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arXiv:1107.2840 · Physical Review B

Anomalies in the entanglement properties of the square-lattice Heisenberg model

Ann B. Kallin, Matthew B. Hastings, Roger G. Melko, Rajiv R. P. Singh

Published Oct 28, 2011• 129 citations
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Abstract

We compute the bipartite entanglement properties of the spin-half square-lattice Heisenberg model by a variety of numerical techniques that include valence-bond quantum Monte Carlo (QMC), stochastic series expansion QMC, high-temperature series expansions, and zero-temperature coupling constant expansions around the Ising limit. We find that the area law is always satisfied, but in addition to the entanglement entropy per unit boundary length, there are other terms that depend logarithmically on the subregion size, arising from broken symmetry in the bulk and from the existence of corners at the boundary. We find that the numerical results are anomalous in several ways. First, the bulk term arising from broken symmetry deviates from an exact calculation that can be done for a mean-field N\'eel state. Second, the corner logs do not agree with the known results for noninteracting Boson modes. And, third, even the finite-temperature mutual information shows an anomalous behavior as $T$ goes to zero, suggesting that the $T\ensuremath{\rightarrow}0$ and $L\ensuremath{\rightarrow}\ensuremath{\infty}$ limits do not commute. These calculations show that entanglement entropy demonstrates a very rich behavior in $d>1$, which deserves further attention.

DOI
10.1103/physrevb.84.165134

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We compute the bipartite entanglement properties of the spin-half square-lattice Heisenberg model by a variety of numerical techniques that include valence-bond quantum Monte Carlo (QMC), stochastic series expansion QMC, high-temperature series expansions, and zero-temperature coupling constant expansions around the Ising limit. We find that the area law is always satisfied, but in addition to the entanglement entropy per unit boundary length, there are other terms that depend logarithmically on the subregion size, arising from broken symmetry in the bulk and from the existence of corners at the boundary. We find that the numerical results are anomalous in several ways. First, the bulk term arising from broken symmetry deviates from an exact calculation that can be done for a mean-field N'eel state. Second, the corner logs do not agree with the known results for noninteracting Boson modes. And, third, even the finite-temperature mutual information shows an anomalous behavior as TTT goes to zero, suggesting that the T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0 and L\ensuremath→\ensuremath∞L\ensuremath{\rightarrow}\ensuremath{\infty}L\ensuremath→\ensuremath∞ limits do not commute. These calculations show that entanglement entropy demonstrates a very rich behavior in d>1d>1d>1, which deserves further attention.
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.
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.
We analyze the critical properties and the entanglement scaling at the quantum critical points of the spin-half XY model on a two-dimensional square-lattice bilayer and necklace lattice, based on quantum Monte Carlo simulations on finite tori and for different subregion shapes. For both models, the finite-size scaling of the transverse staggered spin structure factor is found in accord with a quantum critical point described by the two-component, three-dimensional ${\ensuremath{\phi}}^{4}$ theory. The second R\'enyi entanglement entropy in the absence of corners along the subsystem boundary exhibits area-law scaling in both models, with an area-law prefactor of 0.0674(7)0.0674(7)0.0674(7) [0.0664(4)][0.0664(4)][0.0664(4)] for the bilayer (necklace) model, respectively. Furthermore, the presence of ${90}^{\ensuremath{\circ}}$ corners leads to an additive logarithmic term in both models. We estimate a contribution of $\ensuremath{-}0.010(2)$ $[\ensuremath{-}0.009(2)]$ due to each ${90}^{\ensuremath{\circ}}$ corner to the logarithmic correction for the bilayer (necklace) model, and compare our findings to recent numerical linked-cluster calculations and series expansion results on related models.
Using a modified spin-wave theory which artificially restores zero sublattice magnetization on finite lattices, we investigate the entanglement properties of the N\'eel ordered ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg antiferromagnet on the square lattice. Different kinds of subsystem geometries are studied, either corner-free (line, strip) or with sharp corners (square). Contributions from the nG=2{n}_{G}=2nG​=2 Nambu-Goldstone modes give additive logarithmic corrections with a prefactor nG/2{n}_{G}/2nG​/2 independent of the R\'enyi index. On the other hand, $\ensuremath{\pi}/2$ corners lead to additional (negative) logarithmic corrections with a prefactor lqc{l}_{q}^{c}lqc​ which does depend on both nG{n}_{G}nG​ and the R\'enyi index qqq, in good agreement with scalar field theory predictions. By varying the second neighbor coupling J2{J}_{2}J2​ we also explore universality across the N\'eel ordered side of the phase diagram of the ${J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnet, from the frustrated side 0<J2/J1<1/20<{J}_{2}/{J}_{1}<1/20<J2​/J1​<1/2 where the area law term is maximal, to the strongly ferromagnetic regime $\ensuremath{-}{J}_{2}/{J}_{1}\ensuremath{\gg}1$ with a purely logarithmic growth Sq=nG2lnN{S}_{q}=\frac{{n}_{G}}{2}lnNSq​=2nG​​lnN, thus recovering the mean-field limit for a subsystem of NNN sites. Finally, a universal subleading constant term ${\ensuremath{\gamma}}_{q}^{\mathrm{ord}}$ is extracted in the case of strip subsystems, and a direct relation is found (in the large-SSS limit) with the same constant extracted from free lattice systems. The singular limit of vanishing aspect ratios is also explored, where we identify for ${\ensuremath{\gamma}}_{q}^{\text{ord}}$ a regular part and a singular component, explaining the discrepancy of the linear scaling term for fixed width vs fixed aspect ratio subsystems.
In a d -dimensional topological insulator of order d , there are zero-energy states on its corners which have close relationship with its entanglement behaviors. We studied the bipartite entanglement spectra for different subsystem shapes and found that only when the entanglement boundary has corners matching the lattice, exact zero modes exist in the entanglement spectrum corresponding to the zero-energy states caused by the same physical corners. We then considered finite-size systems in which cases these corner states are coupled together by long-range hybridizations to form multipartite entangled states. We proposed a scheme to calculate the quadripartite entanglement entropy on the square lattice, which is well described by a four-sites toy model and thus provides another way to identify the higher order topological insulators from the multipartite entanglement point of view.
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.