Advertisement

Communications in Mathematical Physics

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

The classical limit of quantum spin systems

  • Elliott H. Lieb
Article

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 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).

Keywords

Neural Network Free Energy Statistical Physic Complex System Lower Bound 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Millard, K., Leff, H.: J. Math. Phys.12, 1000–1005 (1971).Google Scholar
  2. 2.
    Arecchi, F.T., Courtens, E., Gilmore, R., Thomas, H.: Phys. Rev. A6, 2211–2237 (1972).Google Scholar
  3. 3.
    Radcliffe, J.M.: J. Phys. A4, 313–323 (1971).Google Scholar
  4. 4.
    Kutzner, J.: Phys. Lett. A41, 475–476 (1972).Google Scholar
  5. 4a.
    Atkins, P.W., Dobson, J.C.: Proc. Roy. Soc. (London) A, A321, 321–340 (1971).Google Scholar
  6. 5.
    Golden, S.: Phys. Rev. B137, 1127–1128 (1965).Google Scholar
  7. 6.
    Griffiths, R.B.: J. Math. Phys.5, 1215–1222 (1964).Google Scholar
  8. 7.
    Hepp, K., Lieb, E. H.: The equilibrium statistical mechanics of matter interacting with the quantized radiation field. Preprint.Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

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

Personalised recommendations