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arXiv:2011.11977 · Journal of Mathematical Physics

Entanglement entropy bounds in the higher spin XXZ chain

Christoph Fischbacher, Oluwadara Ogunkoya

Published Oct 1, 2021• 3 citations
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Abstract

We consider the Heisenberg XXZ spin-J chain (J∈N/2) with anisotropy parameter Δ. Assuming that Δ > 2J and introducing threshold energies EK≔K1−2JΔ, we show that the bipartite entanglement entropy of states belonging to any spectral subspace with energy less than EK+1 satisfies a logarithmically corrected area law with the prefactor (2⌊K/J⌋ − 2). This generalizes the previous results by Beaud and Warzel [J. Math. Phys. 59, 012109 (2018)], as well as by Abdul-Rahman, Fischbacher, and Stolz [Ann. Henri Poincaré 21, 2327 (2020)], who covered the spin-1/2 case.

DOI
10.1063/5.0042861

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We consider the Heisenberg XXZ spin-J chain (J∈N/2) with anisotropy parameter Δ. Assuming that Δ > 2J and introducing threshold energies EK≔K1−2JΔ, we show that the bipartite entanglement entropy of states belonging to any spectral subspace with energy less than EK+1 satisfies a logarithmically corrected area law with the prefactor (2⌊K/J⌋ − 2). This generalizes the previous results by Beaud and Warzel [J. Math. Phys. 59, 012109 (2018)], as well as by Abdul-Rahman, Fischbacher, and Stolz [Ann. Henri Poincaré 21, 2327 (2020)], who covered the spin-1/2 case.
We consider a class of one-dimensional quantum spin systems on the finite lattice Λ⊂Z, related to the XXZ spin chain in its Ising phase. It includes in particular the so-called droplet Hamiltonian. The entanglement entropy of energetically low-lying states over a bipartition Λ = B ∪ Bc is investigated and proven to satisfy a logarithmic bound in terms of min{n, |B|, |Bc|}, where n denotes the maximal number of down spins in the considered state. Upon addition of any (positive) random potential, the bound becomes uniformly constant on average, thereby establishing an area law. The proof is based on spectral methods: a deterministic bound on the local (many-body integrated) density of states is derived from an energetically motivated Combes–Thomas estimate.
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.
We study the relationship between entanglement and spectral gap for local Hamiltonians in one dimension (1D). The area law for a 1D system states that for the ground state, the entanglement of any interval is upper bounded by a constant independent of the size of the interval. However, the possible dependence of the upper bound on the spectral gap Δ is not known, as the best known general upper bound is asymptotically much larger than the largest possible entropy of any model system previously constructed for small Δ. To help resolve this asymptotic behavior, we construct a family of 1D local systems for which some intervals have entanglement entropy, which is polynomial in 1/Δ, whereas previously studied systems, such as free fermion systems or systems described by conformal field theory, had the entropy of all intervals bounded by a constant time log(1/Δ).
We study the relationship between entanglement and spectral gap for local Hamiltonians in one dimension. The area law for a one-dimensional system states that for the ground state, the entanglement of any interval is upper-bounded by a constant independent of the size of the interval. However, the possible dependence of the upper bound on the spectral gap {Delta} is not known, as the best known general upper bound is asymptotically much larger than the largest possible entropy of any model system previously constructed for small {Delta}. To help resolve this asymptotic behavior, we construct a family of one-dimensional local systems for which some intervals have entanglement entropy which is polynomial in 1/{Delta}, whereas previously studied systems had the entropy of all intervals bounded by a constant times log(1/{Delta}).
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.