Absence of Goldstone bosons on the Bethe lattice

C. R. Laumann, S. A. Parameswaran, Shivaji Lal Sondhi

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11 Scopus citations


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 Nf limit of ferromagnetic models with O (Nf) symmetry. We show that the approach to quantum criticality is characterized by the vanishing of a gap to just the global modes so that 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.

Original languageEnglish (US)
Article number144415
JournalPhysical Review B - Condensed Matter and Materials Physics
Issue number14
StatePublished - Oct 19 2009

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics


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