The Mass Gap of the Nonlinear Sigma Model through the Finite Temperature Effective Action

TL;DR

This paper investigates the temperature-dependent mass gap in the disordered phase of the $O(3)$ nonlinear sigma model using effective action techniques, providing analytical results and comparisons with experimental data.

Contribution

It introduces a calculation of the finite temperature effective potential for the Lagrange multiplier, revealing the mass gap behavior across dimensions 1, 2, and 3.

Findings

Mass gap as a function of temperature in 1, 2, and 3 dimensions.

Identification of a nontrivial minimum indicating a disordered phase.

Comparison of the 1D Haldane gap with experimental results.

Abstract

The $O(3)$ nonlinear $\sigma$ model is studied in the disordered phase, using the techniques of the effective action and finite temperature field theory. The nonlinear constraint is implemented through a Lagrange multiplier. The finite temperature effective potential for this multiplier is calculated at one loop. The existence of a nontrivial minimum for this potential is the signal of a disordered phase in which the lowest excited state is a massive triplet. The mass gap is easily calculated as a function of temperature in dimensions 1, 2 and 3. In dimension 1, this gap is known as the Haldane gap, and its temperature dependence is compared with experimental results.