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Conformal Field Theory Applied to Loop Models

  • Jesper Lykke Jacobsen
Part of the Lecture Notes in Physics book series (LNP, volume 775)

The application of methods of quantum field theory to problems of statistical mechanics can in some sense be traced back to Onsager's 1944 solution [1] of the two-dimensional Ising model. It does however appear fair to state that the 1970's witnessed a real gain of momentum for this approach, when Wilson's ideas on scale invariance [2] were applied to study critical phenomena, in the form of the celebrated renormalisation group [3]. In particular, the so-called ε expansion permitted the systematic calculation of critical exponents [4], as formal power series in the space dimensionality d, below the upper critical dimension d c . An important lesson of these efforts was that critical exponents often do not depend on the precise details of the microscopic interactions, leading to the notion of a restricted number of distinct universality classes.

Meanwhile, further exact knowledge on two-dimensional models had appeared with Lieb's 1967 solution [5] of the six-vertex model and Baxter's subsequent 1971 generalisation [6] to the eight-vertex model. These solutions challenged the notion of universality class, since they provided examples of situations where the critical exponents depend continuously on the parameters of the underlying lattice model. On the other hand, the techniques of integrability used relied crucially on certain exact microscopic conservation laws, thus placing important restrictions on the models which could be thus solved.

Keywords

Partition Function Critical Exponent Vertex Operator Continuum Limit Operator Product Expansion 
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.

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© Canopus Academic Publishing Limited 2009

Authors and Affiliations

  • Jesper Lykke Jacobsen
    • 1
  1. 1.Laboratoire de Physique ThéoriqueEcole Normale Supérieure (LPTENS)France

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