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arXiv:1211.5211 · Annals of Physics

Quantum mechanical Hamiltonians with large ground-state degeneracy

Choonkyu Lee, Kimyeong Lee

Published Jan 11, 2013• 0 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2013.01.003

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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.
Counting ground state degeneracy of a k-local Hamiltonian is important in many fields of physics. Its complexity class is harder than that of finding the ground state of a k-local Hamiltonian. Very few methods can efficiently count the degeneracy of ground state while many are known for finding the ground state. This Letter proposes an algorithm for mapping the problem of counting ground state degeneracy into one for finding a special ground state of a k-local super Hamiltonian in an enlarged system with twice as many qubits or spins. We show that all traditional methods, designed for finding the ground state by evolution under a function of a Hamiltonian, can be employed to find the special ground state of the super Hamiltonian from a well-designed initial state and thus can be used to count the ground degeneracy. We combine our algorithm with the power method, the Lanczos method, and the quantum imaginary time evolution method for selective examples to illustrate the detection of phase boundaries, competition between frustration and quantum fluctuation, and potential implementation with quantum circuits.