The role of a form of vector potential - normalization of the antisymmetric gauge

TL;DR

This paper compares different gauge choices for the vector potential in magnetic systems, emphasizing the importance of normalization, magnetic cells, and gauge dependence of factor systems and translation commutators.

Contribution

It introduces a normalization procedure that unifies various approaches and clarifies the gauge dependence of factor systems and the gauge independence of magnetic translation commutators.

Findings

Normalization aligns Brown's, Zak's, and other approaches.

Magnetic cells are crucial for proper boundary conditions.

Factor systems are gauge-dependent, but magnetic translation commutators are gauge-independent.

Abstract

Results obtained for the antisymmetric gauge A=[Hy,-Hx]/2 by Brown and Zak are compared with those based on pure group-theoretical considerations and corresponding to the Landau gauge A=[0,Hx]. Imposing the periodic boundary conditions one has to be very careful since the first gauge leads to a factor system which is not normalized. A period N introduced in Brown's and Zak's papers should be considered as a magnetic one, whereas the crystal period is in fact 2N. The `normalization' procedure proposed here shows the equivalence of Brown's, Zak's, and other approaches. It also indicates the importance of the concept of magnetic cells. Moreover, it is shown that factor systems (of projective representations and central extensions) are gauge-dependent, whereas a commutator of two magnetic translations is gauge-independent. This result indicates that a form of the vector potential (a gauge) is also important in physical investigations.