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

Exercises
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.

Content:
  1. Introduction
    - Classical phase transitions - phase diagram of water - Ehrenfest classification - continuous phase transitions - quantum phase transitions
    [ Notes | Slides | Suppl. Mat. | Lecture 01 | 02 | 03 ]
  2. 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 ]
  3. 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 ]
  4. 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. ]
  5. 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 ]

Prerequisites:
Skriptum :
  • 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.

Literature:

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 ]
Reviews on critical phenomena and (quantum) phase transitions General texts on statistical mechanics
  • 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 ]
General texts on quantum field theory
  • 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 ]
Additional material
Exercises:

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.)

Problem sets:

    Sheet 01:
    Exact solution of the one-dimensional Ising model - Spontaneuous magnetization in one-dimensional Ising-like models
    Sheet 02:
    Mean-field analysis of the Ising model in a transverse field - Infinite-range Ising model - Spin-1 Ising model and upper critical dimension
    Sheet 03:
    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
    Sheet 04:
    One-loop flow equations for the O(N)-symmetric φ4-theory
    Sheet 05:
    Jordan-Wigner transformation of the transverse Ising model - XXZ Hamiltonian and spinless Hubbard model


Exam:

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.