Canonical Transformations in Crystals

TL;DR

This paper presents exact solutions for canonical transformations in crystal lattices, enabling precise analysis of particle dynamics under electric fields, especially in triangular lattices, expanding the solvable models in quantum crystal physics.

Contribution

It introduces exact solutions to Mello-Moshinsky equations for discrete symmetric problems, extending the class of solvable models beyond bilinear Hamiltonians in crystal systems.

Findings

Exact propagators for Wannier-Stark ladders in 1D and 2D crystals.

Detailed analysis of particle behavior in a triangular lattice under time-dependent electric fields.

Generalized Mello-Moshinsky equations for arbitrary lattice structures.

Abstract

The space representations of linear canonical transformations were studied by Moshinsky and Quesne in 1972. For a few decades, the bilinear hamiltonian remained as the only exactly solvable representative for such problems. In this work we show that the Mello-Moshinsky equations can be solved exactly for a class of problems with discrete symmetry, leading to exact propagators for Wannier-Stark ladders in one and two dimensional crystals. We give a detailed study for a particle in a triangular lattice under the influence of a time-dependent electric field. A more general set of Mello-Moshinsky equations for arbitrary lattices is presented.