Pseudo-Goldstone Gaps and Order-by-Quantum Disorder in Frustrated Magnets

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 ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Ca}}_3{\mathrm{Fe}}_2{\mathrm{Ge}}_3{\mathrm{O}}_{12}$.

Figure 1

Schematic illustration of the semiclassical energy density $\epsilon (\varphi ,\theta )$, and the curvatures ${[({\partial }^2\epsilon /\partial {\theta }^2)]}_0$ and ${[({\partial }^2\epsilon /\partial {\varphi }^2)]}_0$ about the semiclassical ground state, which are related to the pseudo-Goldstone gap, $\mathrm{\Delta }$, via Eq. (1). The nearly soft manifold associated with a type I pseudo-Goldstone mode is shown, with the two principal curvatures about a global minimum indicated.

Figure 2

Schematic form of the spectrum of (a) type I and (b) type II pseudo-Goldstone modes. Type I modes have $\omega \sim |\mathit{k}|$ at $O(S)$, with the pseudo-Goldstone gap scaling as $\mathrm{\Delta }\sim O(S^{1/2})$, while type II modes have $\omega \sim |\mathit{k}{|}^2$ at $O(S)$, with $\mathrm{\Delta }\sim O(S^0)$.

Figure 3

The three classes of (Hugenholtz) diagrams contributing to the $O(S^0)$ magnon self-energy, ${\mathbf{\Sigma }}^R(\mathit{k},\omega )$. The (free) propagators (dark line) include normal and anomalous parts, and are connected to external legs (dull line). The first diagram involves a single four-magnon interaction (filled rectangle), while the second and third involve a pair of three-magnon interactions (filled circle).