On the binding of polarons in a mean-field quantum crystal

TL;DR

This paper investigates the binding properties of polarons within a mean-field quantum crystal, proving the existence of bound states for single and multiple polarons under specific conditions.

Contribution

It establishes the existence of bound states for single polarons and provides conditions for multi-polaron binding in a mean-field crystal model.

Findings

Single polaron always binds, with a proven minimizer for N=1.

Multi-polaron binding depends on quantized HVZ-type inequalities.

Conditions for multi-polaron binding are explicitly characterized.

Abstract

We consider a multi-polaron model obtained by coupling the many-body Schr\"odinger equation for N interacting electrons with the energy functional of a mean-field crystal with a localized defect, obtaining a highly non linear many-body problem. The physical picture is that the electrons constitute a charge defect in an otherwise perfect periodic crystal. A remarkable feature of such a system is the possibility to form a bound state of electrons via their interaction with the polarizable background. We prove first that a single polaron always binds, i.e. the energy functional has a minimizer for N=1. Then we discuss the case of multi-polarons containing two electrons or more. We show that their existence is guaranteed when certain quantized binding inequalities of HVZ type are satisfied.