The Casimir effect for nonlinear sigma models and the Mermin-Wagner-Hohenberg-Coleman theorem

TL;DR

This paper investigates how nonlinearities and dimensionality influence the Casimir effect in quantum field theories, revealing complex behaviors and long-range forces related to the Mermin-Wagner-Hohenberg-Coleman theorem.

Contribution

It demonstrates the impact of nonlinearities on the Casimir effect within a nonlinear sigma model, connecting it to fundamental theorems and showing modified force behaviors.

Findings

Vacuum-induced force remains long-ranged at large distances.

Force exhibits complex behavior at small separations.

Nonlinearities modulate the force based on coupling and temperature.

Abstract

The quantum vacuum (Casimir) energy arising from noninteracting massless quanta is known to induce a long-range force, while decays exponentially for massive fields and separations larger than the inverse mass of the quanta involved. Here, we show that the interplay between dimensionality and nonlinearities in the field theory alters this behaviour in a nontrivial way. We argue that the changes are intimately related to the Mermin-Wagner-Hohenberg-Coleman theorem, and illustrate this situation using a nonlinear sigma model as a working example. We compute the quantum vacuum energy, which consists of the usual Casimir contribution plus a semiclassical contribution, and find that the vacuum-induced force is long-ranged at large distance, while displays a complex behaviour at small separations. Finally, even for this relatively simple set-up, we show that nonlinearities are generally responsible for modulations in the force as a function of the coupling constant and the temperature.