Local gauge and magnetic translation groups

Authors:

• Wojciech Stefan Florek

Abstract

The magnetic translation group was introduced as a set of operators T(R) = exp[-iR-(p-eA/c)/h]. 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 a factor system ω(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 depends only on the magnetic field and not on the gauge.