Multipolarons in a Constant Magnetic Field

Abstract

The binding of a system of $N$ polarons subject to a constant magnetic field of strength $B$ is investigated within the Pekar-Tomasevich approximation. In this approximation the energy of $N$ polarons is described in terms of a non-quadratic functional with a quartic term that accounts for the electron-electron self-interaction mediated by phonons. The size of a coupling constant, denoted by $\alpha$, in front of the quartic is determined by the electronic properties of the crystal under consideration, but in any case it is constrained by $0<\alpha<1$. For all values of $N$ and $B$ we find an interval $\alpha_{N,B}<\alpha<1$ where the $N$ polarons bind in a single cluster described by a minimizer of the Pekar-Tomasevich functional. This minimizer is exponentially localized in the $N$-particle configuration space $\R^{3N}$.