Quantum Phase Transitions, SUPA Doctoral Training Center, @StAndrews

  • Course Outline and Lecture Notes


    • What are Quantum-Phase Transitions?
    • Basic Concepts of Phase Transitions and Critical Behavior
    • Critical Exponents, Scaling and Hyperscaling
    • How Important is Quantum Mechanics?
    • Sketch of the Quantum-to-Classical Mapping
    • What's Different about Quantum-Phase Transitions?


    Ising Chain in a Transversal Field
    • Quantum Ising Model
    • Strong Coupling
    • Weak Coupling
    • Exact Spectrum from Jordan-Wigner Transformation
    • Equal Time Correlations of Order Parameter
    • Finite Temperature Crossovers
    • Quantum-to-Classical Mapping
    • Quantum Field Theory


    Bose-Hubbard Model
    • Hamiltonian and Mott-Insulator/Superfluid Transition
    • Coherent States for Bosons
    • Coherent-State Path Integral Representation of the Partition
    • Hubbard-Stratonovich Transformation
    • Mean-Field Phase Diagram
    • Dynamical Critical Exponent and
      Upper Critiacal Dimension

    Chapter4(a), Chapter4(b):

    Frustrated Quantum
    • Hamiltonian, Classical Groundstates and Linear
      Spin-Wave Theory
    • Classical Non-Linear Sigma Model
    • Renormalization-Group (RG) Analysis
    • Spin-Coherent States
    • Coherent-State Path Integral for a Single Quantum Spin
    • Quantum Non-Linear Sigma Model and Berry Phases
    • RG of Quantum Non-Linear Sigma Model
    • Phase Diagrams for d=2 and d=3
    • Finite Temperature Crossovers
    • Crucial Role of Berry Phases

  • Assessment is based on three homework problem sets (set1, set2, set3).

  • Books recommended background reading

    • Quantum Phase Transitions, Subir Sachdev, Cambridge Univ. Press
      Very nice book. In particular the first chapters give a very nice introduction.

    • A Modern Approach to Critical Phenomena, Igor Herbut, Cambridge Univ. Press
      Short and to the point.

    • Scaling and Renormalization in Statistical Physics, John Cardy, Cambridge Univ. Press
      Excellent introduction to the renormalization group