Pseudo-Goldstone gaps and order-by-quantum-disorder in frustrated magnets

TL;DR

This paper presents an exact method to compute pseudo-Goldstone gaps in frustrated magnets caused by order-by-quantum-disorder, confirmed through theoretical calculations and applicable to real materials.

Contribution

It introduces a leading-order $1/S$ spin-wave theory approach to calculate pseudo-Goldstone gaps without considering spin-wave interactions, validated by higher-order calculations.

Findings

Exact computation of pseudo-Goldstone gaps from zero-point energy curvature.

Confirmation of the method through $O(1/S^2)$ calculations in various models.

Application to real materials like Er$_2$Ti$_2$O$_7$ and Ca$_3$Fe$_2$Ge$_3$O$_{12}$.

Abstract

In systems with competing interactions, continuous degeneracies can appear which are accidental, in that they are not related to any symmetry of the Hamiltonian. Accordingly, the pseudo-Goldstone modes associated with these degeneracies are also unprotected. Indeed, through a process known as "order-by-quantum-disorder", quantum zero-point fluctuations can lift the degeneracy and induce a gap for these modes. We show that this gap can be exactly computed at leading order in $1/S$ in spin-wave theory from the mean curvature of the classical and quantum zero-point energies - without the need to consider any spin-wave interactions. We confirm this equivalence through direct calculations of the spin-wave spectrum to $O(1/S^2)$ in a wide variety of theoretically and experimentally relevant quantum spin models. We prove this equivalence through the use of an exact sum rule that provides the required mixing of different orders of $1/S$. Finally, we discuss some implications for several leading order-by-quantum-disorder candidate materials, clarifying the expected pseudo-Goldstone gap sizes in Er$_2$Ti$_2$O$_7$ and Ca$_3$Fe$_2$Ge$_3$O$_{12}$.