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arXiv:2305.00649 · Journal of Mathematical Physics

Entanglement entropy bounds for droplet states of the XXZ model on the strip

Christoph Fischbacher, Lee Fisher

Published Feb 1, 2024• 0 citations
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Abstract

The scaling behavior of the entanglement entropy of droplet states in Heisenberg spin-1/2 XXZ model defined on a strip of width M under the presence of a non-negative background magnetic field is investigated. Without any assumptions on V, a logarithmically corrected area law is shown. Assuming that the values of V are i.i.d. random variables, an area law in expectation is obtained.

DOI
10.1063/5.0156546

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The scaling behavior of the entanglement entropy of droplet states in Heisenberg spin-1/2 XXZ model defined on a strip of width M under the presence of a non-negative background magnetic field is investigated. Without any assumptions on V, a logarithmically corrected area law is shown. Assuming that the values of V are i.i.d. random variables, an area law in expectation is obtained.
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.
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.
The entanglement entropy measures quantum correlations and it can be seen as the uncertainty on a quantum state. In one spatial dimension, the entanglement entropy scales as the boundary that divides two subsystems, so an area law has been proposed. However, the entanglement entropy diverges logarithmically at conformally invariant critical points, so the area law does not hold. The purpose of the work is to find a way to get more information about a critical state. The ground state of the Heisenberg XXZ model at criticality is analyzed by means of critical Ising eigenstates. Two ways of analysis are followed: a basis made of Ising eigenstates is built up and used to represent the XXZ ground state, then the Shannon entropy in the new basis is computed; the adiabatic evolution from the Ising ground state to the XXZ ground state. The result is that the Shannon entropy in the Ising basis scales linearly with the length of the system, while a phase transition is encountered during the adiabatic evolution. The conclusion is that there is no net gain in information after the procedure and possibly it is related to the fact the two systems stand in different phases.
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 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 introduce a computable estimator of block entanglement entropy for many-body spin systems admitting total singlet ground states. Building on a simple geometrical interpretation of entanglement entropy of the so-called valence bond states, this estimator is defined as the average number of common singlets to two subsystems of spins. We show that our estimator possesses the characteristic scaling properties of the block entanglement entropy in critical and noncritical one-dimensional Heisenberg systems. We invoke this new measure to examine entanglement scaling in the two-dimensional Heisenberg model on a square lattice revealing an "area law" for the gapped phase and a logarithmic correction to this law in the gapless phase.