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.

  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

Skriptum :

Textbooks on critical phenomena and (quantum) phase transitions Reviews on critical phenomena and (quantum) phase transitions General texts on statistical mechanics General texts on quantum field theory Additional material

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.)


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.