The Strongly Coupled Polaron on the Torus: Quantum Corrections to the Pekar Asymptotics

Abstract

We investigate the Fröhlich polaron model on a three-dimensional torus, and give a proof of the second-order quantum corrections to its ground-state energy in the strong-coupling limit. Compared to previous work in the confined case, the translational symmetry (and its breaking in the Pekar approximation) makes the analysis substantially more challenging.

Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron

“…Now inserting (1.24) and (1.25) into (1.21), and based on the expectation that the quasi-particle regime is restricted to |P | 2M eff (α)/α ∼ α, it is clear that the Bogoliubov energy needs to be taken into account in order to see the quasi-particle energy shift given by P 2 /(2α 4 M LP ) α −2 . Mathematically, the validity of (1.25) has been established only for confined polaron models [13,16]. The corresponding upper bound for the unconfined model is a corollary of our main result.…”

“…The effective mass has been studied in [4,10,12,33,34,48]. Further improvements have been obtained for confined polarons or polaron models with more regular interaction [13,16,43]. For completeness, let us also mention recent progress in the understanding of the polaron path measure [3,42] as well as the increased interest in the analysis of the Schrödinger time evolution of strongly coupled polarons [11,14,15,23,30,31,38].…”

“…Even for P = 0 it remains a challenging problem to improve the Lieb-Thomas bound (1.15) such that it includes the quantum corrections of order α −2 . Progress in this direction has been achieved in [13,16] for simplified polaron models in which the electron and the quantum field are confined to suitable finite size regions.…”

The Fröhlich Polaron at Strong Coupling -- Part II: Energy-Momentum Relation and Effective Mass

“…If follows from the analysis in [9] that there exists a unique radial minimizer ϕ Pek of the functional F Pek . With this minimizer at hand, we can introduce the constant m := 2 3 ∇ϕ Pek 2 in Eq. (1.1).…”

“…This was first shown in [4] for a confined polaron model on a bounded domain, but the method extends in a straightforward way to the model on R 3 , as shown in [13] (see also [2] for the corresponding result for a polaron model on a torus). In the following, let Ψ ǫ be a state satisfying…”