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The Hamiltonian Limit and Universality

  • Malte Henkel
Chapter
  • 541 Downloads
Part of the Texts and Monographs in Physics book series (TMP)

Abstract

We have already discussed in Chap. 1 the close analogy between statistical mechanics and field theory. Reinterpreting the transfer matrix as an Euclidean time evolution operator, we were led to define a corresponding quantum Hamiltonian. While in general, this will have a rather complicated form, there is a limit procedure, called the Hamiltonian limit, to simplify the structure of the transfer matrix considerably without affecting the universal properties of the critical behaviour under study [582, 242] . As compared to the isotropic transfer matrix, the resulting quantum Hamiltonian is much sparser, which is useful for numerical investigations. The quantum Hamiltonian contains the information about the universal critical behaviour in condensed form. To illustrate this, we show that for the Ising spin system and for the scalar φ4 field theory, which are in the same universality class, the same quantum Hamiltonian is obtained. For reviews, see [411, 312] .

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Malte Henkel
    • 1
  1. 1.Laboratoire de Physique des Matériaux CNRS-UMR 7556Université Henri Poincaré Nancy IVandœuvre lès NancyFrance

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