Teacher(s)

Hagendorf Christian; Ruelle Philippe;

Language

French

Content

• Introduction : classical statistical mechanics in d dimensions and quantum systems in d-1 dimensions, transfer matrices: spectrum and correlation functions, a renormalisation group reminder, scaling relations;
• The two-dimensional Ising model : duality and the critical point, disorder operators, lattice fermions, transfer matrix, Hamiltonian limit, spectrum of the quantum Hamiltonian : the Jordan-Wigner transformation, diagonalisation, scaling limit: the free fermion, conformal Hamiltonians;
• The conformal Ward identity :  conformal invariance in d > 2 dimensions, the energy-momentum tensor, conformal invariance in d = 2 dimensions, the Ward identity, the Virasoro algebra, central charge, quasi-primary and primary fields, conformal families, the operator product expansion;
• Free-field theories in two dimensions : the massless Gaussian free field in two dimensions, propagators, correlations functions and the Wick theorem, vertex operators; the massless free fermion in two dimensions, the fermionic Wick theorem;
• Minimal models — an introduction : the operator formalism, representations of the Virasoro algebra, unitarity, the Kac determinant, reducibility and singular vectors, differential equations for correlation functions, fusion rules, minimal models: examples and related models in statistical mechanics, the critical Ising model : correlation functions in the scaling limit.

Teaching methods

The learning activities consist of lectures. The lectures introduce fundamental concepts of the theory of nonlinear systems and their motivation through concrete examples from various scientific disciplines.

Evaluation methods

The evaluation is based on an oral exam. The students are asked to present their personal work on a physical or mathematical problem that is related to the course’s topics. The evaluation tests the student’s knowledge and his understanding of the notions seen in the theoretical course, his ability to apply them to new problems and his oral presentation skills.

Bibliography

• J. Cardy, Scaling and renormalisation in statistical physics. Cambridge lecture notes in statistical physics (1996). 
• Ph. Di Francesco, P. Mathieu, D. Sénéchal, Conformal field theory. Springer (1997).
• P. Ginsparg, Applied conformal field theory. arXiv:hep-th/9108028 (1991).
• C. Itzykson, J.M. Drouffe, Théorie statistique des champs. EDP Sciences (1989).
• G. Mussardo, Statistical field theory. Oxford University Press (2010).

Faculty or entity

PHYS