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

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${}_{12}$.

5 + 26 pages, 3 figures. Corrected discussion of cubic garnet, expanded supplemental material

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• Quantum Phases in the Honeycomb-Lattice $J_1$--$J_3$ Ferro-Antiferromagnetic Model
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• Fingerprinting Triangular-Lattice Antiferromagnet by Excitation Gaps
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• Flavor-wave theory with quasiparticle damping at finite temperatures: Application to chiral edge modes in the Kitaev model
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• Magnon interactions in the quantum paramagnetic phase of CoNb${}_2$O${}_6$
• Insights into the Anisotropic Spin-$S$ Kitaev Chain
• Testing Ising Topological Order in $\alpha$-RuCl${}_3$ Under In-Plane Magnetic Fields
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• Partially disordered Heisenberg antiferromagnet with short-range stripe correlations
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• Demystifying quantum escapism on the honeycomb lattice
• Non-linear spin wave theory in the strong easy-axis limit of the triangular XXZ model
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• Programmable order by disorder effect and underlying phases through dipolar quantum simulators
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• Projected Holstein-Primakoff boson representation of quantum spins for spin wave theory
• Finite-Size Spectral Signatures of Order by Quantum Disorder: A Perspective from Anderson's Tower of States
• Robust semiclassical magnetization plateau in the kagome lattice