Entanglement and locality in quantum many-body systems (winter term 14/15)

Lecture "Entanglement and locality in quantum many-body systems" (winter term 2014/15)

Lecturer: Norbert Schuch

(This is a mirror copy of the original course website at RWTH Aachen.)

Description

Interacting quantum many-body systems appear in all areas of physics, from condensed matter systems to high-energy physics. A characteristic feature of these systems is the local nature of their interactions.

In the first part of this lecture, I will discuss the consequences of this locality, especially in the context of condensed matter systems. Most importantly, locality implies a finite propagation speed of excitations (the so-called Lieb-Robinson bounds), which have a number of remarkable consequences regarding the behavior of correlations, the stability of topological topological order, the classification of quantum phases, and the nature of the entanglement structure of these systems.

The second part of the lecture will focus on the consequences of the specific entanglement structure of many-body systems. This covers in particular their description in terms of Tensor Network States, such as Matrix Product States and Projected Entangled Pair States, and the resulting class of simulation methods, most importantly the Density Matrix Renormalization Group (DMRG) algorithm.

Material

Lecture notes

Lecture 1 (17.10.): Quantum many-body systems.

Further reading: W. Nolting and A. Ramakanth, Quantum Theory of Magnetism (Springer, 2009).

Lecture 2 and 3 (24./31.10.): Lieb-Robinson bounds.

Further reading: [Hastings]; T. Koma and M.B. Hastings, Spectral gap and exponential decay of correlations;
R. Sims and B. Nachtergaele, Lieb-Robinson bounds and the exponential clustering theorem;
K. Fredenhagen, A Remark on the Cluster Theorem.

Lecture 4 and 5 (7./14.11.): Exponential clustering of correlations.

Further reading: [Hastings]; T. Koma and M.B. Hastings, Spectral gap and exponential decay of correlations;
R. Sims and B. Nachtergaele, Lieb-Robinson bounds and the exponential clustering theorem.

Lecture 6 (21.11.): Quasi-adiabatic evolution.

Further reading: [Hastings]; S. Bravyi, M.B. Hastings, and S. Michalakis, Topological quantum order: stability under local perturbations (Section 7).

Lecture 7 and 8 (28.11./5.12.): Applications of quasi-adiabatic evolution.

Further reading: [Hastings]; M.B. Hastings and X.-G. Wen, Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance;
S. Bravyi, M. B. Hastings, F. Verstraete, Lieb-Robinson bounds and the generation of correlations and topological quantum order.

Lecture 9 and 10 (12./19.12.): Entanglement in many body systems: The area law, Matrix Product States.