Quantum mechanical Hamiltonians with large ground-state degeneracy

• Published in: Annals of physics Vol. 331; pp. 258 - 268
• Main Authors: Lee, Choonkyu, Lee, Kimyeong
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01-04-2013; Elsevier BV
• Subjects: 3-dim Landau level; Dirac operator; Large ground-state degeneracy; Mechanical properties; Nonrelativistic quantum mechanics; Quantum field theory; Nonrelativistic quantum mechanics; Large ground-state degeneracy; Dirac operator; 3-dim Landau level
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2013.01.003

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.
► We study a class of nonrelativistic Hamiltonian which is a square of the Dirac operator. ► Some of them has a large number of degenerate ground states. ► 2-dim Pauli Hamiltonian and 3-dim Landau problem are a part of this class. ► In a version, all 2-dim Landau levels become degenerate ground states, stacking along the magnetic direction.