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Local tensor network for strongly correlated projective states B. Béri and N. R. Cooper Theory of Condensed Matter Group, Cavendish Laboratory, J. J. Thomson Ave., Cambridge CB3 0HE, UK (Dated: January 2011) The success of tensor network approaches in simulating strongly correlated quantum systems cruciallydependsonwhetherthemanybodystatesthatarerelevantfortheproblemcanbeencoded inalocal tensornetwork. Despitenumerousefforts,strongly correlated projectivestates,fractional quantum Hall states in particular, have not yet found a local tensor network representation. Here we show that one can encode the calculation of averages of local operators in a Grassmann tensor 1 networkwhichislocal. Ourconstructionisexplicit,andallows theuseofphysicallymotivatedtrial 1 wavefunctions as starting pointsin tensor network variational calculations. 0 2 PACSnumbers: 73.43.-f,71.27.+a n a Approximating ground states of strongly correlated ter demonstrate the contrary: we show thatFQH states, J quantum systems is one of the central challenges in con- or more generally, states from the so-called projective 8 densed matter physics. In recent years there have been construction[21,22]doadmitalocaltensornetworkrep- 2 several promising proposals aimed at tackling this prob- resentation. Tensor network based variational schemes ] lem for gapped spin[1–4], as well as fermionic[5–9] sys- should thus be able to find such ground states. l e tems with local interactions. Common to these ap- To explain our idea, we start by first describing - proaches is that they are all based on a class of states a non-interacting system: the many electron ground r t which can be parametrized with tensor networks. For a state Ψ of a gapped quadratic Hamiltonian H = s | i . latticesystem,thetensorsliveonthesites,whicharecon- 1 (ψ†,ψ )h (ψ ,ψ†)T, where the indices k,l label t 2 kl k k kl l l a nectedbyasetoflinkssuchthattheindicesoftensorsat sitesofalatticeandψ(†) arethefermionicfieldoperators m the ends of the links are contracted (possibly according P l on these sites. (Writing H, in terms of a Bogoliubov de - to a metric defined by link tensors). An approximation d Gennes Hamiltonian h with a 2 2 electron-hole struc- ofthe groundstateisobtainedbytreatingthe tensorsas × n ture for each kl is redundant for Hamiltonians without o variational degrees of freedom. In order for tensor net- pairing,butitallowsustotreatpairingstatesandstates c work approaches to be efficient, it is important that, for withfixedparticlenumberonequalfooting.) Intheusual [ an accurate representation of the state, the number of tensor networkapproaches,the tensornetwork represen- 1 degrees of freedom per tensor does not grow exponen- tation of Ψ would be a preparatory step in express- v tially with the system size. Tensor networks with this | i ing averagesof operators as traces over tensor networks. 0 property are local tensor networks. There exist various The key observation of our work is that, instead of first 1 schemes for efficiently calculating averagesof local oper- 6 representing Ψ , one can construct tensor networks for ators with respect to local tensor network states[1–9]. | i 5 the averagesdirectly. We consider operators of the form 1. Strongly correlatedsystems can realize gapped phases A= j∈KA(ψj†)k¯jψjkj, where kj,k¯j =0,1 and KA is the 0 showing topological order, with features (ground state set of sites j where A acts, i.e., where k +k¯ = 0. (A 1 degeneracy, braiding statistics, etc.) that depend only generQal local operator can be obtained ajs a lijne6ar com- 1 on the topology of the configuration space, not its lo- bination of such products.) Using a result of Bravyi[23] : v cal details[10, 11]. Due to the topological nature of we can express ΨAΨ as a Grassmann integral, i h | | i X these phases, it is a nontrivial question whether one can ar cteanpstourrenetotwpoolrokgsi.calFlyororsdpeinredsygstreomunsd, Rsteaftse.s w[2i,th12l–o1ca5]l hΨ|A|Ψi=N dφjdφ¯j e−21(φ¯j,φj) h(jfkl)(φk,φ¯k)T Z j jk have shown that the ground states in a class of time- Y Y reversalinvarianttopologicalphases, so called string-net ω[(ψ†)k¯jψkj], (1) × j j condensates[16, 17], admit a local tensor network repre- j∈YKA sentation. The experimentally most accessible topologi- where is an A-independent normalization cally ordered states, the fractional quantum Hall (FQH) N factor[24]. (We omit henceforth, as it drops states, however, are fermionic, break time-reversal sym- from ΨAΨ / ΨΨ .) NHere φ¯ , φ are Grass- metry. Their existing tensor network representations in h | | i h | i j j the literature turn out to be nonlocal[7, 9, 18]. Does mann variables, and ω[ψj(†)] is φj/√2 (φ¯j/√2) and this mean that these states, and their spin-system de- ω[ψ†ψ ] = exp(φ¯ φ )/2. The matrix h(fl) is the “flat j j j j scendants (such as the chiral spin liquid state[19, 20]) band Hamiltonian”, which has the same eigenvectors as lie outside of the scope of tensor network based varia- h,buthastheeigenvaluesE replacedbysgn(E )[25,26]. j j tional simulations? Fortunately, the results of this Let- Using the Grassmann Leibniz rule (see e.g. Ref 27), 2 Eq. (1) can be transformed from an integral over FQH effect, spin-liquids, high T superconductors, and c fermions φ living on lattice sites to one over fermions such novel states of matter as the d=3 fractional topo- k φ living on the legs (i.e., ends of links) K of sites k, logical insulator state[29]. K Theprojectiveconstructiondefinesstronglycorrelated ΨAΨ = P T G . (2) trial ground states of the form[21, 22] 0 j kl h | | i Z j kl Y Y 1 Ψ( n )= 0 (ψ˜ )nj Θ . (5) Here, the symbol P represents a projection of the re- j j 0 { } n !h | | i sultofthe integraltothe termcontainingnoGrassmann j j Yj variables. The factors Gkl =gklglk with qQ Thestateiswrittenintheoccupationnumberrepresenta- gkl =exp[−21(φ¯K,φK) h(kfll)(φL,φ¯L)T] (3) itsiotnh,ewgirtohunnjdbsetiantgetohfeaogccaupppaetdioqnuoafdsriatteicj.HTamheilsttoantiea|nΘhi of P flavors of fermions (partons) living in a configura- belongtolinksklbetweensiteskandl. Thepreciseform tionspaceidenticaltothephysicalone. Theannihilation of the site factor T depends on the operators of A on j; j operatorsofthepartonsare c α=1...P . Thestate in the case that j / K it is given by αj A { | } ∈ Ψ can be bosonic or fermionic. For bosonic (fermionic) Tj =( dφJ)( dφ¯J)−(h(jflj))11, (4) |staites, the operators ψj are even (odd) polynomials of J∈j J∈j the operators c . The operator ψ should not be con- X X αj j fused with the field opeerator ψ : the former acts in the where the indices 11 refer to the electron-hole structure j parton Fock space, and is not a fieeld operator, while the and denotes summation over the legs J of site j. J∈j latter acts in the physical Fock space, and is a fermionic Why did we go from Eq. (1) to Eq. (2)? Because with P or bosonic field. Eq. (2), ΨAΨ takes the form of a tensor trace in an h | | i A well known projective state is the Laughlin state at existing tensor network scheme, namely the scheme of Grassmann tensor networks introduced by Gu et al.[9]. filling fraction ν = 1/3[30]. Its wavefunction in position representation is Ψ( z ) Ψ ( z )3, where Ψ ( z ) Our construction has a saliently pleasant feature: it i 1 i 1 i { } ∼ { } { } is the wavefunction of the integer quantum Hall (IQH) follows from h explicitly. This is to be compared with state at ν =1. In terms of Eq. (5), Ψ corresponds to the usual situation in tensor network approaches, where 3 ifonehasanexplicittensornetworkforastate Ψ ,often P =3, ψj =c1jc2jc3j and Θ = Ψ1 1 Ψ1 2 Ψ1 3. This it is not derived from a Hamiltonian, but a Ha|miiltonian Laughlin state is a membe|r oif a c|lasis o|f stiate|s wiith[22] (for which Ψ is the ground state) is derived from it. e | i P The opposite direction is viable only numerically. The ψ˜ = χ c ...c , (6) key feature of our construction, however, is rooted in j α1...αS α1j αSj the fact that for gapped systems, the matrix elements αXj=1 hT(ihjfli)sdimecpalyieesxtphoanteonutiraltleynasosrthneetfwuonrcktioisnloocfa|li.−Injd|[e2e5d,,2i8n]-. wrivheeraetSva≤rioPusanFQinHtegsetar.teDs,epsuecnhdiansgtohne|ZΘi,poanreafcearmniaorn- k troducing a cutoff length l, the relative error introduced states[31–33], or the states based on the composite by neglecting a link longer than l is overestimated by fermion picture[34]. exp( l/ξ),whereξ isthe decaylengthofh(fl). The total To use our tensor network construction for projective − number of links is N2, thus N2exp( l/ξ) largely over- states, we need to convert averages of the form ΨAΨ estimates the error due the neglected−links. Requiring into averages with respect to the state Θ . (Ashb|efo|rei, thattheerrorremainsfixedimpliesthatwehavetoscale we consider A= (ψ†)k¯jψkj.) In t|heibosonic case, the cutoff at most as l lnN. The number of legs at j∈KA j j ∼ we have site j is estimated as ld (lnN)d in a d dimensional Q ∼ system. Since the number of variables M per leg is in- n !(n +k¯ k )! dependent of the system size, the number of degrees of (ψ†)k¯jψkj n = j j j − j n +k¯ k (7) freedom in Tj (equal to the number of coefficients in its j j | ji q (nj −kj)! | j j − ji expansion in dφ , etc.[9]) grows at most as 2[M(lnN)d], K ifn k and0otherwise. Thisleadsto(withk¯ =k = i.e., quasi-polynomially. j ≥ j j j 0 for j / K ) Solving quadratic Hamiltonians with tensor networks ∈ A is certainly not a practical strategy. The usefulness of (ψ˜†)nj+k¯j−kj ourconstructionlieselsewhere: aswenowexplain,itcan ΨAΨ = Θ Y X j 0 0(ψ˜j)nj Θ . be employedto describe an important subset of strongly h | | i h | j "nj≥kj (nj −kj)! | ih | #| i correlatedstates,thestatesfromtheprojectiveconstruc- (8) tion. These states play an important role in several are- The key feature is that the expectation value of an op- nas of strongly correlated quantum systems, such as the erator factorizes into a product over different j-s. This 3 is essential for obtaining a local tensor network in the our construction can be built up straightforwardly. Just subsequent stages. asinEq.(1),wehaveaverageswithrespecttotheground In the fermionic case, the formula similar to Eq. (7) state Θ ofagappedquadraticfermionHamiltonian. In | i contains a sign factor depending on the occupations of both the boson and the fermion case, we have a prod- sites j < l, preventing such a factorization. This can uct between Θ and Θ which we denote f . The h | | i j j be remedied by using objects with anticommuting ingre- average will then have the same form as in Eq. (1), dtoiernRtsj. wMhoicrhe pdreepceisnedlys,osnulpypoonsepoanretocnasnafitnjd, aanndopceorma-- bfaucttowritωh(fj)jωis(fojb)taininsteedadsimofilarljy∈KaAsωb[e(fψoj†r)ek¯Q:jψojknje].taTkhees bines with ψ˜† to mimic field operators in the following f (c† ,c Q), expands it as a nQormal ordered polynomial j j αj αj sense: {Rj,Rj′} = 0, and for j′ 6= j {Rj,ψ˜j†′} = 0 and inthepartonoperators. Ifinatermbothc†αj andcαj are R ψ˜† 0 = 0 , R 0 =0. One can then show that present, they are brought next to each other. Single fac- j j| i | i j| i torsc (c† )arethensubstitutedbyφ /√2(φ¯ /√2), αj αj αj αj ΨA(ψ†,ψ)Ψ = ΘA(ψ˜†,R) and the product c†αjcαj by exp[φ¯αjφαj]/2. h | | i h | The GrassmannLeibniz rule leads again to an expres- (ψ˜†)nj 0 0(ψ˜ )nj Θ , (9) sionfor ΨAΨ intermsofatensornetworkoftheform ×  j | ih | j | i (2). Thehlin|k|faictors are now given by G =g g , Yj njX=0,1 kl kl lk   where the ordering in the product is arbitrary, since the ffaaccttoorriszaesr,eweveennoiwnphaavrteoannopaveeraratogreso.vSeirncaepAro(ψd˜u†,cRt a)salwsoe gkl=exp"−21Xαβ(φ¯αK,φαK)[h(kfll)]αβ(φβL,φ¯βL)T#, (10) wanted. The order of factors matters only for j K A with odd k¯ +k . The question is whether we can∈find suchanRj.jWepjresentherethesolutionofthisrepresen- where h(kfll) is a 2P ×2P block between partons at sites tationproblemfortheclassofstatesdescribedbyEq.(6). i and j of the flat band Hamiltonian associated to Θ . In that case, Rj = ψ˜j satisfies all the requirements, if χ The Grassmann variables φ¯αK,φαK live on the leg o|fik is chosen to satisfy χ χ∗ =1/S!. ending the link lk, etc. {αl} α1...αS α1...αS Expressions (8),(9) give a starting point from which To obtain the site factors, one makes the expansion P 1 ω(fj)exp"−2Xαβ(φ¯αj,φαj)[h(jflj)]αβ(φβj,φ¯βj)T#={παXj,π¯αj}Fj{παj,π¯αj}αY,incφ¯πα¯αjjφπααjj, (11) where we introduced the expansion coefficients F{πj,π¯j}. The product arranges the α-s in increasing order. A j straightforwardcalculation leads to T = F{παj,π¯αj} (1 2π )1−π¯αj( dφ )1−παj( dφ¯ )1−π¯αj. (12) j j − αj αJ αJ {παXj,π¯αj} αY,inc XJ∈j XJ∈j This tensor network for the states of the projective con- tensor network representation has struction has all the discussed advantages: it is formulated in the framework of an existing scheme, it is ex- gkl=Ye−Hkαlφ¯αKφαL, H1,2=(h(1fl))11, H3 =(h(mfl))11, α plicitly constructed, and most importantly, it is local. (13) Eqs. (2), (10) and (12) are the main results of the Let- (fl) where h is the flat band Hamiltonian obtained from ter. m the IQH Hamiltonian h realizing the state Ψ . The m m | i site factors for j / K are given by A ∈ We now use our results to present the link and site 1 factors for some concrete projective states. We first Tj=4(Yaαj+Xaαj), aαj=X dφαJdφ¯αK−Hjαj. (14) consider a fermionic FQH state Ψ on the lattice α α J,K∈j CF | i fromthecompositefermionserieswithHallconductance If the operator content of A on site j is ψ , we have j σ =m/(1+2m)(withm integer). Itis ageneralization 12 1 oΨf th-es bνy=Ψ1/3,satanteI,QoHbtsatianteedwbityhrσepla=cimng[3o4n,e35o]f. tIhtes Tj =−23/2 Y(Xdφ¯αJ). (15) 1 m 12 α J∈j | i | i 4 The other types of site factors can be found similarly. In the CF state, ψ˜ was a product of c-s. As pointed j outinRef.[9],calculatingaverageswithrespecttostates [1] F. Verstraeteand J. I. Cirac, cond-mat/0407066. with such ψ˜ is also possible by variational Monte Carlo j [2] F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. techniques, but these techniques fail once ψ˜j is a sum of Cirac, Phys.Rev.Lett. 96, 220601 (2006). products of c-s. It is thus important that our construc- [3] G. Vidal, Phys.Rev. Lett.99, 220405 (2007). tionincludesthelattercaseaswell. Asanillustrationwe [4] M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 consider(thelatticeversionof)thebosonicPfaffianstate. (2007). [5] P.CorbozandG.Vidal,Phys.Rev.B80,165129(2009). IntermsofEq.(5),thisstatehas Θ = Ψ Ψ Ψ Ψ and ψ˜j=c1jc4j−c3jc2j[22]. Its te|nsoir n|et1wio|rk1ir|ep1rie|sen1i- [6] CPh.yVs..RKerva.usA, N81.,S0c5h2u3c3h8, (F2.0V10e)r.straete, and J. I. Cirac, tation has [7] H.J.Changlani,J.M.Kinder,C.J.Umrigar,andG.K.- L. Chan, Phys. Rev.B 80, 245116 (2009). 4 gkl = Y e−Hkαlφ¯αKφαL, Hα =(h(1fl))11. (16) [8] PR.eCv.orAbo8z1,,G0.1E03v0e3nb(l2y0,1F0.)V. erstraete,andG.Vidal,Phys. α=1 [9] Z.-C.Gu,F.Verstraete,andX.-G.Wen,arXiv:1004.2563 The site factors, for example for j / K , are given by (2010). A ∈ [10] X. Wen,Int.J. Mod. Phys. B 4, 239 (1990). 1 [11] X. G. Wen and Q. Niu, Phys.Rev.B 41, 9377 (1990). Tj = [Y(aαj +1)+4(a1ja4j +1)(a2ja3j +1) [12] M. Aguado and G. Vidal, Phys. Rev. Lett. 100, 070404 16 α (2008). 4 X (dφ¯1Jdφ2Kdφ3Ldφ¯4M+dφ1Jdφ¯2Kdφ¯3Ldφ4M)]. [13] R. König, B. W. Reichardt, and G. Vidal, Phys. Rev. B − 79, 195123 (2009). J,K,L,M∈j [14] Z.-C. Gu, M. Levin, B. Swingle, and X.-G. Wen, Phys. Once the site and link factors T and G are known, Rev. B 79, 085118 (2009). j kl the calculation of ΨAΨ amounts to the evaluation of [15] O.Buerschaper,M.Aguado,andG.Vidal,Phys.Rev.B h | | i 79, 085119 (2009). the tensor trace Eq. (2). In tensor network approaches, [16] X.-G. Wen,Phys. Rev.D 68, 065003 (2003). evaluatingtensortracesisachallengebyitself,whichre- [17] M. A. Levin and X.-G. Wen, Phys. Rev. B 71, 045110 quires approximations. Depending on the concrete ten- (2005). sor network framework at hand, one can choose from [18] S. Iblisdir, J. I. Latorre, and R. Orus, Phys. Rev. Lett. existing approximation schemes to tackle this task[1–9]. 98, 060402 (2007). As Eq. (2) is formulated entirely in terms of the frame- [19] V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. 59, work of Ref. 9, it can be subjected to the approximation 2095 (1987). [20] X. G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B 39, schemedevelopedthere,theGrassmanntensorentangle- 11413 (1989). ment renormalization group. The concrete implementa- [21] J. K.Jain, Phys. Rev.B 40, 8079 (1989). tion of this renormalization procedure for our networks [22] X.-G. Wen,Phys. Rev.B 60, 8827 (1999). is however beyond the scope of this Letter. [23] S. Bravyi, Quantum Inf.and Comp. 5, 216 (2005). Inconclusion,wehavedemonstratedthatstronglycor- [24] The formula (1) follows from the results of Ref. 23 after related projective states can be represented using local a unitary changeof variables. Grassmanntensornetworks. Theresultsarevalidforthe [25] A. Kitaev, Ann.Phys.(N.Y.) 321, 2 (2006). wholescopeoftheprojectiveconstruction,whichinclude [26] In terms of the eigenvectors |ϕji of h one has FQH states, spin-liquids, fractional topological insula- h(fl) =1−2PEj<0|ϕjihϕj|. Alternatively, h(fl) can be approximated by a power expansion in h.[25, 28]. tors,etc. Onthe levelof principles, ourwork showsthat [27] J. Zinn-Justin, Quantum field theory and critical phe- strongly correlated quantum systems where the ground nomena (OUP,USA,2002). stateisclosetoaprojectivestatecanbemeaningfullyap- [28] Z. Ringel and Y.E. Kraus, arXiv:1010.5357 (2010). proachedusingtensornetworkbasedalgorithms. Froma [29] J. Maciejko, X.-L.Qi,A.Karch,and S.-C.Zhang,Phys. computationalpointofview,ourdevelopmentbringsthe Rev. Lett.105, 246809 (2010). prospects of variational simulations with projective trial [30] R. B. Laughlin, Phys.Rev.Lett. 50, 1395 (1983). states qualitatively closer: first, it breaks an exponen- [31] G. Moore and N.Read, Nucl.Phys. B360, 362 (1991). [32] N. Read and E. Rezayi, Phys.Rev.B 59, 8084 (1999). tial bound faced in previous efforts in representing such [33] M. Barkeshli and X.-G. Wen, Phys. Rev. B 81, 155302 states via tensor networks; and, second, it does so with- (2010). outgivingupontheexplicitnatureoftherepresentation. [34] J. K.Jain, Phys. Rev.Lett. 63, 199 (1989). This allows to use physically motivated trial wavefunc- [35] A. Kol and N. Read, Phys.Rev.B 48, 8890 (1993). tionsasstartingpoints,whichcouldspeedupvariational calculations dramatically. This work was supported by EPSRC Grant EP/F032773/1.