Entanglement entropy for a Maxwell field: Numerical calculation on a two dimensional lattice

TL;DR

The paper develops an algebraic framework to define and compute entanglement entropy for a Maxwell field in $2+1$ dimensions on a two-dimensional lattice, where local algebras can have centers due to constraints. It extends Gaussian-state entropy formulas to algebras with center and arbitrary commutators, enabling detailed lattice calculations that reveal a universal continuum limit for mutual information independent of the chosen algebra, while single-region entropies depend on center type and boundary geometry. A secondary novelty is the identification of a universal logarithmic term in the entropy, comprising an angle-dependent ultraviolet piece and a topological contribution proportional to the number of connected components, with the latter not captured by mutual information. The work further proposes an algebraic formulation of strong subadditivity (SSA) that remains valid for algebras with center and illustrates its behavior in lattice examples, linking the results to broader questions about topological entanglement and c-theorems in gauge theories.

Abstract

We study entanglement entropy (EE) for a Maxwell field in 2+1 dimensions. We do numerical calculations in two dimensional lattices. This gives a concrete example of the general results of our recent work on entropy for lattice gauge fields using an algebraic approach. To evaluate the entropies we extend the standard calculation methods for the entropy of Gaussian states in canonical commutation algebras to the more general case of algebras with center and arbitrary numerical commutators. We find that while the entropy depends on the details of the algebra choice, mutual information has a well defined continuum limit. We study several universal terms for the entropy of the Maxwell field and compare with the case of a massless scalar field. We find some interesting new phenomena: An "evanescent" logarithmically divergent term in the entropy with topological coefficient which does not have any correspondence with ultraviolet entanglement in the universal quantities, and a non standard way in which strong subadditivity is realized. Based on the results of our calculations we propose a generalization of strong subadditivity for the entropy on some algebras that are not in tensor product.

Figures (15)

• Figure 1: The magnetic field is assigned to the center of the plaquette and the electric fields to the links.
• Figure 2: Dual lattices: The magnetic field coincides with the momentum operator of the scalar field, and the electric field $E$ in some link is equal to a difference of scalar fields across the link in the dual lattice which is perpendicular to the one corresponding to $E$.
• Figure 3: Some algebra choices for a square region. The upper three figures correspond to the gauge model and the ones at the bottom to the truncated scalar representation of the same algebras. Links with dashed lines mean the corresponding electric operator does not belong to the algebra. Marked dots correspond to magnetic operators in the algebra in the gauge model, and momentum operators in the scalar one. The left panel shows the electric center choice, where all electric and magnetic operators on the square belong to the algebra. Because of constraints the algebra also automatically contains the links coming out of the square, and there are more independent electric generators than magnetic ones. The central panel shows a trivial center choice, with balanced number of electric and magnetic degree of freedom. The panel on the right shows the magnetic center choice. Here, all electric operators on the boundary are missing and there is one more magnetic degree of freedom than the number of electric degrees of freedom.
• Figure 4: A circle on a square lattice. A maximal tree of links inside the region gives all linearly independent link variables in the truncated scalar model. We keep only variables in an arbitrary maximal internal tree to make actual computations. Analogously, in the gauge model we have to keep only the electric fields that are orthogonal to this tree in the dual lattice.
• Figure 5: We compute the mutual information for two squares with different algebra choice. Upper panel: Trivial center, where operators $\phi_{i}$ and $\pi_{i}$ are attached to each vertex (black dots). Empty dots are shown with the purpose to describe the position of the squares in the lattice and no operator are attached to them. Middle panel: Non trivial center. Operators $\phi_{i}$ and $\pi_{i}$ are attached to black vertices. At gray vertices the corresponding $\pi$ operators are removed in ${\cal A}_\phi$ and the $\phi$ operators are removed in ${\cal A}_\pi$. The center is generated by operators remaining at gray vertices. Lower panel: The algebra $\tilde{{\cal A}}$ is the full algebra of the central square of size $n-2$ points.