiral spin liquid Hamiltonian given inEquation 12 is 64-dimensional.The code next loops over all possible values of i and j and does the following. First it computes R ij = (cid:8)(cid:2) { s } m · (cid:8) i − , j − (cid:9) , { s } n · (cid:8) i − , j − (cid:9)(cid:3) → J ij Ω mn } . (43)All this means is to compute the contribution of the twospins at sites i and j to the binary number that will labelthe entire state. For our example, assuming that we areat a point in the loop where i = 3 and j = 7, this gives R = (cid:26) [0 , → J , (cid:2) , (cid:3) → − J , (cid:2) , (cid:3) → J , (cid:2) , (cid:3) → − J , (cid:2) + 2 , + 2 (cid:3) → J (cid:27) . (44)The code next computes the contributions to the binarynumbers labeling the states from all the sites that are notinvolved in the interaction. There are 2 N − p of these andfor our two-site example this list is B ij = N X l = i,j s l l − . (45)Finally, one updates the Hamiltonian according to H = H + R ij ⊗ B ij (46)where the addition means to add the matrix defined bythese rules and the generalized outer product means R ij ⊗ B ij = (cid:8)(cid:2) { s } m · (cid:8) i − , j − (cid:9) + b, { s } n · (cid:8) i − , j − (cid:9) + b (cid:3) → J ij Ω mn (cid:9) (47)for b an element of B ij . In this way one may construct theHamiltonian extremely quickly since all the steps involvelist operations and there is only a single loop over the N choose 2 ways to pick the sites i and j . (In practice,one uses translational invariance to fix i = 1 and, for atwo-site operator as in this example, the loop is then overthe N − D ij = Ω i − Ω j , we split up the Hamiltonian into H = X h ij i Ω z, † ij Ω zij + 12 (cid:16) Ω + , † ij Ω + ij + Ω − , † ij Ω − ij (cid:17) , (48)where the z -component as well as the ladder componentsof the vector operators can be written out in terms of spinoperators S z , S + = S x + iS y , and S − = S x − iS y . Asthe treatment is very similar, we constrain our attentionto the contribution P h ij i Ω + , † ij Ω + ij , where for clarity we8gain write out the + ladder operator explicitly: Ω + j = ′ X i,k = j K ijk (cid:20) i ( S zj S + k − S + j S zk ) + 45 ( S j · S k ) S + i − 15 ( S k · S i ) S + j − 15 ( S i · S j ) S + k (cid:21) + X i = j U ij S + i . (49)Using the notation analogous to Eq. 45 B k ijk = N X l = i,j,k s l l − B u i = N X l = i s l l − , (50)we can write Ω + j = X ii = j X kk = i,j R k ijk ⊗ B k ijk + R u i ⊗ B u i , (51)where R k ijk and R u i relate to the first and second sumof Eq. 49, respectively. Given these 3-body operators inabove notation, the total 6-body interaction can be con-veniently computed. The implementation of the tensorHamilton operator Eq. 14 is completely analogous. VII. NUMERICAL CONFIRMATION Using the method outlined in Section VI above, themodels in Eq. 12 and Eq. 14 have been solved by exactdiagonalization on 16-site lattices with periodic boundaryconditions. We start by considering the vector Hamilto-nian given by Eq. 12. The spectrum is shown in Fig. 3;the points in the Brillouin zone which label the axis ofthis figure are shown in Fig. 2. We find the spectrum tobe positive semi-definite, with a doubly-degenerate zero-energy state at the Γ point. The rest of the spectrumis well separated from the ground state by a gap that is k x k y Γ ∆ X Σ KM FIG. 2: A plot of the symmetry points in the first Brillouinzone. The arrows show the path taken in plotting the energyspectra in Fig. 3, starting from the origin at Γ = (0 , Γ ∆ X K M Σ FIG. 3: Low energy spectrum of the Hamiltonian Eq. 12,scaled down to order of unity. There are two E = 0 eigenval-ues at the Γ point. substantial and we believe not due to finite size effects inthe calculation. This claim is based on the fact that itexceeds the finite size level splitting of the spectrum by afactor of ∼ 15. The presence of a gap is expected betweenthe chiral spin liquid ground state and what should be atwo-spinon excited state. The spinon excitations of thismodel will be addressed in future work.We now discuss the two orthogonal zero-energy eigen-states. For comparison, we construct the CSL state Eq. 2explicitly and find a two-dimensional subspace of func-tions with the center of mass variable being treated asan external parameter. We have computed the overlapof the Hamiltonian ground state subspace and the CSLsubspace and find that they match perfectly. Therefore,the ground state of this Hamiltonian is indeed the two-fold degenerate CSL state. Additionally, we have onlytwo zero-energy states, by which follows that the CSLstate is the only ground state of the model, a statementwhich cannot be achieved analytically.For the tensor Hamiltonian, however, we find thatthe zero-energy subspace is massively degenerate. It ofcourse contains the CSL, in accord with the analyticalproof, but also many additional states. While the restric-tion to small system sizes prevents us from studying thethermodynamic limit precisely, our numerical findings in-dicate that the Hamiltonian Eq. 14 does not stabilize the9SL state as the unique ground state, which thus singlesout the model in Eq. 12 to be subject of further study. VIII. CONCLUSION In this work we have shown a method for constructingparent Hamiltonians for the chiral spin liquid. We havecomputed the spectra of the Hamiltonians by use of aKernel-sweeping method in exact diagonalization. There,for the Hamiltonian operator composed of the sphericalvector component of the CSL destruction operator, weobserve that the CSL states are the only ground statesof the model. We conclude that this model is a promisingcandidate to also study the elementary excitations of themodel, i.e. , spinons, and many other questions in the fieldof two-dimensional fractionalization of quantum numbersin spin systems. Acknowledgments RT was supported by a PhD scholarship from the Stu-dienstiftung des deutschen Volkes; DS acknowledges sup-port from the Research Corporation under grant CC6682.We would like to thank J.S. Franklin, R. Crandall, andR.B. Laughlin for many useful discussions. APPENDIX A: TENSOR DECOMPOSITION The operators ω j introduced in Section IV may be de-composed into irreducible spherical tensors of ranks 1 and 3. We write these irreducible operators as T qm ; q and m correspond to angular momentum and its z compo-nent respectively. We wish to write ω = P c q T q , where T q is the collection of all operators which transform as aspherical tensor of rank q . Here we have suppressed thesite index on the operator ω .The operator in Eq. 7 that is not manifestly the com-ponent of a vector is S zi S zj S zk , which is a component ofa third-rank Cartesian tensor. In order to keep the nota-tion manageable, we start by considering the direct prod-uct of two operators U and V with angular momentum j and j respectively. An element in the direct productspace of these operators may be written as U j m V j m = j + j X j = | j − j | j X m = − j C m j m j m j T j m (A1)in terms of irreducible spherical tensors T j m carryingangular momentum j with z -component m = m + m . Eq. A1 may be inverted to give T j m = j X m = − j j X m = − j C m j m j m j U j m V j m . (A2)Using these equations, one may construct correspond-ing expressions for the product of three vector operatorsby applying Eq. A1 twice: U j m V j m W j m = j + j X j = −| j − j | j X m = − j C m j m j m j T j m W j m = j + j X j = −| j − j | j X m = − j C m j m j m j j + j X j = | j − j | j X m = − j C m j m j mj T j ( j ) m . (A3)The second superscript on the tensor T in the last line distinguishes between the different tensors of the same rankthat appear when combining three vector operators; since 1 ⊗ ⊗ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 0, there are two rank-2spherical tensors and three vector operators that can be formed. For the case of interest m = m = m = 0 and j = j = j = 1, the expression reduces to U z V z W z = X j =0 j +1 X j = | j − | C j C j j T j ( j )0 = − √ T − √ T + r T , (A4)which shows that the operator contains only vector and rank-3 tensor components, but no scalar or rank-2 tensorcomponents. Note that the second index on the rank-3 tensor has been suppressed since the construction of thisobject is unambiguous. 10pplying Eq. A2 twice, the rank-3 tensor component is T = X m = − X m = − C m m T m W m (A5)= X m ,m ,m = − C − m m C m m − m U m V m W m (A6)= 5 U z V z W z − ( U · V ) W z − ( V · W ) U z − ( W · U ) V z √ , (A7)where we have used the fact that the dot product is U · V = P m ( − m U m V − m in the spherical representation. Asimilar construction can be used to find the vector operator or, one may note from Eqs. A4 and A7 that the vectorcomponent is equivalent to U z V z W z − r T = ( U · V ) W z +( V · W ) U z +( W · U ) V z x and y ) components of the vector operator in Eq. A8 is straightforward since onemerely replaces z with either x or y . In order to construct the remaining six components of the rank-3 tensor operatorone simply applies Eq. A2 twice without specifying m = 0: T m = X m ,m ,m = − C m − m m m C m m m − m U m V m W m . (A9)The explicit form of these components are T = − √ (cid:2) (5 V z W z − V · W ) U + +(5 U z W z − U · W ) V + +(5 U z V z − U · V ) W + (cid:3) (A10) T = 12 √ (cid:2) U + V + W z + U + V z W + + U z V + W + (cid:3) (A11) T = − √ U + V + W + , (A12)with the remaining three components obtained from T q − m = ( − m ( T qm ) † . APPENDIX B: SUM RULE The sum rule used in Section V, on which the proofthat ω destroys the ground state hinges, is given bylim R →∞ X ≤| z | 1. As shown in Figure 2, these sitesdefines a sublattice with twice the original lattice spacing. F ( c ) = X z e c z − π | z | − X z ′ e c z ′ − π | z ′ | . (B3)Setting z ′ = 2 z we can write this as F ( c ) = X z e c z − π | z | − X z e c z − π | z | , (B4)where both sums now run over the entire lattice. Writing z = x + i y this function can be factored into four sums11ver the integers x and y : F ( c ) = X x e ( c x − π x ) ! X y e ( i c y − π y ) ! − X x e c x − π x ! X y e i c y − π y ! . (B5)In terms of the third Jacobi theta function θ ( z | τ ) = ∞ X n = −∞ e i π n τ e i n z , (B6)this function may be recast as F ( c ) = ϑ (cid:18) − i c (cid:12)(cid:12)(cid:12) i (cid:19) ϑ (cid:18) c (cid:12)(cid:12)(cid:12) i (cid:19) − ϑ (cid:16) − i c (cid:12)(cid:12)(cid:12) i (cid:17) ϑ (cid:16) c (cid:12)(cid:12)(cid:12) i (cid:17) . (B7)The fact that the two terms in this expression preciselycancel is a result of Jacobi’s imaginary transformation , θ ( z | τ ) = 1 √− i τ e z /i π τ ϑ (cid:18) ± zτ (cid:12)(cid:12)(cid:12) − τ (cid:19) , (B8)and the fact that the third Jacobi theta function is even.Application of this identity to either product of thetafunctions in Eq. B7 shows that the two terms preciselycancel, proving that F ( c ) = 0. This in turn proves the n = 0 case of Eq. B1 by simply setting c = 0. 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