### Advanced Methods in Quantum Many-Body Theory (PH2297)

**Lecturers:** Alvise Bastianello, Sanjay Moudgalya, Leo Mangeolle, Masataka Kawano (coordination: Johannes Knolle)

**Lectures:** Mon 16:00 - 18:00 (PH 3344), Tue 16:00 - 18:00 (PH 1121 Minkowski Raum)

**Tutorials:** Tue 10:00 - 12:00 (PH 1121 Minkowski Raum)

**Outline**:

The course will cover a number of advanced topics in theoretical quantum many-body physics. This Summer Semester the focus will be on non-equilibrium many-body physics and transport. It will consist of four main parts:

**1. Semiclassical dynamics and Boltzmann transport**

Kinetic equations provide an intuitive framework to the theory of transport, in both classical and quantum systems. This lecture will focus on Boltzmann's equation, first reviewing a few standard applications to the transport of particles, charge, and energy. We will see that this "classical" equation can be derived quite systematically for quantum systems, and how it allows one to deal with interactions. We will review the well known (and a few less known) reciprocity, symmetry, conservation and sum rules of linear response as derived from the kinetic formalism. We will eventually see how inclusion of topological properties of bands in Boltzmann's equation provides an alternative route towards some "purely quantum" effects.

**2. Nonequilibrium and transport in integrable models**

These lectures introduce the basic concepts of integrability applied to out-of-equilibrium phenomena and transport. The final goal is to give a taste of the recent developments in the hydrodynamic theory of integrable models, known as Generalized Hydrodynamics. For the sake of concreteness, we focus on the working example of the one-dimensional Bose gas, but the techniques are of much wider applicability.

**3. Hydrodynamic transport in constrained systems**

Hydrodynamics has been used to describe low-energy and long-wavelength dynamics of conserved quantities in many-body systems. Conserved quantities, such as charge and spins, typically follow diffusive hydrodynamics at late times. However, constrained systems, e.g., systems with a dipole conservation law, often exhibit unconventional hydrodynamic behavior, and there has been growing interest in studying the hydrodynamic transport in such systems. This lecture will provide an overview of the methods used to describe the hydrodynamic transport in constrained quantum many-body systems.

**4. Quantum Many-Body Dynamics and Thermalization **

Generic quantum many-body systems at any non-zero temperature are observed to undergo thermalization, which ultimately justifies the application of the standard theory of thermodynamics in quantum systems. In this course, we will discuss various aspects of this phenomenon, such as its signatures and impact on physical quantities, conditions under which it happens, and important universal conjectures such as the Eigenstate Thermalization Hypothesis (ETH) that have connections to Random Matrix Theory. Thermalization is also a major obstacle to building useful quantum devices. Later in the course we will also discuss known routes to avoid thermalization such as via Many-Body Localization (MBL) and the recently discovered phenomena such as Quantum Many-Body Scars and Hilbert Space Fragmentation. In this context, we will also briefly discuss the connection to cold atom experiments done here in Munich.

**Literature:**

- An introduction to integrable techniques for one-dimensional quantum systems, F. Franchini arxiv.org/abs/1609.02100
- Lecture notes on Generalized Hydrodynamics, B. Doyon SciPost Phys. Lect. Notes 18 (2020)
- Quantum field theory of many-body systems. Xiao-Gang Wen
- P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, 1995), pp 369-389.
- C. L. Henley, Relaxation time for a dimer covering with height representation, J. Stat. Phys. 89, 483 (1997).