Path-integral methods for treating quantal behavior in solids: Mean-field theory and the effects of fluctuations

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Abstract
Recent developments in the chemical literature have shown how discretized path integrals may be used to treat the quantum-mechanical internal degrees of freedom of molecules in liquids. This paper suggests that analogous methods might also be useful for quantum-mechanical solid-state problems, such as those posed by lattice-spin systems. The idea, illustrated in this paper by a common model for tunneling in solids, the transverse Ising model, is to transform the quantal spins into classical spins with ‘‘internal structure.’’ The resulting classical system can then be treated by analytical methods. For the transverse Ising model, both the traditional mean-field theory and a new approximation (which includes a fluctuation correction) are derived in this way. The effects of thermal and quantum-mechanical fluctuations are shown to be largely identical for the example considered.