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arXiv:2104.00580

Entanglement in the ground state of clusters joined by a single bond

Barry Friedman, Alyssa Horne

Published Apr 1, 2021• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2104.00580

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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 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.
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.
In this thesis we examine methods for measuring entanglement entropy in spin-1/2 Heisenberg systems using quantum Monte Carlo in the valence bond basis. We begin by presenting the quantum Monte Carlo techniques used in this research. We then use these techniques to directly compare the recently proposed valence bond entanglement entropy to the standard definition of entanglement entropy: the von Neumann entanglement entropy. We find that the valence bond entanglement entropy does not give a bound on the von Neumann entanglement entropy, and that it exhibits a multiplicative logarithmic correction to the area law that is not present in the scaling of the von Neumann entanglement entropy. We then present a method to measure higher orders of the generalized Renyi entanglement entropies using valence bond quantum Monte Carlo, and show results for the second Renyi entropy. We find the results converge to the exact results for one dimensional Heisenberg spin-1/2 chains, and see that the scaling of the second Renyi entropy follows an area law in the two dimensional Heisenberg ground state.
We calculate the bipartite von Neumann and second R\'enyi entanglement entropies of the ground states of spin-1/2 dimerized Heisenberg antiferromagnets on a square lattice. Two distinct dimerization patterns are considered: columnar and staggered. In both cases, we concentrate on the valence bond solid (VBS) phase and describe such a phase with the bond-operator representation. Within this formalism, the original spin Hamiltonian is mapped into an effective interacting boson model for the triplet excitations. We study the effective Hamiltonian at the harmonic approximation and determine the spectrum of the elementary triplet excitations. We then follow an analytical procedure, which is based on a modified spin-wave theory for finite systems and was originally employed to calculate the entanglement entropies of magnetic ordered phases, and calculate the entanglement entropies of the VBS ground states. In particular, we consider one-dimensional (line) subsystems within the square lattice, a choice that allows us to consider line subsystems with sizes up to ${L}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}1000$. We combine such a procedure with the results of the bond-operator formalism at the harmonic level and show that, for both dimerized Heisenberg models, the entanglement entropies of the corresponding VBS ground states obey an area law as expected for gapped phases. For both columnar-dimer and staggered-dimer models, we also show that the entanglement entropies increase but they seem to not diverge as the dimerization decreases and the system approaches the N\'eel-VBS quantum phase transition. Finally, the entanglement spectra associated with the VBS ground states are presented.
The valence-bond structure of spin-1/2 Heisenberg antiferromagnets is closely related to quantum entanglement. We investigate measures of entanglement entropy based on transition graphs, which characterize state overlaps in the overcomplete valence-bond basis. The transition graphs can be generated using projector Monte Carlo simulations of ground states of specific Hamiltonians or using importance sampling of valence-bond configurations of amplitude-product states. We consider definitions of entanglement entropy based on the bonds or loops shared by two subsystems (bipartite entanglement). Results for the bond-based definition agree with a previously studied definition using valence-bond wave functions (instead of the transition graphs, which involve two states). For the one-dimensional Heisenberg chain, with uniform or random coupling constants, the prefactor of the logarithmic divergence with the size of the smaller subsystem agrees with exact results. For the ground state of the two-dimensional Heisenberg model (and also N\'eel-ordered amplitude-product states), there is a similar multiplicative violation of the area law. In contrast, the loop-based entropy obeys the area law in two dimensions while still violating it in one dimension---both behaviors in accord with expectations for proper measures of entanglement entropy.
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.