# 101-0190-07L Two-dimensional Phase Transitions

Semester | Spring Semester 2017 |

Lecturers | M. Henkel |

Periodicity | non-recurring course |

Language of instruction | English |

Abstract | This course aims at an introduction to two-dimensional phase transitions and the techniques of conformal invariance required for their descriptions. Some familiarity with equilibrium statistical mechanics will be assumed, but a prior knowledge of quantum-field theory is not required. Explicit applications to specific models will be given to show how the methods actually work in practise. |

Objective | Interacting many-body systems acquire many new properties, which can be qualitatively different from the properties of a single individual degree of freedom. Phase transitions are paradigmatic examples of the collective behaviour where strong fluctuations preclude the use of simplistic mean-field methods. On the other hand, the underlying field-theories are characterised not only by scale-invariance, but in many cases by the larger symmetry of conformal invariance. Furthermore, phase transitions in two dimensions have a genuine physical interest and are found in many practically relevant real-world applications. One can then use the powerful techniques of two-dimensional conformal invariance for a deep understanding of their behaviour and for very detailed predictions. |

Content | 1. Examples of two-dimensional critical phenomena Anti-ferromagnets, adsorption, superconductivity,... background on critical exponents, scaling relations, relationship with critical quantum chains 2. Scale-invariance Renormalisation group, invariance of partition function, co-variance of correlators 3. Conformal transformations in d dimensions Conformal group and Lie algebra, quasi-primary scaling operators, shape of correlators; does scale-invariance imply conformal invariance ? 4. Two-dimensional conformal transformations and complex analyticity Primary scaling operators, energy-momentum tensor & the Virasoro algebra, radial quantisation, the free boson, measuring scaling dimensions and central charges 5. Representation theory of the Virasoro algebra Null vectors, Kac formula, unitary minimal models, rational CFT 6. Operator product expansions Four-point correlators from null vectors, modular invariance 7. The two-dimensional Ising model Conformal invariance as spectrum-generating symmetry, operator content 8. Extensions and applications e.g. geometric phase transitions (percolation), logarithmic CFT ; relevant perturbations, c-theorem and delta-theorem |

Literature | 1. P. Di Francesco, P. Mathieu, D. Sénéchal, Conformal Field-Theory, Springer (1997) 2. M. Henkel, Conformal Invariance and Critical Phenomena, Springer (1999) 3. R. Blumenhagen, E. Plauschinn, Introduction to Conformal Field-Theory, Springer (2009) 4. S. Rychkov, EPFL lectures on CFT, CERN-TH/2016-012 arxiv:1601.05000 5. H. Nishimori, G. Ortiz, Elements of Phase Transitions and Critical Phenomena, Oxford (2011) 6. M. Henkel, D. Karevski (éds), Conformal Invariance: Loops, Interfaces ..., Springer (2012) |