Explore the interface of condensed matter physics, quantum information theory and high-energy physics, a highly active area of current research, through the lens of quantum many-body physics. Study correlation structures and their role in determining the physical properties of these systems. Understand the special role of ground states and their entanglement properties, such as entanglement area laws and how correlations decay over distance. Learn how to sidestep the complexity of many-body systems and efficiently describe their properties using approximation tools, such as tensor networks and several renormalization methods that have become standard workhorses in the modern literature. Review standard phase transitions and find out how symmetries lead to a novel notion of symmetry protected phases and topological phase transitions. Time permitting, discover how to compute entanglement entropies in the presence of gauge symmetries and in quantum field theory. Practice these findings in exercises and journal club presentations and explain them in a final oral exam.

- Review of reduced density matrix, entanglement for pure and mixed states, entanglement (Renyi) entropies and mutual information.

- Why is many-body entanglement a hard problem?

- Introduction to tensor networks, matrix-product and projected entangled-pair states.

- Tensor network renormalization

- The multi-scale entanglement renormalization ansatz

- Correlation properties in many-body systems (area laws, Lieb-Robinson bounds, etc.)

- Thermodynamic properties of many-body systems and phases

- (Gauge) symmetries in many-body physics

- Quantum correlations in QFT (replica trick, lattice models, vacuum states, entanglement entropies, mutual information, gauge symmetry)

Homework 30%, exam 35%, small projects & presentation 35%

1. Roman Orus, "A Practical Introduction to Tensor Networks: MatrixProduct States and Projected Entangled Pair States”, Pre-print: 1306.2164, Journal: Annals of Physics 349 (2014) 117-158

2. J. Ignacio Cirac David Perez-Garcia, Norbert Schuch and Frank Verstraete, “Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems”, Pre-print: 2011.12127

3. Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, Luca Tagliacozzo, Gang Su, Maciej Lewenstein, "Tensor Network Contractions", Lecture Notes in Physics 964, Springer Cham (2020)

Simone Montangero, "Introduction to Tensor Network Methods", Springer Cham (2018)

Hal Tasaki, "Physics and Mathematics of Quantum Many-Body Systems", Springer (2020)