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Advanced Methods in Quantum Many-Body Theory

Module PH2297

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Module version of SS 2021 (current)

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
SS 2021SS 2020

Basic Information

PH2297 is a semester module in English language at Master’s level which is offered in summer semester.

This Module is included in the following catalogues within the study programs in physics.

  • Specific catalogue of special courses for condensed matter physics
  • Focus Area Theoretical Quantum Science & Technology in M.Sc. Quantum Science & Technology
  • Complementary catalogue of special courses for nuclear, particle, and astrophysics
  • Complementary catalogue of special courses for Biophysics
  • Complementary catalogue of special courses for Applied and Engineering Physics
  • Specialization Modules in Elite-Master Program Theoretical and Mathematical Physics (TMP)

If not stated otherwise for export to a non-physics program the student workload is given in the following table.

Total workloadContact hoursCredits (ECTS)
300 h 90 h 10 CP

Responsible coordinator of the module PH2297 is Michael Knap.

Content, Learning Outcome and Preconditions

Content

• 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

Learning Outcome

• Learn a versatile semi-classical approach to the dynamics of many-body systems. Simple applications for experimentally relevant systems: sine-Gordon model and transverse field Ising chain. • Learn analytical techniques to exactly compute thermodynamic properties of certain models of 2D classical statistical mechanics and 1D quantum spin chains. • Learning exact techniques to study integrable models out of equilibrium, with emphasis on transport and hydrodynamics. • Becoming acquainted with the 1d interacting Bose gas, a paradigmatic model for cold atom experiments. • Construct a mean-field theory for quantum spin liquids using PSG and then solve this mean-field theory self-consistently. • Construct a free energy of multicomponent-superconductivity and then connect the free energy with microscopic details.

Preconditions

No preconditions in addition to the requirements for the Master’s program in Physics.

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

TypeSWSTitleLecturer(s)DatesLinks
VO 4 Advanced Methods in Quantum Many-Body Theory Knap, M.
Assistants: Bastianello, A.Jin, H.Lovas, L.Roy, A.Sim, G.
Tue, 16:00–18:00, PH 3344
Mon, 16:00–18:00, PH 3344
eLearning
UE 2 Exercise to Advanced Methods in Quantum Many-Body Theory Bastianello, A. Jin, H. Lovas, L. Roy, A. Sim, G.
Responsible/Coordination: Knap, M.
Thu, 16:00–18:00, PH 3344

Learning and Teaching Methods

The lecture is designed for the presentation of the subject, usually by blackboard presentation. The focus resides on theoretical foundations of the field, presentation of methods and simple, illustrative examples. Command of basic methods is deepened and practised through homework problems, which cover important aspects of the field. The homework problems should develop the analytic skills of the students and their ability to perform calculations. The homework problems are discussed in the exercise classes by the students themselves under supervision in order to develop the skills to explain a physics problem logically.

Media

Blackboard lectures, written notes for download, exercise sheets, course homepage

Literature

• Phase space representation of quantum dynamics, Anatoli Polkovnikov, Annals of Phys. 325, 1790 (2010). https://arxiv.org/abs/0905.3384 • Quantum corrections to the dynamics of interacting bosons: beyond the truncated Wigner approximation, Anatoli Polkovnikov, Phys. Rev. A 68, 053604 (2003). https://arxiv.org/abs/cond-mat/0303628 • 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 https://arxiv.org/abs/1609.02100 • 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) https://arxiv.org/abs/1307.2142 • 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

Module Exam

Description of exams and course work

The achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using presentations independently prepared by the students. The exam of 25 minutes consists of the presentation and a subsequent discussion.

For example an assignment in the exam might be:

  • Sketch the derivation of the (standard phase space) TWA from the Keldysh path integral.
  • Describe the main steps of the discrete (or cluster) truncated Wigner approximation for a spin-1/2 chain.
  • Sketch the derivation of the diagonalization of the Hamiltonian of the XXZ chain using Algebraic Bethe Ansatz.
  • Calculate the ground state from the Bethe equations.
  • Explain why we can use emergent hydrodynamics to study integrable models
  • Argue how to solve the algebraic PSG for square lattice antiferromagnetic Heisenberg model
  • Sketch how to solve the Kitaev honeycomb model using projective construction
  • Explain the energetical stabilization of specific pairing

Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.

There will be a bonus (one intermediate stepping of "0,3" to the better grade) on passed module exams (4,3 is not upgraded to 4,0). The bonus is applicable to the exam period directly following the lecture period (not to the exam repetition) and subject to the condition that the student passes the mid-term of active participation in the tutorials and at least 70% of exercise points

Exam Repetition

The exam may be repeated at the end of the semester.

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