Pseudo-Goldstone Modes at Finite Temperature

TL;DR

This paper derives a general formula for the finite-temperature gap of pseudo-Goldstone modes in magnetic systems, validating it with models and applying it to the XXZ model on a triangular lattice, revealing a linear temperature dependence.

Contribution

It introduces a universal curvature formula for PG gaps at finite temperature applicable to various magnetic orders, validated against known models and applied to complex frustrated systems.

Findings

PG gap decreases linearly with temperature due to magnon entropy effects

The formula applies to both collinear and noncollinear magnetic orders

Results contrast with high-temperature scaling in single magnon band systems

Abstract

Goldstone's theorem and its extension to pseudo-Goldstone (PG) modes have profound implications across diverse areas of physics, from quantum chromodynamics to quantum magnetism. PG modes emerge from accidental degeneracies lifted by quantum and thermal fluctuations, leading to a finite gap--a phenomenon known as "order by disorder." In this paper, we derive a general curvature formula for the PG gap at finite temperature, applicable to both collinear (e.g., ferromagnets and anti-ferromagnets) and noncollinear magnetic orders (e.g., coplanar orders in frustrated magnetic systems). After validating our formula against known models, we apply it to the XXZ model on the triangular lattice, which hosts coplanar magnetic orders in equilibrium and is relevant to materials such as Na2BaCo(PO4)2 and K2Co(SeO3)2, known for their supersolid phases and giant magnetocaloric effects. Our results reveal a distinct scaling behavior: a linear decrease of the PG gap with temperature, driven by entropy effects from magnon scattering across multiple bands. This stands in stark contrast to the high-temperature scaling recently proposed for systems with a single magnon band. This work establishes a general framework for investigating PG modes at finite temperatures and opens an avenue to explore rich quantum phases and dynamics in frustrated systems with noncollinear magnetic orders.