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

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.

arXiv comment14 pages

References in corpus (16)

• High temperature fractional quantum Hall states
• Fractional quantum Hall states at zero magnetic field
• Nearly-flat bands with nontrivial topology
• Fractional quantum Hall effect in the absence of Landau levels
• Fractional Chern Insulator
• Fractional Quantum Hall Effect of Hard-Core Bosons in Topological Flat Bands
• Fractional Chern Insulators in Topological Flat bands with Higher Chern Number
• Zoology of Fractional Chern Insulators
• Fractional topological liquids with time-reversal symmetry and their lattice realization
• Emergent Many-Body Translational Symmetries of Abelian and Non-Abelian Fractionally Filled Topological Insulators
• Fractional Quantum Hall Effect in Topological Flat Bands with Chern Number Two
• Fractional Chern Insulators and the W-Infinity Algebra
• Non-Abelian Quantum Hall Effect in Topological Flat Bands
• Fractional quantum-Hall liquid spontaneously generated by strongly correlated t_2g electrons
• Noncommutative geometry for three-dimensional topological insulators
• From Fractional Chern Insulators to a Fractional Quantum Spin Hall Effect

Cited by in corpus (10)

• Fractional Quantum Hall Physics in Topological Flat Bands
• Hamiltonian theory of fractionally filled Chern bands
• Fractional Chern insulators with strong interactions far exceeding bandgaps
• The single-mode approximation for fractional Chern insulators and the fractional quantum Hall effect on the torus
• Combined topological and Landau order from strong correlations in Chern bands
• Fractional Chern Insulator phase at the transition between checkerboard and Lieb lattice
• Bosonization of Fermi liquids in a weak magnetic field
• Topological BF theory of the quantum hydrodynamics of incompressible polar fluids
• Weyl Semimetal and Nonassociative Nambu Geometry
• Interband excitations in the 1D limit of two-band fractional Chern insulators