Communications in Mathematical Physics

, Volume 31, Issue 4, pp 327–340 | Cite as

The classical limit of quantum spin systems

  • Elliott H. Lieb


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 momentumJ → ∞ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality isZ C (J)≦Z Q (J)≦Z C (J+1).


Neural Network Free Energy Statistical Physic Complex System Lower Bound 
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Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  1. 1.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance

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