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arXiv:cond-mat/0304125 · Physical Review Letters

Bohr-Sommerfeld Quantization of Spin Hamiltonians

Anupam Garg, Michael Stone

Published Jan 8, 2004• 23 citations
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Abstract

The Bohr-Sommerfeld rule for a spin system is obtained, including the first quantum corrections. The rule applies to both integer and half-integer spin. It is tested for various models, in particular, the Lipkin-Meshkov-Glick model, and found to agree very well with exact results.
DOI
10.1103/physrevlett.92.010401

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With the use of the Weyl correspondence between quantum mechanical operators and classical dynamical functions, an exact quantization rule is derived for a system with one degree of freedom and arbitrary Hamiltonian. In the semiclassical limit of small Planck's constant, this reduces to the Bohr-Sommerfeld quantization rule as derived by the WKB approximation. Higher order terms are also derived. It is further proven that for a simple harmonic oscillator the Bohr-Sommerfeld rule must give the exact energy eigenvalues for all states.
Abstract Part I: Quantum Collective Spin Systems We use a coherent spin state formalism in order to study collective spin models, which have many applications in physics. In particular, the Lipkin-Meshkov-Glick (LMG) has been analyzed in the thermodynamic limit. The method developed during this work can, in principle, be used for more general Hamiltonians written as a function of the su(2) algebra generators. We have derived exact expressions for the integrated density of states for this model. The nature of the density of states singularities has been detailed. The first order finite size corrections as well as observables expectation values were also computed. The standard Bohr-Sommerfeld quantization approach, adapted to the spin case, is no longer valid near the spectral singularities. In order to treat these cases, we have developed a new approach that permits to describe the spectrum in the neighbourhood of the critical points. Part II: Adiabatic Quantum Computation We have proposed a simple model which highlights the relations between quantum phase transitions and adiabatic (quantum) computation. This model puts in evidence the importance of the choice of the initial Hamiltonian and of the adiabatic path in the parameter space; it should be helpful as a toy model for more realistic cases. We have also studied the state populations dynamics when a quantum phase transition point is crossed, for the LMG model studied in the first part. Numerical simulations show that the dynamics of the populations is very sensitive to the presence of exceptional points in the spectrum, which a simplified model for the quantum evolution already suggests.
In this paper we theoretically report an unconventional quantum phase transition of a simple Lipkin–Meshkov–Glick model: an interacting collective spin system without external magnetic field. It is shown that this model with integer-spin can exhibit a first-order quantum phase transition between different disordered phases, and more intriguingly, possesses a hidden supersymmetry at the critical point. However, for half-integer spin we predict another first-order quantum phase transition between two different long-range-ordered phases with a vanishing energy gap, which is induced by the destructive topological quantum interference between the intanton and anti-instanton tunneling paths and accompanies spontaneously breaking of supersymmetry at the same critical point. We also show that, when the total spin-value varies from half-integer to integer this model can exhibit an abrupt variation of Berry phase from π to zero.