Renormalization and Quantum Scaling of Frenkel-Kontorova Models

TL;DR

This paper extends the classical understanding of the Frenkel-Kontorova models to quantum and finite temperature regimes using renormalization techniques, revealing how classical phenomena like hierarchical melting persist in quantum settings.

Contribution

It introduces a quantum and finite temperature generalization of the renormalization approach for Frenkel-Kontorova models, connecting classical and quantum phenomena.

Findings

Classical results are extended to the quantum regime.

Scaling laws for low frequency and quantum effects are derived.

Hierarchical melting phenomena are shown to persist in quantum regimes.

Abstract

We generalise the classical Transition by Breaking of Analyticity for the class of Frenkel-Kontorova models studied by Aubry and others to non-zero Planck's constant and temperature. This analysis is based on the study of a renormalization operator for the case of irrational mean spacing using Feynman's functional integral approach. We show how existing classical results extend to the quantum regime. In particular we extend MacKay's renormalization approach for the classical statistical mechanics to deduce scaling of low frequency effects and quantum effects. Our approach extends the phenomenon of hierarchical melting studied by Vallet, Schilling and Aubry to the quantum regime.