Pseudo-Goldstone modes and dynamical gap generation from order-by-thermal-disorder

TL;DR

This paper identifies a unique temperature-dependent signature of order-by-thermal-disorder (ObTD) through the fluctuation-induced pseudo-Goldstone gap, demonstrated via simulations and mean-field calculations in a 2D model.

Contribution

It introduces a clear, qualitative signature of ObTD based on the temperature dependence of the pseudo-Goldstone gap, supported by a minimal model and computational analysis.

Findings

Pseudo-Goldstone gap scales as √T at low temperatures.

Power-law temperature dependence is a general feature of ObTD.

Key physics captured by a simple particle-in-potential model.

Abstract

Accidental ground state degeneracies -- those not a consequence of global symmetries of the Hamiltonian -- are inevitably lifted by fluctuations, often leading to long-range order, a phenomenon known as "order-by-disorder" (ObD). The detection and characterization of ObD in real materials currently lacks clear, qualitative signatures that distinguish ObD from conventional energetic selection. We show that for order-by-thermal-disorder (ObTD) such a signature exists: a characteristic temperature dependence of the fluctuation-induced pseudo-Goldstone gap. We demonstrate this in a minimal two-dimensional model that exhibits ObTD, the ferromagnetic Heisenberg-compass model on a square lattice. Using spin-dynamics simulations and self-consistent mean-field calculations, we determine the pseudo-Goldstone gap, $\Delta $, and show that at low temperatures it scales as the square root of temperature, $\sqrt{T} $. We establish that a power-law temperature dependence of the gap is a general consequence of ObTD, showing that all key features of this physics can be captured in a simple model of a particle moving in an effective potential generated by the fluctuation-induced free energy.