Quantum Field Theory of Many-Body Systems
Introduction to Quantum Phase Transitions
Lecture (MVSpec)Thomas Gasenzer
Tuesday, 11:15-13:00 (starting on 14/04); Thursday, 11:15-13:00 (during odd weeks, starting on 23/04); INF 227 (KIP), SR 1.404. [LSF]
Note: Exam: Tue, 28/07, 11:00-13:00 hrs, INF 227, SR 1.404
Tutor: Asier Piñeiro Orioli
Register and view group list here.
Classes take place in general during even weeks on Thursdays, 11:15-12:45 hrs, starting on 30/04: INF 227 (KIP), SR 1.404.
Written exam on Tue., 28/07/15, 11:00-13:00 hrs, INF 227 (KIP), SR 1.404.
Content - Prerequisites - Script - Literature - Supplementary materials - Exercises - Exam
The lecture course provides an introduction to field theoretic methods for systems with many degrees of freedom. A focus will be set on quantum phase transitions, with special emphasis on applications to ultracold, mostly bosonic, atomic gases as they are the subject of many fore-front present-day experiments. The course will introduce to the basis of the theory of classical and quantum phase transitions, with a special emphasis on simple model applications. Methodologically, the lecture will build on the basics of the operator as well as the path-integral approach to quantum field theory. Knowledge of the basics of quantum mechanics, statistical mechanics, and quantum field theory is presumed.
- Classical phase transitions - phase diagram of water - Ehrenfest classification - continuous phase transitions - quantum phase transitions
[ Notes | Slides | Suppl. Mat. | Lecture 01 | 02 | 03 ]
Phase transition in the classical Ising model
- Ising Hamiltonian - Spontaneous symmetry breaking - Thermodynamic properties - Phase transitions in the Ising model - Landau mean-field theory - Mean-field critical exponents - Correlation functions - Hubbard Stratonovich transformation - Functional-integral representation - Ginzburg-Landau-Wilson functional - Saddlepoint approximation and Gaussian effective action - Ginzburg criterion
[ Notes | Slides | Suppl. Mat. | Lecture 03 | 04 | 05 | 06 | 07 ]
Renormalisation-group theory in position space
- Block-spin transformation - Transfer-matrix solution of the 1D Ising chain - RG stepping for the 1D and 2D Ising models - Critical point - RG fixed points - Relevant and irrelevant couplings - Universality and universality class - Renormalisation-group flows - Scaling properties of the free energy and of the two-point correlation function - Scaling relations between critical exponents - The scaling hypothesis
[ Notes | Lecture 08 | 09 | 10 ]
Wilson's Renormalisation Group
- Perturbation theory - Linked-Cluster and Wick's theorems - Dyson equation - One-loop critical properties - Dimensional analysis - Momentum-scale RG - Gaussian fixed point - Wilson-Fisher fixed point - Epsilon-expansion - Critical exponents - Wave function renormalisation and anomalous dimension - Suppl. Mat.: Asymptotic expansions
[ Notes | Slides | Lecture 11 | 12 | 13 | 14 | Suppl. Mat. ]
Quantum phase transitions
- Quantum Ising model - Mapping of the classical Ising chain to a quantum spin model - Universal scaling behaviour - Thermal as time-ordered correlators - Quantum to classical mapping - Perturbative spectrum of the transverse-field Ising model - Jordan Wigner transformation and exact spectrum - Universal crossover functions near the quantum critical point - Anomalous scaling dimension - Low-temperature and quantum critical regimes - Conformal mapping - Spectral properties close to criticality - Structure factor, susceptibility, and linear response - Relaxational response in the quantum critical regime
[ Notes | Slides | Lecture 14 | 15 | 16 | 17 | 18 | 19 ]
- Quantum Mechanics (PTP 3), Statistical Mechanics (PTP 4); Quantum Field Theory I or Quantum Field Theory of Many-Body Systems
- The notes are available for download above, separately for each chapter.
- The Script of the lecture on QFT of Many-Body Systems in WT 14/15 (with a different focus) can be found here.
Textbooks on critical phenomena and (quantum) phase transitions
- D. Belitz und T.R. Kirkpatrick, in J. Karkheck (Hrsg.), Dynamics: Models and kinetic methods for non-equilibrium many-body systems. Kluwer, Dordrecht (2000). [ Google books | HEIDI ]
- John Cardy, Scaling and renormalization in statistical physics. CUP, Cambridge, 2003. [ Google books | HEIDI ]
- Peter Kopietz, Lorenz Bartosch, Florian Schütz, Introduction to the Functional Renormalization Group. Springer, Berlin Heidelberg, 2010. [ Google books | HEIDI (online) | Errata and Addenda ]
- Lincoln D. Carr (Ed.), Understanding quantum phase transitions. CRC-Press, Boca Raton, 2011. [ Google books | HEIDI ]
- Nigel Goldenfeld, Lectures on phase transitions and the renormalization group. Addison-Wesley, Reading, 1992. [ Google books | HEIDI ]
- Igor Herbut, A modern approach to critical phenomena. CUP, Cambridge, 2007. [ Google books | HEIDI ]
- Subir Sachdev, Quantum Phase Transitions. CUP, Cambridge, 2011. [ Google books | HEIDI (incl. online) ]
- S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Continuous quantum phase transitions. Rev. Mod. Phys. 69, 315 (1997). [ arXiv:cond-mat/9609279 ]
- Jean Zinn-Justin, Quantum field theory and critical phenomena. Clarendon, Oxford, 2004. [ Google books | HEIDI ]
- Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral, Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008). [arXiv:quant-ph/0703044v3]
- Kerson Huang, Statistical Mechanics. Wiley, 1987. [ Google books | HEIDI ]
- Linda E. Reichl, A Modern Course in Statistical Physics. Wiley Interscience, 2nd edition 1998. [ Google books (3rd ed.) | HEIDI ]
- Frederick Reif, Fundamentals of Statistical and Thermal Physics McGraw-Hill, New York, 1987. [ Google books | HEIDI ]
- Franz Schwabl, Statistische Mechanik. Springer, Heidelberg, 2000. [ Google books | HEIDI ]
- M. Toda, R. Kubo, N. Saito, Statistical Physics, Equilibrium Statistical Mechanics, Springer, 2nd edition 1992. [ Google books | HEIDI ]
- Brian Hatfield, Quantum Field Theory of Point Particles and Strings. Addison Wesley, Oxford, 2010. [ Google books | HEIDI ]
- Michael E. Peskin, Daniel V. Schroeder An introduction to quantum field theory. Westview, Boulder, 2006. [ Google books | HEIDI ]
- Xiao-Gang Wen, Quantum Field Theory of Many-Body Systems. OUP, Oxford, 2010. [ Google books | HEIDI ]
- Experiments realising critical opalescence and supercritical fluidity in Water and Carbon Dioxide (further, commercial video with explanations).
- Applet simulating the 2D Ising model (You may have to adapt your Java settings to not block this page as well as isingApplet.html and isingText.html in the same path).
Exercises will be held in general (exceptions posted above) on Thursdays during even weeks, 11:15-12:45 hrs, in SR 1.404, INF 227 (KIP), starting on 30/04/15. Tutor: Asier Piñeiro Orioli (Please register here.)
Exact solution of the one-dimensional Ising model - Spontaneuous magnetization in one-dimensional Ising-like models
Mean-field analysis of the Ising model in a transverse field - Infinite-range Ising model - Spin-1 Ising model and upper critical dimension
Real-space RG of the one-dimensional Ising model - Migdal-Kadanoff RG for the 2D Ising model - Dangerously irrelevant coupling in φ4-theory for d > 4
One-loop flow equations for the O(N)-symmetric φ4-theory
Jordan-Wigner transformation of the transverse Ising model - XXZ Hamiltonian and spinless Hubbard model
Passing the written exam, which will prospectively take place on Tue, 28/07/15, 11:00-13:00 hrs, INF 227 (KIP), SR 1.404, will be the condition to obtain 6 CPs for the lecture.
Rules for the exam: You are allowed to use one A4 two-sided and handwritten sheet. No electronic devices of any kind are permitted. The exam lasts 120 mins. Please bring enough paper to be able to start every problem on a new sheet of paper.