Scaling of Entanglement Entropy for the Heisenberg Model on Clusters Joined by Point Contacts
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Noname manuscript No. (will be inserted by the editor)
B. A. Friedman · G. C. Levine
Scaling of Entanglement Entropy forthe Heisenberg Model on ClustersJoined by Point Contacts
Possible Violation of the Area Law in DimensionsGreater than One
Received: date / Accepted: date
Abstract
The scaling of entanglement entropy for the nearest neighbor an-tiferromagnetic Heisenberg spin model is studied computationally for clustersjoined by a single bond. Bisecting the balanced three legged Bethe Cluster,gives a second Renyi entropy and the valence bond entropy which scales asthe number of sites in the cluster. For the analogous situation with squareclusters, i.e. two L × L clusters joined by a single bond, numerical resultssuggest that the second Renyi entropy and the valence bond entropy scalesas L . For both systems, the environment and the system are connected bythe single bond and interaction is short range. The entropy is not constantwith system size as suggested by the area law. Keywords
Entanglement entropy · Area law · Valence Bond Monte Carlo
This paper is a numerical investigation of entanglement entropy of the nearestneighbor isotropic Heisenberg spin 1/2 model on clusters, in particular, aBethe cluster and two L × L square clusters joined by a single bond. We are B. A. FriedmanDepartment of PhysicsSam Houston State UniversityHuntsville, Texas 77341-2267, USATel.: +936-294-1604Fax: +936-294-1585E-mail: phy [email protected]. C. LevineDepartment of Physics and AstronomyHofstra UniversityHempstead, New York 11549, USATel.:+516-463-5583E-mail: [email protected]
Fig. 1
14 site three branch Bethe cluster. studying the ground state quantum mechanical properties of these modelsand the coupling J is taken to be the same on every bond (including thesingle bond joining the clusters), that is the interaction is antiferromagnetic.Recall that for a large Bethe cluster and for the square lattice there is verygood numerical evidence that there is long range antiferromagnetic order(suitably defined for a Bethe cluster), though to the best of our knowledgethere is no proof. The numerical methods used are spin wave theory [1],direct diagonalization and valence bond Monte Carlo [2,3]. There is no signproblem associated with the models so Monte Carlo is an effective numericalmethod. The quantities calculated by valence bond Monte Carlo are thevalence bond entropy [4,5] and the n = 2 Renyi generalized entropy S [3].Both these quantities are straightforward (given the techniques in ref. [3] )to calculate by valence bond Monte Carlo. Note that S , the Von Neumannis not so easy to calculate, however, it is believed that the same essentialphysics is contained in S (but see ref. [6]). Generically, we will refer to allthese entropies as entanglement entropy. We only use ”balanced” clusters,where the number of even sites is equal to the number of odd sites, thus theground state has spin 0 [7] and the ground state can be represented by asuperposition of valence bond states [8,2].In particular, for the present study one of the clusters we shall consider isthe three branched bond centered Bethe clusters, see figure 1 for an illustra-tion of the 14 site 3 branched bond centered cluster [9]. As discussed in [9], ifyou bisect such a cluster, a simple argument shows, the valence bond entropymust scale as the number of sites N . In contrast, as we will later numeri-cally demonstrate, the spin-wave technique gives an entropy proportional tolog N . Because of this apparent contradiction it is important to use an un-biased numerical method, valence bond Monte Carlo, to calculate the Renyigeneralized entropy S . Due to recent advances in numerical techniques[3,11]such calculations can be done accurately for large clusters. A priori, however, one does not know if the clusters one can treat are large enough to see theasymptotic scaling law. One of the objects of the current investigation is todetermine if one can realize the asymptotic regime with existing numericaltechniques and hardware.Why are we interested in the scaling law for entanglement entropy?Firstly, if the entanglement entropy scales as ln N , DMRG (Density MatrixRenormalization Group) is an effective, unbiased method to calculate theground state properties. That is, one expects the number of states neededto describe the DMRG blocks to go as e S ≈ O ( N ) not say e N . A still out-standing issue in condensed matter physics is how Neel order is destroyed bythe addition of holes and ultimately is transformed into superconductivity.Since the Bethe cluster has Neel order at 1/2 filling [12,13] , the Hubbard ort-J model on the Bethe cluster would be an effective way to study this issueassuming the DMRG blocks scale with the number of sites, not the numberof states, in the blocks.Secondly, unlike in one dimension, the status of the area law for theentanglement entropy is not as clear [14]. What are the conditions on theHamiltonian and the cluster for the validity of the area law for a non onedimensional system? Naively, since a single bond connects the two halves ofthe Bethe cluster, one would expect either a constant or a logarithmic depen-dence of entanglement entropy on system size. However, for non interactingfermions, one sees N dependent entropy for the Bethe cluster and √ N orlog N (depending on boundary conditions) for square clusters separated by asingle bond [10,9]. Is this a pathological feature of non interacting systems?There have been a number of very interesting papers [15,16,17,18] whereexamples of models exhibiting large entanglement entropies or violating thearea law, are investigated. To best of our knowledge, the work developedin these papers does not apply directly to the particular systems we con-sider. Broadly speaking, it seems in the above papers, the lattice or clusteris straightforward to experimentally realize, while the Hamiltonian is diffi-cult to realize in a practical situation. For our models, the Hamiltonian isphysically realistic, however, a very large Bethe cluster is hard to realize inan experiment [19]. However, there is no such difficulty in realizing the two L × L cluster system. S and the second generalized en-tropy S from a spin-wave approximation we use the approach of [1] whichis easily applied to the Bethe Cluster. Let us consider initially a differentsituation from that considered in DMRG, namely, we take the subsystem in-side the system. As an example, consider in Figure 1, a three site subsystem,consisting of the sites labelled 1,2,3 inside the 14 site bond centered cluster.Figure 2 is the Von Neumann entropy for subsystems of size 3 to 63 for the254 and 510 site clusters. S Fig. 2 S vs. sites in the interior subsystem. The red circles are for the 254 sitecluster and the blue diamonds are for 510 sites. The curve is a linear fit to thepoints for the 510 site cluster. We see the Von Neumann entropy scales with the number of sites inthe interior cluster assuming one is sufficiently far from the boundary ofthe cluster. Note for the 254 site cluster the 63 site interior cluster is quitefar from the linear fit in figure 2 while for the 510 site cluster the 63 siteinterior cluster is on the linear fit. The linear scaling is consistent with whatone expect from the area law, since for an interior cluster, the number ofboundary points scales with the number of sites in the cluster.Let us now examine a situation of greater similarity to that encounteredin the blocking procedure in DMRG. Take a bond centered cluster (figure 1)and pick the subsystem to be the left half of the cluster.This is done in figure 3, the Von Neumann entropy and S are plottedvs. the logarithm of the system size. We see both quantities scale as thelogarithm of the system size, similar to a one dimensional system. However,as previously mentioned, by the argument of ref. [9], the valence bond entropyscales as the number of sites. Naively, one would expect, say the valence bondentropy and S (or S ) to scale the same way with system size ( at least up tologarithms [20]). Thus either the valence bond entropy scales differently fromother entropies or the spin wave calculation gives an incorrect result. In figure3, the green squares refer to exact diagonalization results for S for systemsizes 6, 14, and 30. The system sizes accessible to direct diagonalization (the30 site cluster has state space of dimension approximately 1 . × ) are toosmall to infer the scaling of S with system size. Nonetheless, some insightcan be gained from plotting, in figure 4, the entanglement entropy for the30 site cluster vs. the number of sites in the interior subsystem. Perhapsnot surprisingly, the entanglement entropy increases as the number of bonds en t ang l e m en t en t r op y Fig. 3
Entanglement Entropy vs. ln(cluster size) for subsystems bisecting the sys-tem. The red circles are for S while the blue diamonds are for S . The greensquares are exact diagonalization results for S . connecting the subsystem to the system increases (up to 7 sites), after whichthe number of connections decreases and the entropy decreases.2.2 Valence Bond Monte CarloWe thus turn to valence bond Monte Carlo [2] as a method to compute,essentially exactly, the properties of the Heisenberg model on a Bethe Cluster.In figure 5, the valence bond entropy is plotted vs. system size for a bisectedsystem. The valence bond entropy [4,5] is a natural quantity to computewith valence bond Monte Carlo, as the basis consists of valence bonds andthe valence bond entropy counts the number of valence bonds leaving thesubsystem. From the figure, one sees that the valence bond entropy scaleswith system size. Of course, given the argument in [9], this is no surprise.Further insight can be obtained by looking at the value of the valence bondentropy. By the argument of [9], at least 1/3 of the valence bonds (for 1/2the cluster, i.e. for a 1022 site cluster, roughly 170 bonds) must connect thetwo halves of the cluster; from figure 3, we see for large clusters, very closeto 1/3 of the bonds connect the two halves.Let us now consider the n = 2 Renyi generalized entropy S . Due torecent advances in computational technique this quantity can be calculatedwith valence bond Monte Carlo. We apply the methods developed in ref. [3],see also [11]; S ( ρ A ) is calculated as − ln ( h Swap A i ) where A is the subsystemand Swap A is a swap operator. h Swap A i is calculated in the simplest formu- en t ang l e m en t en t r op y Fig. 4
Entanglement Entropy vs. number of sites in the interior subsystem for a30 site cluster. The red circles are for S while the green squares are for S . Thepoints were calculated by exact diagonalization. S v b / l n2 Fig. 5
Valence Bond Entropy vs. system size for a bisected system. S Fig. 6 S vs. sites in the subsystem for a 254 site cluster. The red crosses arecalculated from the r = 1 ratio method, while the blue diamonds use the shelltechnique. Statistical errors are smaller than the symbols in the figure. lation by a double projection Monte Carlo algorithm. In a more sophisticatedapproach the ratios h Swap A i+r ih Swap A i i (1)are computed from Monte Carlo; from these ratios h Swap A i is then cal-culated. Here r + i is a symbolic notation for a subsystem bigger than thesubsystem i . For system sizes 6,14, 30 and 62 sites we have used three ”differ-ent” approaches: the naive (no ratio) approach, the sophisticated approachwhere i + r consists of the next shell in the Bethe cluster and the brute force”sophisticated” approach where ”r” is only one site. Recall there is a shellor layer structure for Bethe clusters, i.e. for the 30 site cluster (take 1/2 thecluster) the first layer has 1 site, layer 2 has 2 sites, layer 3 has 4 sites, layer 4has 8 sites. All three approaches agree, to within the statistical errors. Exactdiagonalization results for 6,14 and 30 site clusters are also in agreement withthe Monte Carlo calculations.In figure 6, we plot S vs sites in the subsystem for a 254 site cluster. Thesubsystem is taken to be an interior subsystem as in figure 4. The red crossesare calculated from the r=1 ratio method, while the blue diamonds use theshell technique. Statistical errors are smaller than the symbols in the figure.We see that S appears to be a continuous piece wise linear function of thenumber of sites in the subsystem. It is linear within the shell with a kink ingoing from one shell to another. The kink (discontinuity in the derivative) issmaller in the shells near the center of the cluster. The decrease in entropy S Fig. 7 S vs. sites in the subsystem. Green squares are for 126 site system, redcircles are for 254 site system and blue diamonds are for 510 site system. Statisticalerrors are smaller than the symbols in the figure. for the final shell is presumably caused by the spins on the boundary of thecluster that are no longer connected by bonds to spins outside the subsystem.We next consider a situation similar to figure 2. In figure 7, we takean interior subsystem and calculate S via valence bond Monte Carlo forsystems of size 126 sites , green squares, 254 sites, red circles and 510 sites,blue diamonds. Error bars would be smaller than the symbols in the figure.It appears that S scales with number of sites in the interior subsystem for asufficiently small subsystem relative to the system as expected from the arealaw.Finally in figure 8, we plot S vs. system sizes for subsystems that consistof half the system. The blue diamonds are the spin wave results, while thered circles are calculated by valence bond Monte Carlo, r=1. The green crossis for the shell valence Monte Carlo method but breaking up the large shellsto 32 sites. For system sizes less than or equal to 62, at least qualitatively,Monte Carlo and spin wave agrees; however as the system sizes grow pastthis point there is an increasing discrepancy. It appears the spin wave resultincreases logarithmically with the number of sites while the Monte Carloresults give linear dependence on the number of sites in the system. Doesthis crossover with system size make sense?One can present a simple minded argument that rationalizes this behav-ior. The relevant ”aspect” ratio α is the ratio of the number of generationsto the number of boundary points, for example for the 6 site cluster α = ,for 14 sites α = for 30 sites α = etc. . One would expect α must be small S Fig. 8 S vs. sites in the system for subsystems bisecting the system. The bluediamonds are the spin wave results, while the red circles are calculated by valencebond Monte Carlo, r = 1. The green cross is for the shell method but breaking upthe large shells to 32 sites. to see Bethe cluster rather than 1 dimensional behavior. Hence one wouldanticipate one needs to study system sizes substantially larger than 30 sites.A heuristic argument can be made for the magnitude of the slope of S vs. system size. Recall that S = − ln h φ | Swap A | φ i where φ = | ψ i| ψ i i.e. thereplicated ground state. Note that | ψ i = P α f α | α i where | α i is a valencebond state and f α ≥
0. Then the normalization1 = h φ | φ i = X α,β,α ′ ,β ′ f α f β f α ′ f β ′ h β ′ |h α ′ | α i| β i (2)and h φ | Swap A | φ i = (3) X α,β,α ′ ,β ′ f α f β f α ′ f β ′ h β ′ |h α ′ | Swap A | α i| β i (4)All the valence bond states have at least N/ N sites in A and an equal numberin B are connected by valence bonds in both | α i and | β i . Assume states h β ′ | and h α ′ | with the same valence bonds dominate the normalization andthe expectation value of Swap A . The term h β ′ |h α ′ | Swap A | α i| β i is reduced incomparison to h β ′ |h α ′ | α i| β i by a factor of 2 − N due to these bonds. Hence S ≈ − ln 2 − N = N
54 ln 2 (5) Fig. 9 i.e. S is proportional to N with a slope of about 1/78. This is comparisonto the numerical results in figure 8 which give a slope of roughly . As in [9] , consider two L × L clusters linked by a single bond. Here the bondis a spin-spin interaction of strength J , the same strength as all the otherspin-spin interactions. The single bond is chosen to be as close to the middleof the side as possible. For L even (odd) this bond joins the L/ L + 1) / L even, there exists valence bond patterns whereno valence bond need to go from one L × L cluster to the other. Lastly, fornon interacting fermions, at 1/2 filling, the entanglement entropy scales as L , not as a constant or ln L . This is consistent with the Bethe cluster results,both for free fermions and the Heisenberg model, in that the ”perimeter” ofthe model (the sites with only one bond) scales as the number of sites in thecluster.Figure 10 is a plot of the second Renyi entropy S vs. L for L even takingthe subsystem as half the cluster calculated using valence bond Monte Carlo.The values of S are noticeably smaller than for comparably sized BetheClusters. However, S does not appear to be a constant or depend on ln L .The calculation appears, to within the large statistical errors and the limitedsystem sizes, to be consistent again with S scaling as L . To make the casefor L more convincing, figure 11 is a plot of valence bond entropy vs. L forvarious system sizes and L even. One sees a rather clear indication of thevalence bond entropy scaling with L . The same quantities are plotted for L odd in figure 12. Again one sees the valence bond entropy scaling as L . The L S Fig. 10 S vs L for L × L - L × L clusters, L even L S v b / l n ( ) Fig. 11
Valence Bond Entropy vs. system size for a bisected system, L even. Errorbars are smaller than the data symbols. structure in the graph can be rationalized by noting that for L = 2 n + 1 n even, there is a valence bond pattern with only one valence bond joiningthe two L × L squares and that bond can be chosen to be the middle bond(where the interaction joining the squares is located). Numerical evidence has been presented that for a Heisenberg model on clus-ters joined by a single interaction, the entanglement entropy scales as theperimeter, not as a constant, as suggested by the area law. This is consis-tent with work on non interacting fermions and demonstrates these earlier L S v b / l n ( ) Fig. 12