Broken Symmetries in statistical Field Theory

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Faculty : Marc GABAY and Pascal SIMON
Tutor : Lih-King Lim
Condensed Matter Physics : Compulsory
Macroscopic Physics and Complexity : Option
Quantum Physics : Option
Theoretical Physics : Option
ECTS credits : 6
Language of instruction : French
Web site :

Description

Phase transitions are ubiquitous in nature. They involve broad classes of transformations of matter : structural, electric, magnetic, superfluid, superconducting to name a few. The pioneering insights of P. Curie and subsequently of L. Landau led to the understanding that seemingly unrelated types of transitions shared in fact common characteristics (belong to the same universality class). A global or local (gauge) symmetry can be broken during the course of the phase change and if a given symmetry is at play, then no matter what the system is or what the nature of the transformation consists of, the same thermodynamic characteristics apply. The so called order parameter, an extensive quantity, quantifies the concept of symmetry breaking. The Curie-Landau theory of phase transitions is extremely rich and fruitful, yet it suffers from several limitations. One is that it treats collective phenomena at the mean-field level, i.e. it neglects fluctuations near the transition point. For critical phenomena, significant deviations from the thermodynamic predictions of the model are observed. These critical fluctuations can be handled by means of a very power tool of field theory, viz by the renormalization group. A statistical field theory treatment of the transition can be set up and deviations from the mean-field results systematically computed. Scaling and critical exponents are hallmarks of this approach. The latter depend, inter alia, on the dimensionality of system, on the range of the interactions and on the characteristics of the order parameter. Another limitation arises when either no clear symmetry appears to be broken at the transition (e.g. in the liquid to gas case) or if no symmetry can be broken, as in the Kosterlitz-Thouless case in two dimensions.

1. Introduction to phase transitions Ehrenfest and Curie-Landau classifications. The concept of order parameter and its relation to phase transitions. Link between the order parameter and symmetry breaking. Expansion of the free energy in powers of the order parameter. Examples : solid-liquid, isotropic-nematic, paramagnetic-ferromagnetic, normal-superfluid, normal-superconducting transitions.

2. Mean-field theory and beyond Mean-field theories for the Ising model. Mean-field critical exponents. Fluctuations as seen on exact solutions of the Ising model in d=1 and d=2 (d is the dimensionality of the system). Bethe lattice, infinite d limit.

3. Landau-Ginzburg free energy functional Explicit construction of the functional for the Ising case. General case : functionals for first order, second order, tricritical transitions.

4. Gaussian model Gaussian approximation and Ginzburg criterion. Correlation functions. Linear response in the thermodynamic limit. Static structure factor. Harmonic phonon fluctuations in a crystal.

5. Renormalization group Real-space renormalization : the Migdal-Kadanoff construction. Renormalization in momentum space : Wilson scheme. Explicit implementation in the "phi^4" case. Loop expansion and diagrams. Epsilon expansion.

6. From order to disorder : approching the critical point from the ordered phase. O(N) model. N=0 - polymer case - limit. Nematic liquid crystals. Non-linear sigma model and renormalisation group.

7. XY model and the Berezinskii-Kosterlitz-Thouless (BKT) transition. Topological defects. BKT transition. Coulomb gas and renormalisation group. Duality.

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