Bootstrapping Bloch bands

TL;DR

This paper extends bootstrap methods to accurately compute the band structure of a quantum particle in a periodic potential, introducing new techniques for dimensionality reduction and probability distribution reconstruction.

Contribution

The authors develop a novel bootstrap approach for continuous Bloch spectra, incorporating position and momentum moments, and introduce methods for dimensionality reduction and probability distribution reconstruction.

Findings

Accurate band structure obtained using bootstrap with position and momentum moments.

New techniques successfully reduce search space dimensionality.

Distribution probability reconstruction enables calculation of Bloch momentum.

Abstract

Bootstrap methods, initially developed for solving statistical and quantum field theories, have recently been shown to capture the discrete spectrum of quantum mechanical problems, such as the single particle Schr\"odinger equation with an anharmonic potential. The core of bootstrap methods builds on exact recursion relations of arbitrary moments of some quantum operator and the use of an adequate set of positivity criteria. We extend this methodology to models with continuous Bloch band spectra, by considering a single quantum particle in a periodic cosine potential. We find that the band structure can be obtained accurately provided the bootstrap uses moments involving both position and momentum variables. We also introduce several new techniques that can apply generally to other bootstrap studies. First, we devise a trick to reduce by one unit the dimensionality of the search space for the variables parametrizing the bootstrap. Second, we employ statistical techniques to reconstruct the distribution probability allowing to compute observables that are analytic functions of the canonical variables. This method is used to extract the Bloch momentum, a quantity that is not readily available from the bootstrap recursion itself.