Local gauge and magnetic translation groups

TL;DR

This paper explores how local gauge choices on a crystal lattice influence magnetic translation operators, showing they commute with the Hamiltonian regardless of the gauge, and that their commutators depend only on the magnetic field.

Contribution

It introduces a local gauge field framework that ensures magnetic translation operators commute with the Hamiltonian for any gauge, revealing gauge-independent properties of their algebra.

Findings

Operators commute with the Hamiltonian for any gauge A.

The factor system depends only on a global gauge.

Commutators depend solely on the magnetic field, not the gauge.

Abstract

The magnetic translation group was introduced as a set of operators T(R)=\exp[-iR.(p-eA/c)/\hbar]. However,these operators commute with the Hamiltonian for an electron in a periodic potential and a uniform magnetic field if the vector potential A (the gauge) is chosen in a symmetric way. It is showed that a local gauge field A_R(r) on a crystal lattice leads to operators, which commute with the Hamiltonian for any (global) gauge field A=A(r). Such choice of the local gauge determines afactor system \omega(R,R')= T(R)T(R') T(R+R')^{-1}, which depends on a global gauge only. Moreover, for any potential A a commutator T(R)T(R')T(R)^{-1}T(R')^{-1} depends only on the magnetic field and not on the gauge.