Quantum Information Methods in Many-Body Physics

Quantum many-body systems exhibit a rich variety of interesting physical phenomena, such as systems with exotic "topological" order which cannot be captured by Landau's theory of phases, and in which the system organizes globally in its quantum correlations rather than locally. The heart of such phenomena is formed by quantum correlations, this is, entanglement, which is at the heart of quantum information theory. This has given rise to a new research area at the intersection of quantum information theory and quantum many-body physics, where methods from quantum information are applied to the structure of these systems. This module will give a comprehensive introduction to this new research field. A special focus will be formed by the field of Tensor Network States, which provide an entanglement based description of quantum many-body states.

Topics covered include:

• Quantum many body systems
• Entanglement and the area law
• Matrix Product States (MPS)
• Variational simulations based on MPS, the DMRG method
• Simulation of thermal states and time evolution
• Projected Entangled Pair States (PEPS)
• topological order
• fermionic systems

After successful completion of the module the students are able to

• understand the entanglement properties of many-body systems.
• identify the properties relevant for a state with low entanglement.
• give the construction of a matrix product state.
• design a variational algorithm based on matrix product states.
• derive the classification of symmetry protected phases in one dimension.
• understand the way in which symmetries are realized in matrix product states.
• provide examples for matrix product states, such as the AKLT state.
• start a research project in the field of tensor network states.

Quantum mechanics,  in particular also many-body quantum mechanics and second quantisation (e.g. PH0007 and PH1002). Knowledge in quantum information and/or condensed matter physics is helpful, but not required.

In the lecture, the learning content will be presented in a form structured according to the different subtopics such basic structure, analytical theory, and numerical realization.  The cross-references between the different topics and the underlying universal principle of entanglement and the area law will be made clear at each level.  Students will be engaged in scientific discussions about the topic and will be encouraged to build their analytical skills as well as their numerical abilities.

In the exercise, the content of the lecture will be deepened by working out specific problems and examples as well as hands-on programming exercise classes, amond other measures.  The students will thus be enabled to carry out research talks in the field on their own.

Blackboard, presentations (slides), electronic board (with handouts).  Exercises will be done in small groups with individual supervision.

https://arxiv.org/abs/1306.2164

https://arxiv.org/abs/1008.3477

There will be an oral exam of 25 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using comprehension questions and sample calculations.

For example an assignment in the exam might be:

• What is the area law?
• What is a Matrix Product State?
• What is the basic idea of the DMRG algorithm?
• How can we use Matrix Product States to simulate time evolution?

Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.