There are no Goldstone bosons on the Bethe lattice

Abstract

We discuss symmetry breaking quantum phase transitions on the oft studied Bethe lattice in the context of the ferromagnetic scalar spherical model or, equivalently, the infinite $N_f$ limit of ferromagnetic models with $O(N_f)$ symmetry. We show that the approach to quantum criticality is characterized by the vanishing of a gap to just the global modes so that {\it all} local correlation functions continue to exhibit massive behavior. This behavior persists into the broken symmetry phase even as the order parameter develops an expectation value and thus there are no massless Goldstone bosons in the spectrum. We relate this feature to a spectral property of the graph Laplacian shared by the set of `expander' graphs, and argue that our results apply to symmetry breaking transitions on such graphs quite generally.

6 pages, 2 figures

References in corpus (3)

• On the path integral representation for quantum spin models and its application to the quantum cavity method and to Monte Carlo simulations
• Cavity method for quantum spin glasses on the Bethe lattice
• The Quantum Transverse Field Ising Model on an Infinite Tree from Matrix Product States

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