What surface maximizes entanglement entropy?

TL;DR

This paper investigates which surfaces maximize entanglement entropy in quantum field theories, revealing a connection to the Willmore energy minimization problem in four dimensions and proposing a generalization for higher dimensions.

Contribution

It establishes a link between entanglement entropy maximization and Willmore energy minimization, introducing a higher-dimensional generalization and analyzing specific topologies.

Findings

In four dimensions, maximizing surfaces relate to Willmore energy minimizers.

Proposes a generalized Willmore energy for higher dimensions.

Analyzes minimizers in topologies like S^m×S^n.

Abstract

For a given quantum field theory, provided the area of the entangling surface is fixed, what surface maximizes entanglement entropy? We analyze the answer to this question in four and higher dimensions. Surprisingly, in four dimensions the answer is related to a mathematical problem of finding surfaces which minimize the Willmore (bending) energy and eventually to the Willmore conjecture. We propose a generalization of the Willmore energy in higher dimensions and analyze its minimizers in a general class of topologies $S^m\times S^n$ and make certain observations and conjectures which may have some mathematical significance.