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 α, 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 < α < 1. For all values of N and B we find an interval α N,B < α < 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 .

On the strong coupling limit of many-polaron systems in electromagnetic fields

“…Hence they are rescaled such that they grow with increasing α. Combining this with the binding of Pekar-Tomasevich N -polarons subject to a constant magnetic field, which was recently established in [3], we prove binding for Fröhlich N -polarons in strong constant magnetic fields for large couplings. In the N -particle case without external fields, similar asymptotic exactness and binding results have recently been derived in [2].…”

“…We already know binding for Pekar-Tomasevich N -polarons in constant magnetic fields for ν in some neighborhood of ν = 2 [3], hence as a corollary of Theorem 1.1, it follows the existence of bound states for Fröhlich N -polarons in strong constant magnetic fields for α large enough. …”

“…This follows from (4) and the binding of N -polarons in the Pekar-Tomasevich model without external fields (see [11] or [3] for A = 0). In other words, in the leading order for α → ∞ the binding energy does not depend on the (non-scaled) external fields.…”

On the dynamics of polarons in the strong-coupling limit

“…The parameter α > 0 is a dimensionless coupling constant and large α means strong coupling. See Section 4 for an explanation of our units in (1). We are interested in the dynamics generated by H F in the case of large α and our work is inspired by the following well-known result on the ground-state energy E F = inf σ(H F ): Let E P be the minimum of ψ, H F ψ with respect to all product states ψ = ϕ ⊗ η of normalized ϕ ∈ L 2 and η ∈ F. Then E P , which is known as the Pekar energy, is an upper bound on E F and E P → E F in the limit α → ∞ [5].…”

The strong-coupling polaron in static electric and magnetic fields

“…The question of binding is not addressed, however, and seems to require a similar analysis of the ground state energy of the Pekar-Tomasevich model. For the binding of N > 2 polarons in the Pekar-Tomasevich model with and without external magnetic fields we refer to [8] and [2], respectively. For the thermodynamic stability, the non-binding, and the binding-unbinding transition of multipolaron systems the reader may consult the short review [6] and the references therein.…”