Quantum Phase Transitions

Lecture (MVSpec)

Thomas Gasenzer
 

Tuesday, 11:15-13:00; Thursday, 11:15-13:00; (starting on 14/10) INF 227 (KIP), SR 3.403. [heiCO / Übungsgruppen] 

Lecture format: In this lecture, I plan to make extensive use of additional online material (videos, quizzes, script). Registered participants gain access to this material as well as to the exercise sheets via the internal webpage of the lecture. The additional material has the purpose to review the content of the lectures presented on site and to test understanding of central aspects. Hence, presence during the lectures and exercises is important and highly recommended. Details will be given during the first lecture on 14/10.

Registration in the exercise system (via heiCO) is required for taking part in the course. I will send more details to registered participants by email.

Exercises: Tutor: Andreea Oros

View group list and further information, and download material here.
Classes take place on Fridays, 14:15-15:45 hrs, starting on 24/10: INF 227 (KIP) SR 1.403.

Content - Prerequisites - Script - Literature - Supplementary materials - Credit Points

The lecture course provides an introduction to the theory of classical and quantum phase transitions, to position-space as well as Wilson renormalisation-group theory. Emphasis will be set on broadly used spin models as well as bosonic field theories relevant in particular for applications in the field of ultracold atomic gases. Methodologically, the lecture will build on the basics of the operator as well as the path-integral approach to quantum field theory. Basic knowledge 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
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
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
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
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

Prerequisites:

• Quantum Mechanics (PTP 3), Statistical Mechanics (PTP 4); Quantum Field Theory I or Quantum Field Theory of Many-Body Systems

Script :

• The Script will be available for download here.
• The Script of the lecture on QFT of Many-Body Systems in ST 25 is available 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

• Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral, Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008). [arXiv:quant-ph/0703044v3]

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 (pdf online) ]
• 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

• The Weirdness of Water, by Rachel Brazil, Chem. World, 6 Apr 2020 -- on Russo et al., Water-like anomalies as a function of tetrahedrality, PNAS 115, E3333 (2018).
• Experiments realising critical opalescence and supercritical fluidity in Water (now only here) and Carbon Dioxide (further, commercial video with explanations).
• The old Java applet simulating the 2D Ising model by Peter Young, which is used in the videos, no longer works in present-day browsers, while its source code is preserved here. A presently working version can be found here.
• 2D Ising model MC-simulator by D. Aleinikava on Wolfram.
• Visualization of the real-space renormalisation group for a 2D Ising system by D. Ashton (Straight to video on youtube).
• Visualization of the scale invariance of critical fluctuations of a 2D Ising system

Credit Points:

Passing the written exam, which will take place, prospectively, on 05/02/2026, 11:15-13:15 hrs, SR 3.403, will be the condition to obtain 8 CPs for the lecture.