**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.
- Lecture 2 and 3 (24./31.10.): Lieb-Robinson bounds.
- Lecture 4 and 5 (7./14.11.): Exponential clustering of correlations.
- Lecture 6 (21.11.): Quasi-adiabatic evolution.
- Lecture 7 and 8 (28.11./5.12.): Applications of quasi-adiabatic evolution.
- Lecture 9 and 10 (12./19.12.): Entanglement in many body systems: The area law, Matrix Product States.
- Lecture 11 and 12 (9.1./16.1.): Matrix Product States: Basic Properties.
- Lecture 13 (30.1.): Numerical simulations with MPS: The DMRG algorithm.
- Lecture 14 (6.2.): Truncation of MPS; real/imaginary time evolution.

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

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.

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.

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

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.

Further reading: [Schuch].

Further reading: [Orus], [Schuch], [Schollwöck].

Further reading: sample DMRG code, and problems 3 and 4 in this exercise sheet.

Exercise sheets

- Sheet 1 (Lectures 1+2, discussed on 31.10.)
- Sheet 2 (Lectures 3+4, discussed on 14.11.)
- Sheet 3 (Lectures 5+6, discussed on 28.11.)
- Sheet 4 (Lectures 7+8, discussed on 12.12.)
- Sheet 5 (Lectures 9+10, discussed on 9.1.)
- Sheet 6 (Lecture 11+12, discussed on 30.1.)

__Literature__

**[Hastings]**M.B. Hastings: Locality in Quantum Systems**[Schuch]**N. Schuch: Condensed Matter Applications of Entanglement Theory**[Orus]**R. Orus: A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States**[Schollwöck]**U. Schollwöck: The Density Matrix Renormalization Group in the age of Matrix Product States

__Organisatorial issues__

The lecture takes place Friday from 14:30 to 16:00 in room 26C401. Lecture notes will be posted on the course L2P site and this website. There is a bi-weekly exercise on Friday 16:00-17:30, as posted on the website. The exercise is voluntary. Examination will be though an oral exam of 30 mins. Please feel free to contact me if you have further questions. |