The ground state of a general electron-phonon Hamiltonian is a spin singlet

TL;DR

This paper proves that the ground state of a broad class of electron-phonon Hamiltonians is a spin singlet, with implications for the uniqueness and smoothness of the ground state across various models and dimensions.

Contribution

It establishes a general spin-singlet ground state theorem for complex electron-phonon systems, including anharmonic and nonlinear couplings, extending previous results to all dimensions and graph structures.

Findings

Ground state is a spin singlet for a wide class of models.

Ground state is unique if hopping has no phonon dependence.

Self-trapping transitions are smooth crossovers, not phase transitions.

Abstract

The many-body ground state of a very general class of electron-phonon Hamiltonians is proven to contain a spin singlet (for an even number of electrons on a finite lattice). The phonons interact with the electronic system in two different ways---there is an interaction with the local electronic charge and there is a functional dependence of the electronic hopping Hamiltonian on the phonon coordinates. The phonon potential energy may include anharmonic terms, and the electron-phonon couplings and the hopping matrix elements may be nonlinear functions of the phonon coordinates. If the hopping Hamiltonian is assumed to have no phonon coordinate dependence, then the ground state is also shown to be unique, implying that there are no ground-state level crossings, and that the ground-state energy is an analytic function of the parameters in the Hamiltonian. In particular, in a finite system any self-trapping transition is a smooth crossover not accompanied by a nonanalytical change in the ground state. The spin-singlet theorem applies to the Su-Schrieffer-Heeger model and both the spin-singlet and uniqueness theorems apply to the Holstein and attractive Hubbard models as special cases. These results hold in all dimensions --- even on a general graph without periodic lattice structure.