101-0190-07L  Two-dimensional Phase Transitions

SemesterSpring Semester 2017
LecturersM. Henkel
Periodicitynon-recurring course
Language of instructionEnglish



Courses

NumberTitleHoursLecturers
101-0190-07 VTwo-dimensional Phase Transitions
Dates: TUE 28.02., THUR 02.03., TUE 07.03., THUR 09.03., TUE 14.03., TUE 21.03., TUE 28.03. and TUE 04.04.2017 (from 16:45 until 18:30 in HIL E7).
16s hrs
28.02.16:45-18:30HIL E 7 »
02.03.16:45-18:30HIL E 7 »
07.03.16:45-18:30HIL E 7 »
09.03.16:45-18:30HIL E 7 »
16.03.16:45-18:30HIL E 7 »
23.03.16:45-18:30HIL E 7 »
30.03.16:45-18:30HIL E 7 »
06.04.16:45-18:30HIL E 7 »
M. Henkel

Catalogue data

AbstractThis 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.
ObjectiveInteracting 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.
Content1. 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
Literature1. 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)

Performance assessment

Performance assessment information (valid until the course unit is held again)
Performance assessment as a semester course
ECTS credits2 credits
ExaminersM. Henkel
Typeungraded semester performance
Language of examinationEnglish
RepetitionRepetition only possible after re-enrolling for the course unit.
Additional information on mode of examinationproject

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