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Reviews in Mathematical Physics

Entropy Density of One-Dimensional Quantum Lattice Systems

Hajime Moriya

Published Apr 1, 1997• 2 citations
Publisher

Abstract

We prove that the entropy density of a KMS state of one-dimensional quantum lattice systems is equal to the thermodynamical limit of the entropy of local Gibbs states.
DOI
10.1142/s0129055x97000130

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Quantum many-body states that frequently appear in physics often obey an entropy scaling law, meaning that an entanglement entropy of a subsystem can be expressed as a sum of terms that scale linearly with its volume and area, plus a correction term that is independent of its size.We conjecture that these states have an efficient dual description in terms of a set of marginal density matrices on bounded regions, obeying the same entropy scaling law locally.We prove a restricted version of this conjecture for translationally invariant systems in two spatial dimensions.Specifically, we prove that a translationally invariant marginal obeying three nonlinear constraints-all of which follow from the entropy scaling law straightforwardlymust be consistent with some global state on an infinite lattice.Moreover, we derive a closed-form expression for the maximum-entropy density compatible with those marginals, deriving a variational upper bound on the thermodynamic free energy.The main assumptions of our construction are satisfied exactly by solvable models of topological order and approximately by finite-temperature Gibbs states of certain quantum spin Hamiltonians.
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