Advanced Methods in Quantum Many-Body Theory (PH2297)

Lecturers: Alvise Bastianello, Hui-Ke Jin, Izabella Lovas, Ananda Roy, and Gibaik Sim

Lectures: Mon, Tue 16:00 - 18:00 
Tutorials: Thu 16:00 - 18:00


Quantum many-body systems can exhbit extremely rich phenomena, ranging from novel phases of matter to exotic non-equilirbrium physics. Tackling such systems theretically is very challenging and requires advanced methods. This module serves as an introduction to several of these advanced analytical methods. The following topics are covered in this module:


  • Introduction to semi-classical methods, focusing on many-body dynamics. Beyond mean-field dynamics: truncated Wigner approximation (TWA).
  • TWA in classical phase space: heuristic arguments, and derivation from non-equilibrium Keldysh path integral. Example: dynamics of the sine-Gordon model.
  • TWA for out-of-equilibrium spin systems. Construction of discrete phase space, and discrete TWA.
  • Introduction to exactly solvable models in classical and quantum statistical mechanics.
  • Algebraic Bethe Ansatz.
  • Introduction to the thermodynamics and out-of-equilibrium properties of 1d quantum integrable models.
  • Quantum quenches, lack of thermalization and relazation to the Generalized Gibbs Ensemble: the example of the 1d Bose gas.
  • Emergent hydrodynamics in integrable models.
  • Introduction to group theory, lattice gauge theory, and projective symmetry group (PSG) theory
  • Projective construction of quantum spin liquids using PSG. Example: Kitaev honeycomb model with (1) four-Majorana exact solution (2) projective construction solution 
  • Beyond mean-field theory: Guztwiller projection.
  • Gorkov equation and anomalous Green’s function
  • Phenomenological theory of  multicomponent-superconductivity
  • Topological aspects of superconductivity


  • Phase space representation of quantum dynamics, Anatoli Polkovnikov, Annals of Phys. 325, 1790 (2010).
  • Quantum corrections to the dynamics of interacting bosons: beyond the truncated Wigner approximation, Anatoli Polkovnikov, Phys. Rev. A 68, 053604 (2003).
  • A Wigner-function formulation of finite-state quantum mechanics, William K. Wootters, Annals of Phys. 176, 1 (1987).
  • Many-body quantum spin dynamics with Monte Carlo trajectories on a discrete phase space, Johannes Schachenmayer, Alexander Pikovski, Ana Maria Rey, 
  • Phys. Rev. X 5, 011022 (2015).
  • Exactly Solvable Models of Statistical mechanics, R. J. Baxter
  • Quantum inverse scattering method and correlation functions, V. Korepin et al
  • Algebraic Bethe Ansatz, N. A. Slavnov
  • An introduction to integrable techniques for one-dimensional quantum systems, F. Franchini
  • Lecture notes on Generalized Hydrodynamics, B. Doyon SciPost Phys. Lect. Notes 18 (2020)
  • Analytic results for a quantum quench from free to hard-core one-dimensional bosons, M. Kormos, M. Collura, P. Calabrese, Phys. Rev. A 89, 013609 (2014)
  • Quantum field theory of many-body systems. Xiao-Gang Wen
  • Anyons in an exactly solved model and beyond. Alexei Kitaev
  • Quantum orders and symmetric spin liquids. Xiao-Gang Wen
  • Group Theory and Quantum Mechanics. Michael Tinkham
  • Physics of projection wavefunctions. Claudius Gros
  • Introduction to Unconventional Superconductivity, V.P. Mineev and K.V. Samokhin
  • Introduction to Unconventional Superconductivity, M. Sigrist
  • Superconducting classes in heavy-fermion systems, G. E. Volovik and L. P. Gorkov
  • Aspects of Topological Superconductivity, M. Sigrist

TUM Course Website PH2297