Migdal-Eliashberg theory as a classical spin chain

TL;DR

This paper reformulates the Migdal-Eliashberg theory of electron-phonon interactions as a classical spin chain, revealing new solutions and insights into the behavior of strongly coupled superconductors, especially far from equilibrium.

Contribution

It introduces a novel spin chain representation of Eliashberg equations, uncovering multiple new solutions and clarifying properties of strongly coupled superconductors.

Findings

Discovery of infinitely many new solutions at strong coupling

Identification of saddle points as spin flips in the free-energy landscape

Insights into far-from-equilibrium dynamics of superconductors

Abstract

We formulate the Migdal-Eliashberg theory of electron-phonon interactions in terms of classical spins by mapping the free energy to a Heisenberg spin chain in a Zeeman magnetic field. Spin components are energy-integrated normal and anomalous Green's functions and sites of the chain are fermionic Matsubara frequencies. The Zeeman field grows linearly with the spin coordinate and competes with ferromagnetic spin-spin interaction that falls off as the square of the inverse distance. The spin-chain representation makes a range of previously unknown properties plain to see. In particular, infinitely many new solutions of the Eliashberg equations both in the normal and superconducting states emerge at strong coupling. These saddle points of the free-energy functional correspond to spin flips. We argue that they are also fixed points of kinetic equations and play an essential role in far from equilibrium dynamics of strongly coupled superconductors. Up to an overall phase, the frequency-dependent gap function that minimizes the free energy must be non-negative. There are strong parallels between our Eliashberg spins and Anderson pseudospins, though the two sets of spins never coincide.