Conformal Invariance and Critical Phenomena

1. Front Matter

   Pages I-XVII

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2. Critical Phenomena: a Reminder

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   Pages 1-42

3. Conformal Invariance

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   Pages 43-62

4. Finite-Size Scaling

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   Pages 63-82

5. Representation Theory of the Virasoro Algebra

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   Pages 83-100

6. Correlators, Null Vectors and Operator Algebra

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   Pages 101-116

7. Ising Model Correlators

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   Pages 117-126

8. Coulomb Gas Realization

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   Pages 127-140

9. The Hamiltonian Limit and Universality

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   Pages 141-156

10. Numerical Techniques

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    Pages 157-182

11. Conformal Invariance in the Ising Quantum Chain

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    Pages 183-204

12. Modular Invariance

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    Pages 205-218

13. Further Developments and Applications

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    Pages 219-260

14. Conformal Perturbation Theory

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    Pages 261-278

15. The Vicinity of the Critical Point

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    Pages 279-320

16. Surface Critical Phenomena

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    Pages 321-368

17. Strongly Anisotropic Scaling

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    Pages 369-384

18. Back Matter

    Pages 385-417

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Critical phenomena arise in a wide variety of physical systems. Classi­ cal examples are the liquid-vapour critical point or the paramagnetic­ ferromagnetic transition. Further examples include multicomponent fluids and alloys, superfluids, superconductors, polymers and fully developed tur­ bulence and may even extend to the quark-gluon plasma and the early uni­ verse as a whole. Early theoretical investigators tried to reduce the problem to a very small number of degrees of freedom, such as the van der Waals equation and mean field approximations, culminating in Landau's general theory of critical phenomena. Nowadays, it is understood that the common ground for all these phenomena lies in the presence of strong fluctuations of infinitely many coupled variables. This was made explicit first through the exact solution of the two-dimensional Ising model by Onsager. Systematic subsequent developments have been leading to the scaling theories of critical phenomena and the renormalization group which allow a precise description of the close neighborhood of the critical point, often in good agreement with experiments. In contrast to the general understanding a century ago, the presence of fluctuations on all length scales at a critical point is emphasized today. This can be briefly summarized by saying that at a critical point a system is scale invariant. In addition, conformal invaTiance permits also a non-uniform, local rescal­ ing, provided only that angles remain unchanged.

• Application of integrable systems in statistical physics
• Computational Physics
• Critical Phenomena
• Infinite-dimensional Lie algebras
• Phase Transitions
• Quantum Field Theory
• algebra
• numerical methods

• Laboratoire de Physique des Matériaux CNRS-UMR 7556, Université Henri Poincaré Nancy I, Vandœuvre lès Nancy, France

  Malte Henkel

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