Instantons, colloids and convergence of the 1/N expansion for the homogeneous electron gas

Tom Banks, Bingnan Zhang

Research output: Contribution to journalArticle

1 Scopus citations


We investigate non-perturbative corrections to the large N expansion of the homogeneous electron gas. These are associated with instanton solutions to the effective action of the plasmon field. We show that, although the large field behavior of that action dominates the quadratic bare Coulomb term, there are no solutions at large field, and consequently none at large density. We argue that solutions would exist at low density if the large N theory had a Wigner crystal (WC) phase. However, we argue that this is not the case. Together with the implied convergence of the large N expansion, this implies that the homogeneous electron gas with N component spins and a Coulomb interaction scaling like 1∕N can only have a WC phase below a curve in the plane of N and density, which asymptotes to zero density at infinite N. We argue that for systems with a semi-classical expansion for order parameter dynamics, and a first order quantum transition between fluid and crystal phases, there are instantons associated with the decays of meta-stable fluid and crystal phases in the appropriate regions of the phase diagram. We argue that the crystal will decay into one or more colloidal or bubble phases (Kivelson and Spivak, 2006) rather than directly into the fluid. It is possible that the bubble phases remain stable all the way to the density where the crystal solution disappears. The bubbles of fluid will expand as the density is raised and gradually convert the system into a fluid filled with chunks of crystal. The transition to a translationally invariant phase is likely to be second order. Unfortunately, the HEG does not have a crystal phase at large N, where these semi-classical ideas could be examined in detail. We suggest that the evidence for negative dielectric function at intermediate densities for N=2 is an indicator of this second order transition. While the infinite N limit does not have a negative dielectric function at any density, it is possible that the closed large N equation for the plasmon two point function, derived in Banks (2018) might capture at least the qualitative features of the second order transition.

Original languageEnglish (US)
Pages (from-to)212-227
Number of pages16
JournalAnnals of Physics
StatePublished - Aug 2019
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)


  • 1/N expansion
  • Colloid
  • Homogeneous electron gas
  • Instanton
  • Wigner crystal

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