Nodal Line Entanglement Entropy: Generalized Widom Formula from Entanglement Hamiltonians

TL;DR

This paper generalizes the Widom formula for entanglement entropy to systems with nodal lines of arbitrary codimension, providing a new method to analyze quantum entanglement in fermionic systems with complex Fermi surface geometries.

Contribution

The authors extend the Widom formula to arbitrary codimension Fermi surfaces, including nodal lines, and develop a local entanglement Hamiltonian approach for these systems.

Findings

The generalized formula applies to nodal lines in three dimensions.

The entanglement entropy coefficient depends on the shape and orientation of the nodal line.

Entanglement entropy can diagnose the presence and geometry of nodal lines.

Abstract

A system of fermions forming a Fermi surface exhibits a large degree of quantum entanglement, even in the absence of interactions. In particular, the usual case of a codimension one Fermi surface leads to a logarithmic violation of the area law for entanglement entropy, as dictated by the Widom formula. We here generalize this formula to the case of arbitrary codimension, which is of particular interest for nodal lines in three dimensions. We first rederive the standard Widom formula by calculating an entanglement Hamiltonian for Fermi surface systems, obtained by repurposing a trick commonly applied to relativistic theories. The entanglement Hamiltonian will take a local form in terms of a low-energy patch theory for the Fermi surface, though it is nonlocal with respect to the microscopic fermions. This entanglement Hamiltonian can then be used to derive the entanglement entropy, yielding a result in agreement with the Widom formula. The method is then generalized to arbitrary codimension. For nodal lines, the area law is obeyed, and the magnitude of the coefficient for a particular partition is non-universal. However, the coefficient has a universal dependence on the shape and orientation of the nodal line relative to the partitioning surface. By comparing the relative magnitude of the area law for different partitioning cuts, entanglement entropy can be used as a tool for diagnosing the presence and shape of a nodal line in a ground state wavefunction.