Frustration, Area Law, and Interference in Quantum Spin Models
Aditi Sen De, Ujjwal Sen, Jacek Dziarmaga, Anna Sanpera, Maciej Lewenstein
aa r X i v : . [ qu a n t - ph ] O c t Frustration, Area Law, and Interference in Quantum Spin Models
Aditi Sen(De) ⋆ , Ujjwal Sen ⋆ , Jacek Dziarmaga ♮ , Anna Sanpera † , ‡ , and Maciej Lewenstein ⋆, † ⋆ ICFO-Institut de Ci`encies Fot`oniques, 08860 Castelldefels (Barcelona), Spain ♮ Institute of Physics and Centre for Complex Systems, Jagiellonian University, 30-059 Krak´ow, Poland † ICREA-Instituci´o Catalana de Recerca i Estudis Avan¸cats, Lluis Companys 23, 08010 Barcelona, Spain ‡ Grup de F´ısica Te`orica, Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Spain
We study frustrated quantum systems from a quantum information perspective. Within thisapproach, we find that highly frustrated systems do not follow any general “area law” of blockentanglement, while weakly frustrated ones have area laws similar to those of nonfrustrated systemsaway from criticality. To calculate the block entanglement in systems with degenerate ground states,typical in frustrated systems, we define a “cooling” procedure of the ground state manifold, andpropose a frustration degree and a method to quantify constructive and destructive interferenceeffects of entanglement.
The study of entanglement in spin lattice models haveprovided a novel insight into the complexity of magneticordering and quantum phase transitions in a large vari-ety of models [1]. In particular, in bipartite partitionsof certain many-body systems, the block entanglement(BE) evaluated in the (gapped) ground state (GS) of thesystem, is proportional to the area of the block boundary.More complex relations arise at criticality (non-gappedsystems). These generic relations between entanglementand area are known under the name of area laws [2].There are several intriguing questions concerning theserelations [3], with respect to systems with unique groundstates. Within this context, one may ask whether thescaling of entanglement can also help to characterize frus-trated many body systems [4]. Typically, frustrated sys-tems exhibit a large GS degeneracy, or quasi-degeneracy,and a rich phase diagram [5]. Frustration in spin systemsarise when the GS spin configuration does not simulta-neously satisfy all the constraints imposed by the Hamil-tonian; it may be caused by the lattice geometry (as inIsing antiferromagnets (AF) on a triangular lattice), orby the presence of disorder (as in spin glasses). Recently,the relation of frustration to high- T c superconductivity,as well as the discovery of “exotic” frustrated phases hastriggered a renewed interest in the subject [5, 6, 7]. Ul-tracold atomic gases offer unprecedented ways to controlmany-body systems, and provide a perfect playgroundto study disordered and frustrated systems. In partic-ular, this should allow for experimental study of effectsdiscussed in this paper (cf. [8], see [9] for a review).The main thesis of this paper is that highly frustratedsystems do not follow any general area law, while weaklyfrustrated systems have area laws similar to those ofnonfrustrated systems (away from criticality). We intro-duce three tools to characterize frustrated systems anddeal with the above thesis: (i) a paradigm of a certain“cooled state” for finding area laws for systems whichhave a degenerate ground state manifold, including frus-trated systems, (ii) a “frustration degree” for quantifyingfrustration in a system, and (iii) a method for quantify-ing constructive and destructive interference of entan- glement. Using these tools, we prove the above thesisfor six prototype frustrated spin models, each having adifferent type of frustration. More specifically, we showthat for the Ising model with long-range (LR) AF in-teractions (1) , and the LR AF Heisenberg model (2) ,with frustration degrees close to unity, there is a com-plete departure from the usual area law for nonfrustratedsystems. However, when we consider systems whose frus-tration degree is much smaller, we find areas laws similarto those of non-frustrated systems (away from critical-ity), i.e. the entropy of a block of characteristic length L , in a system of N particles in dimension D , scales as L D − , when L D ≪ N . Among this class, we have an-alyzed a two-dimensional (2D) J − J − J Heisenbergmodel with resonating valence bond (RVB) states as GSs (3) , the Shastry-Sutherland model (in 2D) (4) [10], theone-dimensional (1D) J − J Heisenberg model at theMajumdar-Ghosh (MG) point (5) [11], and finally anIsing chain with a single disordered interaction (6) . Cooling procedure.
To treat the problem of degener-acy of the GS manifold and calculate bipartite entangle-ment on this manifold, we introduce a “cooling” method.Suppose that a system, made up of several subsystems(e.g. spins), is described by the Hamiltonian H . Wechoose some suitable initial product state | Ψ i = Q i | ψ i i (where i runs over all subsystems) and “cool” | Ψ i tobelow a desired energy level E , by projecting | Ψ i ontothe subspace that is spanned by all energy eigenstates of H whose energies are below E . This quenching methodis a caricature of evaporative cooling of trapped atomicgases, where atoms having energy above a certain valueare removed from the trap. So, the resulting state is | Φ E i = (1 / √ Z ) P [ ≤ E ] | Ψ i , (1)where P [ ≤ E ] = P E i ≤ E P i [ E i ], with P i [ E i ] being theprojector onto the eigenstates of energy E i , and Z = h Ψ | P [ ≤ E ] | Ψ i . For an N -spin state | Φ E i , we char-acterize the entanglement between k spins and the restof the system by the von Neumann entropy E k : N − k = S (tr k | Φ E ih Φ E | ) – the unique asymptotic measure of en-tanglement for pure states in bipartite splittings [12]. Westudy the scaling of BE E k : N − k , for large N , where theinitial state | Ψ i is chosen to maximize E N/ N/ [13]. Frustration degree.
Before proceeding further, we de-fine a frustration degree ( F ) for quantum spin mod-els. Frustration is a classical concept, and thus con-nected to the classical configurations of the quantumsystem. Given a Hamiltonian H , and a correspondingground state |Gi , we replace the one-body, two-body, . . . terms in H by the corresponding Ising ones (i.e. σ zi , σ zi σ zj , etc.) to obtain H I , discarding any constantterm. Here σ α ( α = x, y, z ) are the Pauli operators, and ~σ = ( σ x , σ y , σ z ). We find the terms H kf ( H lnf ) of H I that gives rise to positive (negative or zero) energies in |Gi . For a given |Gi , H I = P k H kf + P l H lnf . We definethe frustration degree of the system described by H as F = Av P k hG|H kf |Gi P l |hG|H lnf |Gi| , where the average (Av) is takenover all ground states of H . Note that the denominatorcannot vanish. In the case of isotropic Heisenberg ( ~σ i · ~σ j )and XY ( σ xi σ xj + σ yi σ yj ) interactions, to find F , one mayreplace ~σ i and ( σ xi , σ yi ) by SU(2) and SO(2) symmet-ric classical vectors, of appropriate lengths, respectively.With these concepts at hand, we move to study the dif-ferent cases. Case 1: Ising model with AF LR interactions:Ising gas.
The governing Hamiltonian, for 2 m spins, is H IsingLR ( λ ) = ( J/ m )( S − mλ ) ≡ ( J/ m )(2 P mi,j =1( i>j ) σ zi σ zj − mλ P mi =1 σ zi + const.),where S = P i σ zi , 0 ≤ λ ≤
1. In the frustrated case(
J > F = (1 + 2 λ − λ − /m ) / (1 + λ ) . Forlarge systems, with λ = 0, F ≈
1. For the initial state | Ψ I i = Q i ( α | i + β | i ) i , with αβ = 0 (with | i and | i being the eigenstates of σ z with eigenvalues ±
1, respec-tively), the cooled state (which is independent of α, β ) is | Φ IsingLR ( λ ) i = superposition of all states with m (1 + λ ) | i ’s and m (1 − λ ) | i ’s, where 2 mλ is a positive integer ≤ m . For arbitrary 2 m and magnetization λ , and forany bipartite splitting k : 2 m − k , the reduced densitymatrix can be found analytically: for even k < m (1 − λ ), ρ k = k X i =0 [ k, i ][2 m − k, mµ − i ] | W k − i i h W k − i | / [2 m, mµ ] , where µ = λ + 1, [ a, b ] = a !( a − b )! b ! , and | W j i denotes thenormalized equal superposition state of j | i ’s and rest | i ’s, i.e. the (generalized) “W” state. Any other par-tition can be found similarly. For a given k , E k :2 m − k increases (and converges to a constant) with 2 m . Fora fixed m , using Stirling approximation, E k :2 m − k ≈− P ki =0 e i log e i , with e i = k (1 + λ ) r (1 − λ ) k − r [ k, i ],for large m and k , such that k ≪ m . In the same limit, E k :2 m − k ≈ log (cid:2) (1 − λ ) k (cid:3) . The leading term to thescaling is log k , independently of λ – a logarithmic di-vergence (LD) of BE for all frustrated AF LR Ising gases.Since this system is effectively in “infinite” dimensions,there is a clear departure from any “area law” (which could be k − /D with D → ∞ ). Similar divergence isalso obtained even if the quenching is performed over afew low lying energy manifolds. In contrast, for the ferro-magnetic ( J <
0) nonfrustrated LR Ising model (say, at λ = 0), the state after cooling is ( | ⊗ m i + | ⊗ m i ) / √ E k :2 m − k =1, irrespective of k and 2 m . Case 2: Heisenberg model with AF LR interactions:Heisenberg gas.
The Hamiltonian, for 2 m spins, is H H LR =( J/ m ) P mi,j =1 ~σ i · ~σ j ( J > m spins as “black”(call them B ) and the remaining as “white” (call them W ), and considering all possible “coverings” by singlets(valence bonds) from black to white spins. Given thata spin-1/2 particle has only two orthogonal states, anupper bound for BE in the cooled state, for an initialstate with two states, is a reasonable approximate up-per bound. Cooling the initial state (cid:12)(cid:12) Ψ H (cid:11) = Q i ∈B (cid:12)(cid:12) ψ B (cid:11) i Q i ∈W (cid:12)(cid:12) ψ W (cid:11) i , we obtain (cid:12)(cid:12) Φ H LR (cid:11) , which is an RVB formedby an equal superposition of all the singlet coverings.Consider a “system” S of k ≪ m spins (containing,say, b black spins and w white ones), and the remain-ing “environment” E . The orthonormal states of S in a S : E Schmidt decomposition must be fully symmetricunder independent permutations in the subsets of blackand white spins. So the basis of symmetric spin multi-plets | b/ , m b i B∩S | w/ , m w i W∩S will span them, where m c = − c , ..., c , c = b, w , so that there are ( b + 1)( w + 1)symmetric basis states. The Schmidt decomposition can-not have more terms than the number of symmetric basisstates, so that E k :2 m − k ≤ log ( b + 1)( w + 1). No matterwhat is b or w , any area law of the form E ∼ k − /D forany D (including D = ∞ ) is out of question. In the casewhen w = k, b = 0 (or w = 0 , b = k ), the cooled state is k/ X M = − k/ | k/ , M i B∩S m/ X p = − m/ ( − ( p − m/ C m,kp,M / √ m + 1 ×| ( m − k ) / , p − M i B∩E | m/ , − p i W , where ( C m,kp,M ) = [ k, M + k/ m − k, p − M + ( m − k ) / / [ m, p + m/
2] defines the Clebsch-Gordon coefficient( C m,kp,M = 0 unless 0 ≤ b ≤ a in all [ a, b ] involved), whenceone can show, with some algebra, that the upper estimateis saturated. Case 3: 2D J − J − J Heisenberg model with RVBground states.
Consider a (2D) square lattice of size2 m × m with periodic boundary conditions (PBC), eachsite being occupied by a spin-1/2 particle. It is governedby a two-body J − J − J AF Heisenberg Hamilto-nian, with exchange constants J for nearest neighbors(NN) and on diagonals in the plaquettes, J for inter-plaquette NNs and diagonals, and J for next-nearestneighbors (NNN) and knight’s-move-distance-away spinson horizontal and vertical ladders formed by the plaque-ttes [14, 15]. A plaquette is a square of four neighboringspins. Realization of such a “quadrumerized” lattice isbeing hotly pursued in current experiments with atoms inoptical lattices (see [16]). For certain choices of the rela-tive strengths of the couplings, it is reasonable to assume[14, 15] that the GS configuration is formed by eithertwo horitzontal singlets | HH i or two vertical ones | V V i in each of the plaquettes, with a fixed density, d = s/m ,of s vertical singlets in the whole state of m plaquettes.A superposition of such states is an example of an RVBstate, whereas ordered configurations of dimers (singlets)correspond to valence bond solids (Peierls order). If d isnot too close to 0 or 1, the dimension of the GS subspacegrows exponentially with system size: for d = 1 /
2, the de-generacy scales as 2 m . Note that in contrast to Cases 1 and , the model here has rather “medium”-range inter-actions, and the frustration degree will correspondinglybe smaller. For instance, for J = J = 0, F = 1 / E p : rest , for p plaquettes . Forthe initial state | Ψ H i = Π m p =1 ( α | HH i + β | V V i ) p (unnor-malized, and αβ = 0), the cooled state (which is inde-pendent of α , β ) is | Φ dRV B i = min( s,k ) X l =0 min( s − l,m − k ) X r =0 | l i S q l [ k, l ] ×| r i E p r [ m − k, r ][ m − l − r, s − l − r ] , where | l i S ( | r i E ) are orthonormal states of a “system” S (“environment” E ) of k ( m − k ) plaquettes. When m → ∞ , the upper limit min( s − l, m − k ) is l -independent, and by Stirling approximation, ln[ m − t, s − t ] ≈ ( m − t ) ln( m − t ) − ( s − t ) ln( s − t ) plus functionsof m and s , where t = l + r . Linearizing the logarithmaround the mean t of t (in | ψ dRV B i ), and exponentiatingback, we obtain a product: [ m − t, s − t ] ≈ q l q r , with q = ( s − t ) / ( m − t ) ≈ ( − p d (1 − d )) / (6(1 − d )).The approximations are systematic, as t ≤ √ k/
2, anddispersion of t ∼ √ t . So the states of E and S canbe well approximated as product states, which furtherimplies that | Φ dRV B i , for any d , can be approximatedby product over plaquettes, where each plaquette is inthe same pure state – a “mean-field picture” is valid,for large m . Consequently, the entanglement of a sys-tem S of size k , plus a boundary B that intersects∆ plaquettes, to the rest of the lattice, will scale as hE pl ( d ) + vE pl (1 − d ), where the boundary intersects h ( v ) plaquettes horizontally (vertically), h + v = ∆, and E pl ( d ) = log (cid:0) q (cid:1) − q q log q .For the most general initial state, | Ψ Hs i = Q m p =1 ( | ψ i| ψ i| ψ i| ψ i ) p , denoting P S =0 (cid:3) ,p for the pro-jector onto the space spanned by {| HH i , | V V i} at the p th plaquette, and {| G i i} for the groundstates, the cooled state is ( P i | G i ih G i | ) | Ψ Hs i =( P i | G i ih G i | ) Q m p =1 P S =0 (cid:3) ,p | Ψ Hs i = ( P i | G i ih G i | ) | Ψ H i ,so that the same area law holds. Surprisingly therefore,the area law depends on the path of the boundary (and not only on its length): it depends on h , even for fixed∆. However, the area law is linear in this 2D system, andsupports our thesis for weakly frustrated systems. Case 4: Shastry-Sutherland model.
It is a 2D NN an-tiferromagnet (with Heisenberg couplings of strength J )on a square lattice ( i, j ), with additional Heisenberg in-teractions of strength J on the diagonals (2 i, j ) ↔ (2 i +1 , j +1) and (2 i, j +1) ↔ (2 i − , j +2), with PBC. Theground state of this model, for J /J < /
2, is a productof dimers along the J diagonals [10]. For J /J < / F ≈ / (1 + (1 / J /J )) < / Case 5: 1D J - J Heisenberg model.
Consider a 1Dsystem with PBC and 2 m sites, having NN and NNNHeisenberg couplings of strengths J and J respectively.For J = 2 J > | G ± MG i = Q mi =1 ( | i i | i i ± − | i i | i i ± ) / √
2. At theMG point, F = 1 /
2. Note however that if we replace ~σ i by a 3D classical vector, the frustration degree is zero.Using the initial state Q m − i =1 | i i − | i i | φ i m − | φ i m ,where | φ i is an arbitrary (qubit) state, we obtain a lowerbound : E k : rest ≥ k . Any cooledstate is of the form | Φ MG i = a | G + MG i + b | G − MG i , whence,after writing in Schmidt decompostion, an upper boundreads: E k : rest ≤ log
3, for even or odd k . Nu-merical simulations at the MG point, show that e.g. for2 m = 8, E rest ≈ .
314 (cf. [17]), and BE after coolingconverges with k . Therefore, the area law is a constantfor this 1D system, in support of our thesis. Note that theabove method of finding bounds can potentially be usedin other models whose GS space is made up of dimers. Case 6: Ising spin chain with NN interactions.
The Hamiltonian, of 2 m spins (with PBC), is H IsingNN = P h ij i J ij σ zi σ zj , with all | J ij | = J , andall except one are negative. F = 1 / (2 m − | Ψ NN i = Q i (( | i + | i ) / √ i (cf. [18]), gives the cooled state | Φ IsingNN i = √ m P mk =0 [( | ⊗ (2 m − k ) ⊗ k i + | ⊗ (2 m − k ) ⊗ k i ) +( | ⊗ ( k +1) ⊗ (2 m − k − i + | ⊗ ( k +1) ⊗ (2 m − k − i )]. Forany k , E k :2 m − k decreases with 2 m , contrary to previouscases. Moreover, E k :2 m − k , for a fixed 2 m is a constantwith k – in support of our thesis. Constructive and destructive interferences.
The abovestudies allows us to identify an interesting interplay be-tween frustration and interference of entanglement. Be-ginning with
Case 3 , for a fixed density d , the latticeis in the state | ψ dRV B i , which is an equal superpositionover all states that have s vertical plaquettes and m − s horizontal ones. If we choose one of the states in thissuperposition, calculate the entropy of the inner area( S ∪ B ) in the chosen state, and then average this en-tropy over all states in the superposition, the averageentropy is E ( d ) = 2 hd + 2 v (1 − d ) . For a square B , Density d
Density d
FIG. 1: Constructive and destructive interference. (Left)
E/E for a square B , so that v = h . (Right) E/E for a horizon-tal (vertical) boundary, so that v = 0 ( h = 0). The verticalaxes are of E/E . the interference is destructive (i.e. E/E <
1) for lowand high densities, but otherwise constructive, and for apurely horizontal (or vertical) B , interference can wipeout the entanglement completely, at low densities (Fig.1). For a square B , E/E is maximal for d = 1 /
2. Cor-respondingly, in the AF LR Ising gas, the highest BEscaling is for λ = 0, when there are an equal number | i sand | i s in the components of the cooled state, and F isalso maximal exactly at λ = 0. We predict that a parentHamiltonian that describes the ground states in Case 3 for fixed density, will be maximally frustrated at d = 1 / Case 1 ),we normalize the entanglement scaling in the frustratedsystem, i.e.
J >
0, with the one in the nonfrustrated one(
J < log k , for large blocks ofsize k , large system size 2 m ( ≫ k ), and for all λ . Thus weobtain constructive interference of BE for all λ , with re-spect to the nonfrustrated situation. For the MG model,let us once again choose any one of the dimer groundstates, find the entanglement in a bipartite split, and av-erage over the two ground states. The result is unity, forany split, so that we again have constructive interferenceof entanglement. For the system described in Case 6 , wehave marginally constructive interference, compared tothe nonfrustrated case (all J ij < Case 2 , we have a rather remarkable example of de-structive interference, because when 2 m ≫ k , E k : rest ∼ k for most elements in the superposition forming the cooledstate, while the latter has only logarithmic scaling atmost. Superposition can therefore give rise to qualita-tive changes in scaling. Summary.
Firstly, our studies show that the dimensionof the GS manifold does not provide a “good” character-ization of frustration. This can be seen by comparing the J − J model at the MG point with the 1D Ising modelwith a single frustrated bond. Secondly, we found thattrying to confer an “area law” on a frustrated system canhave surprising consequences, such as logarithmic diver-gence of BE in an effectively infinite dimensional system,and dependence of BE on the shape (and not only the Eu-clidean area) of the boundary. Interestingly, the seeming independence of area law on dimension, in the long-rangeIsing model, gives rise to the possibility of applying den-sity matrix renormalization group techniques to complexsystems with long-range interactions (cf. [19]). Our re-sults indicate that while weakly frustrated systems followthe usual area law known in the literature, strongly frus-trated systems will each have their own area law. Finally,we have introduced a cooling procedure to study BE indegenerate ground state manifolds, a frustration degree,and a method to quantify constructive and destructiveinterference of entanglement.We acknowledge comments of H.-U. Everts, R. Fazio,T. Vekua, and support of research project of PolishGovernment scientific funds (2005-08), ESF QUDEDIS,Marie Curie ATK project COCOS (MTKD-CT-2004-517186), Spanish MEC (FIS-2005-04627/01368, Con-solider Project QOIT, & Ram´on y Cajal), & EU IPSCALA. [1] L. Amico et al. , quant-ph/0703044.[2] G. Vidal et al. , Phys. Rev. Lett. , 227902 (2003); G.Refael and J.E. Moore, ibid. , 260602 (2004); V.E. Ko-repin, ibid. , 096402 (2004); K. Audenaert et al. , Phys.Rev. A , 042327 (2002); B.-Q. Jin and V.E. Korepin, ibid. , 062314 (2004); B.-Q. Jin and V.E. Korepin, J.Stat. Phys. , 79 (2004); P. Calabrese and J. Cardy,J. Stat. Mech. P06002 (2004); I. Peschel, ibid. P12005(2004); A.R. Its et al. , J. Phys. A , 2975 (2005); M.Cramer and J. Eisert, New J. Phys. , 71 (2006); A. Ki-taev and J. Preskill, Phys. Rev. Lett. , 110404 (2006);A. Riera and J.I. Latorre, quant-ph/0605112.[3] J.I. Latorre et al. , Phys. Rev. A , 064101 (2005);M.M. Wolf, Phys. Rev. Lett. , 010404 (2006); D.Gioev and I. Klich, ibid. etal. , cond-mat/0602077; W. Li et al. , quant-ph/0602094;T. Barthel et al. , Phys. Rev. Lett. , 220402 (2006); J.Vidal et al. , J. Stat. Mech. P01015 (2007).[4] This question was recently adressed for resonating va-lence bond states in F. Alet et al. , cond-mat/0703027.[5] G. Misguich and C. Lhuillier, cond-mat/0310405;C. Lhuillier, cond-mat/0502464; F. Alet et al. ,cond-mat/0511516.[6] M. Rasolt and Z. Tesanovi´c, Rev. Mod. Phys. , 709(1992); M. Sigrist and T.M. Rice, ibid. , 503 (1995).[7] R. Melzi et al. , Phys. Rev. Lett. , 1318 (2000).[8] J. Billy et al. , arXiv:0804.1621; G. Roati et al. ,arXiv:0804.2609.[9] M. Lewenstein et al. , Adv. in Phys. , 243 (2007).[10] B.S. Shastry and B. Sutherland, Physica (Amsterdam) , 1069 (1981).[11] C.K. Majumdar and D.K. Ghosh, J. Math. Phys. ,1388 (1969).[12] C.H. Bennett et al. , Phys. Rev. A , 2046 (1996).[13] If the scaling is not unique after the maximization, wemaximize entanglement in the partition ( N/ − N/
2) + 1, etc., until a unique scaling is obtained.[14] B. Kumar, Phys. Rev. B , 024406 (2002).[15] M. Mambrini et al. , Phys. Rev. B , 144422 (2006). [16] B. Paredes and I. Bloch, arXiv:0711.3796.[17] R.W. Chhajlany et al. , cond-mat/0610379.[18] H.J. Briegel and R. Raussendorf, Phys. Rev. Lett. ,910 (2001). [19] J. Rodr´ıguez-Laguna and G.E. Santoro,cond-mat/0610661; A. Ferraro et al.et al.