260070 VU Entanglement in quantum many-body systems (2021S)

Correlated quantum many-body systems are systems composed of many particles (such as materials) where at low temperatures, complex quantum correlations ("entanglement") between the individual constituents play a central role. These quantum correlations can give rise to rather unconventional physical effects, such as in the Fractional Quantum Hall Effect, which exhibits precisely quantized edge currents and whose excitations carry charges which are e.g. "one third of an electron" and which possess exotic statistics, being neither bosons nor fermions.

This lecture will present a systematic introduction to correlated quantum many-body systems from the perspective of their quantum correlations: entanglement. We will study the structure of the complex entanglement in these systems and see that it naturally gives rise to an efficient description of these systems in terms of so-called Tensor Networks. The core of the lecture will then be devoted to a comprehensive introduction into the area of Tensor Networks: On the one hand, we will study analytical properties of Tensor Networks, in particular Matrix Product States (MPS), and see how they allow to understand the structure of quantum many-body states and to classify the different quantum phases they can exhibit. On the other hand, the lecture will also cover the use of Tensor Networks for the construction of powerful numerical simulation algorithms, most prominently the Density Matrix Renormalization Group (DMRG) method. Finally, the lecture will also touch upon the connections between quantum many-body systems and quantum computation.

The lecture will combine both mathematical and physical aspects of quantum many-body systems and tensor networks, and cover analytical as well as numerical methods and approaches.

Planned topics include:
* Quantum many-body systems and quantum spin systems
* Basics of entanglement theory
* The entanglement area law
* Matrix Product States and their properties
* Parent Hamiltonians and the Affleck-Kennedy-Lieb-Tasaki (AKLT) model
* Classification of phases in one dimension
* The Density Matrix Renormalization Group (DMRG) and other numerical methods
* Projected Entangled Pair States (PEPS)
* Topological order
* Measurement-based quantum computations
* Classical and quantum complexity of many-body systems
* Fermions

The course format will combine lecture units and exercises.

Prerequisites:
Solid knowledge of quantum mechanics, including the basics of quantum spins, is required. (Alternatively, solid knowledge of the basics of quantum information is also sufficient; however, in that case, please let me know beforehand.) Knowledge of quantum condensed matter is useful, but will not be necessary.

Further information on the lecture can also be found on https://schuch.univie.ac.at/ss21-qmb/ .