Angular Momentum on the Lattice: The Case of Non-Zero Linear Momentum

TL;DR

This paper explores how continuum angular momentum states with non-zero linear momentum are represented on the lattice, detailing the mapping between continuum and lattice IRs and illustrating degeneracies in the continuum limit.

Contribution

It provides a detailed analysis of the reduction of continuum angular momentum IRs to lattice IRs for non-zero momentum, including an explicit example with the moving harmonic oscillator.

Findings

Mapping of continuum states to lattice IRs for non-zero momentum

Degeneracies between lattice IR levels in the continuum limit

Explicit example with the moving isotropic harmonic oscillator

Abstract

The irreducible representations (IRs) of the double cover of the Euclidean group with parity in three dimensions are subduced to the corresponding cubic space group. The reduction of these representations gives the mapping of continuum angular momentum states to the lattice in the case of non-zero linear momentum. The continuous states correspond to lattice states with the same momentum and continuum rotational quantum numbers decompose into those of the IRs of the little group of the momentum vector on the lattice. The inverse mapping indicates degeneracies that will appear between levels of different lattice IRs in the continuum limit, recovering the continuum angular momentum multiplets. An example of this inverse mapping is given for the case of the ``moving'' isotropic harmonic oscillator.