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arXiv:1503.07127 · Bulletin of the American Physical Society

A Numerical Study of Entanglement Entropy of the Heisenberg Model on a Bethe Cluster

Barry Friedman, Greg Levine

Published Mar 24, 2015• 0 citations
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Abstract

Numerical evidence is presented for a nearest neighbor Heisenberg spin model on a Bethe cluster, that by bisecting the cluster, the generalized Renyi entropy scales as the number of sites in the cluster. This disagrees with spin wave calculations and a naive application of the area law but agrees with previous results for non interacting fermions on the Bethe cluster. It seems this scaling is not an artifact of non interacting particles. As a consequence, the area law in greater then one dimension is more subtle then generally thought and applications of the density matrix renormalization group to Bethe clusters face difficulties at least as a matter of principle.

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Numerical evidence is presented for a nearest neighbor Heisenberg spin model on a Bethe cluster, that by bisecting the cluster, the generalized Renyi entropy scales as the number of sites in the cluster. This disagrees with spin wave calculations and a naive application of the area law but agrees with previous results for non interacting fermions on the Bethe cluster. It seems this scaling is not an artifact of non interacting particles. As a consequence, the area law in greater then one dimension is more subtle then generally thought and applications of the density matrix renormalization group to Bethe clusters face difficulties at least as a matter of principle.
The ground state of an antiferromagnetic Heisenberg model on L X L clusters joined by a single bond and balanced Bethe clusters are investigated with quantum Monte Carlo and mean field theory. The improved Monte Carlo method of Sandvik and Evertz is used and the observables include valence bond and loop valence bond observables introduced by Lin and Sandvik as well as the valence bond entropy and the second Renyi entropy. For the bisecting of the Bethe cluster, in disagreement with our previous results and in agreement with mean field theory, the valence loop entropy and the second Renyi entropy scale as the log of the number of sites in the cluster. For bisecting the L X L - L X L clusters, all the observables appear to scale as L in agreement with our previous results. This is in disagreement with a straight forward application of the area law.
The ground state of an antiferromagnetic Heisenberg model on L X L clusters joined by a single bond and balanced Bethe clusters are investigated with quantum Monte Carlo and modified spin-wave theory. The improved Monte Carlo method of Sandvik and Evertz is used and the observables include valence bond and loop valence bond observables introduced by Lin and Sandvik as well as the valence bond entropy and the second Renyi entropy. For the bisecting of the Bethe cluster, in disagreement with our previous results and in agreement with modified spin-wave theory, the valence loop entropy and the second Renyi entropy scale as the logarithm of the number of sites in the cluster. For bisecting the L X L - L X L clusters, the valence bond entropy scales as L, however, the loop entropy and the entanglement entropy scale as ln(L). For the entanglement entropy, the coefficient of the logarithm, the number of Goldstone modes/2, is universal and is the analogue of the sub leading term of a LXL cluster. This result was substantiated by a modified spin-wave theory calculation for the ferromagnetic X-Y model. Taken together, the calculations suggest that linking high entanglement objects will not generate much more entanglement. As a consequence, simulating the ground state of clusters joined together by a few bonds should not be much more difficult than simulating the ground state of a single cluster.
The scaling of entanglement entropy is computationally studied in several ($1\ensuremath{\le}d\ensuremath{\le}2$)-dimensional free-fermion systems that are connected by one or more point contacts (PCs). For both the kkk-leg Bethe lattice (d=1)(d=1)(d=1) and d=2d=2d=2 rectangular lattices with a subsystem of Ld{L}^{d}Ld sites, the entanglement entropy associated with a single PC is found to be generically $S\ensuremath{\sim}L$. We argue that the O(L)O(L)O(L) entropy is an expression of the subdominant O(L)O(L)O(L) entropy of the bulk entropy-area law. For d=2d=2d=2 (square) lattices connected by mmm PCs, the area law is found to be $S\ensuremath{\sim}a{L}^{d\ensuremath{-}1}+bm\phantom{\rule{0.16em}{0ex}}logL$ and is thus consistent with the anomalous area law for free fermions ($S\ensuremath{\sim}L\phantom{\rule{0.16em}{0ex}}logL$) as $m\ensuremath{\rightarrow}L$. For the Bethe lattice, the relevance of this result to density-matrix renormalization-group schemes for interacting fermions is discussed.
We examine the entanglement properties of the spin-half Heisenberg model on the two-dimensional square-lattice bilayer based on quantum Monte Carlo calculations of the second R\'enyi entanglement entropy. In particular, we extract the dominant area-law contribution to the bipartite entanglement entropy that shows a nonmonotonous behavior upon increasing the interlayer exchange interaction: a local maximum in the area-law coefficient is located at the quantum critical point separating the antiferromagnetically ordered region from the disordered dimer-singlet regime. Furthermore, we consider subleading logarithmic corrections to the R\'enyi entanglement entropy scaling. Employing different subregion shapes, we isolate the logarithmic corner term from the logarithmic contribution due to Goldstone modes that is found to be enhanced in the limit of decoupled layers. At the quantum critical point, we estimate a contribution of 0.016(1)0.016(1)0.016(1) due to each ${90}^{\ensuremath{\circ}}$ corner. This corner term at the SU(2) quantum critical point deviates from the Gaussian theory value, while it compares well with recent numerical linked cluster calculations on the bilayer model.
Entanglement entropy obeys area law scaling for typical physical quantum systems. This may naively be argued to follow from the locality of interactions. We show that this is not the case by constructing an explicit simple spin chain Hamiltonian with nearest-neighbor interactions that presents an entanglement volume scaling law. This non-translational model is contrived to have couplings that force the accumulation of singlet bonds across the half-chain. This configuration of the couplings is suggested by real-space renormalization group arguments. Computation of the entanglement entropy is performed by mapping the system to free fermions and diagonalizing numerically its correlation matrix. An analytical relationship between the entanglement entropy and the Frobenius norm of the correlation matrix is also established. Our result is complementary to the known relationship between non-translational invariant, nearest-neighbor interacting Hamiltonians and quantum Merlin–Arthur (QMA)-complete problems.
We compute the von Neumann and generalized R\'enyi entanglement entropies in the ground-state of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the square lattice using the modified spin-wave theory for finite lattices. The addition of a staggered magnetic field to regularize the zero modes associated with symmetry breaking is shown to be essential for obtaining well-behaved values for the entanglement entropy. The von Neumann and R\'enyi entropies obey an area law with additive logarithmic corrections, and are in good quantitative agreement with numerical results from valence bond quantum Monte Carlo and density matrix renormalization group calculations. We also compute the spin fluctuations and observe a multiplicative logarithmic correction to the area law in excellent agreement with quantum Monte Carlo calculations.