Magnetic translation algebra with or without magnetic field in the continuum or on arbitrary Bravais lattices in any dimension

TL;DR

This paper demonstrates that magnetic translation algebra can be realized in any dimension using fermionic bilinears, applicable in both continuum and lattice systems, and explores its implications for Hamiltonian representations and electron interactions.

Contribution

It generalizes the realization of magnetic translation algebra to arbitrary dimensions and shows its completeness in even dimensions, providing new tools for analyzing fermionic systems.

Findings

Algebra can be closed in any dimension using fermionic bilinears.

Generators are complete in even dimensions for particle-number conserving Hamiltonians.

Interactions can alter the bandwidth when expressed in this algebra.

Abstract

The magnetic translation algebra plays an important role in the quantum Hall effect. Murthy and Shankar, arXiv:1207.2133, have shown how to realize this algebra using fermionic bilinears defined on a two-dimensional square lattice. We show that, in any dimension $d$, it is always possible to close the magnetic translation algebra using fermionic bilinears, whether in the continuum or on the lattice. We also show that these generators are complete in even, but not odd, dimensions, in the sense that any fermionic Hamiltonian in even dimensions that conserves particle number can be represented in terms of the generators of this algebra, whether or not time-reversal symmetry is broken. As an example, we reproduce the $f$-sum rule of interacting electrons at vanishing magnetic field using this representation. We also show that interactions can significantly change the bare bandwidth of lattice Hamiltonians when represented in terms of the generators of the magnetic translation algebra.