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AIP conference proceedings

Entanglement scaling in classical and quantum harmonic oscillator lattices

K. Audenaert, M. Cramer, Jens Eisert, Martin Bodo Plenio

Published Jan 1, 2006• 0 citations
Publisher

Abstract

We consider entanglement properties of ground and thermal states of harmonic lattice systems. A theorem connecting entanglement between a region and the rest of the lattice with the surface area of the boundary between the two regions is presented for systems in arbitrary spatial dimensions. The behavior of the block entanglement in the field limit is analysed and a logarithmic divergence is recovered.

DOI
10.1063/1.2400881

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We consider entanglement properties of ground and thermal states of harmonic lattice systems. A theorem connecting entanglement between a region and the rest of the lattice with the surface area of the boundary between the two regions is presented for systems in arbitrary spatial dimensions. The behavior of the block entanglement in the field limit is analysed and a logarithmic divergence is recovered.
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We demonstrate that the entropy of entanglement and the distillable entanglement of regions with respect to the rest of a general harmonic-lattice system in the ground or a thermal state scale at most as the boundary area of the region. This area law is rigorously proven to hold true in noncritical harmonic-lattice systems of arbitrary spatial dimension, for general finite-ranged harmonic interactions, regions of arbitrary shape, and states of nonzero temperature. For nearest-neighbor interactions---corresponding to the Klein-Gordon case---upper and lower bounds to the degree of entanglement can be stated explicitly for arbitrarily shaped regions, generalizing the findings of Phys. Rev. Lett. 94, 060503 (2005). These higher-dimensional analogs of the analysis of block entropies in the one-dimensional case show that under general conditions, one can expect an area law for the entanglement in noncritical harmonic many-body systems. The proofs make use of methods from entanglement theory, as well as of results on matrix functions of block-banded matrices. Disordered systems are also considered. We moreover construct a class of examples for which the two-point correlation length diverges, yet still an area law can be proven to hold. We finally consider the scaling of classical correlations in a classical harmonic system and relate it to a quantum lattice system with a modified interaction. We briefly comment on a general relationship between criticality and area laws for the entropy of entanglement.
We study different aspects of quantum von Neumann and R\'enyi entanglement entropy of one-dimensional long-range harmonic oscillators that can be described by well-defined nonlocal field theories. We show that the entanglement entropy of one interval with respect to the rest changes logarithmically with the number of oscillators inside the subsystem. This is true also in the presence of different boundary conditions. We show that the coefficients of the logarithms coming from different boundary conditions can be reduced to just two different universal coefficients. We also study the effect of the mass and temperature on the entanglement entropy of the system in different situations. The universality of our results is also confirmed by changing different parameters in the coupled harmonic oscillators. We also show that more general interactions coming from general singular Toeplitz matrices can be decomposed to our long-range harmonic oscillators. Despite the long-range nature of the couplings, we show that the area law is valid in two dimensions and the universal logarithmic terms appear if we consider subregions with sharp corners. Finally, we study analytically different aspects of the mutual information such as its logarithmic dependence to the subsystem, effect of mass, and influence of the boundary. We also generalize our results in this case to general singular Toeplitz matrices and higher dimensions.
We study the scaling properties of the ground-state entanglement between finite subsystems of infinite two-dimensional free lattice models, as measured by the logarithmic negativity. For adjacent regions with a common boundary, we observe that the negativity follows a strict area law for a lattice of harmonic oscillators, whereas for fermionic hopping models the numerical results indicate a multiplicative logarithmic correction. In this latter case we conjecture a formula for the prefactor of the area-law violating term, which is entirely determined by the geometries of the Fermi surface and the boundary between the subsystems. The conjecture is tested against numerical results and a good agreement is found.
We revisit the question of the relation between entanglement, entropy, and area for harmonic lattice Hamiltonians corresponding to discrete versions of real free Klein-Gordon fields. For the ground state of the d-dimensional cubic harmonic lattice we establish a strict relationship between the surface area of a distinguished hypercube and the degree of entanglement between the hypercube and the rest of the lattice analytically, without resorting to numerical means. We outline extensions of these results to longer ranged interactions, finite temperatures, and for classical correlations in classical harmonic lattice systems. These findings further suggest that the tools of quantum information science may help in establishing results in quantum field theory that were previously less accessible.