Quantum Mechanical Hamiltonians with Large Ground-State Degeneracy

TL;DR

This paper explores Hamiltonians with large or infinite ground-state degeneracy, linking it to Dirac operators, and explicitly constructs ground states for certain classes, including higher-dimensional Landau levels.

Contribution

It identifies a special class of Hamiltonians with explicitly constructible degenerate ground states across any dimension, generalizing Landau level models.

Findings

Explicit ground states for a class of Hamiltonians in any dimension.

Degeneracy related to the asymptotic behavior of background potentials.

Connection between ground-state degeneracy and Dirac operator properties.

Abstract

Nonrelativistic Hamiltonians with large, even infinite, ground-state degeneracy are studied by connecting the degeneracy to the property of a Dirac operator. We then identify a special class of Hamiltonians, for which the full space of degenerate ground states in any spatial dimension can be exhibited explicitly. The two-dimensional version of the latter coincides with the Pauli Hamiltonian, and recently-discussed models leading to higher-dimensional Landau levels are obtained as special cases of the higher-dimensional version of this Hamiltonian. But, in our framework, it is only the asymptotic behavior of the background `potential' that matters for the ground-state degeneracy. We work out in detail the ground states of the three-dimensional model in the presence of a uniform magnetic field and such potential. In the latter case one can see degenerate stacking of all 2d Landau levels along the magnetic field axis.