The classical limit of quantum spin systems

Elliott H. Lieb

doi:10.1007/BF01646493

The classical limit of quantum spin systems

Elliott H. Lieb

Mathematics

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373 Scopus citations

Abstract

We derive a classical integral representation for the partition function, Z^Q, of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentum J → ∞ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality is Z^C(J)≦Z^Q(J)≦Z^C(J+1).

Original language	English (US)
Pages (from-to)	327-340
Number of pages	14
Journal	Communications In Mathematical Physics
Volume	31
Issue number	4
DOIs	https://doi.org/10.1007/BF01646493
State	Published - Dec 1973

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

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title = "The classical limit of quantum spin systems",

abstract = "We derive a classical integral representation for the partition function, ZQ, of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentum J → ∞ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality is ZC(J)≦ZQ(J)≦ZC(J+1).",

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N2 - We derive a classical integral representation for the partition function, ZQ, of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentum J → ∞ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality is ZC(J)≦ZQ(J)≦ZC(J+1).

AB - We derive a classical integral representation for the partition function, ZQ, of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentum J → ∞ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality is ZC(J)≦ZQ(J)≦ZC(J+1).

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The classical limit of quantum spin systems

Abstract

All Science Journal Classification (ASJC) codes

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