Topological order and Berry connection for the Maxwell Vacuum on a four-torus

TL;DR

This paper explores topological contributions to the Maxwell vacuum on a four-torus, revealing new effects beyond photon propagation, with implications for Casimir pressure and analogies to topological insulators.

Contribution

It introduces a topological framework for the Maxwell vacuum on a torus, connecting tunneling events and auxiliary fields to topological insulator concepts.

Findings

Novel topological terms affect the partition function.

These contributions influence Casimir pressure.

The vacuum exhibits properties similar to topological insulators.

Abstract

We study novel type of contributions to the partition function of the Maxwell system defined on a small compact manifold such as torus. These new terms can not be described in terms of the physical propagating photons with two transverse polarizations. Rather, these novel contributions emerge as a result of tunnelling events when transitions occur between topologically different but physically identical vacuum winding states. These new terms give an extra contribution to the Casimir pressure. The infrared physics in the system can be described in terms of the topological auxiliary non-propagating fields $a_i(\mathbf{k})$ governed by Chern-Simons -like action. The system can be studied in terms of these auxiliary fields precisely in the same way as a topological insulator can be analyzed in terms of Berry's connection ${\cal{A}}_i(\mathbf{k})$. We also argue that the Maxwell vacuum defined on a small 4-torus behaves very much in the same way as a topological insulator with $\theta\neq 0$.