Lecture "Entanglement in quantum many-body systems" (summer term 2021)

Lecturer: Norbert Schuch

ufind: 260070-1 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.


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.

Course material

Lecture notes

  1. Background
    1. The quantum many-body problem and quantum spin systems
    2. Entanglement
  2. Matrix Product States
    1. Construction
    2. Truncation of the bond dimension and approximability by MPS
    3. Canonical forms
    4. Periodic, translational invariant, and infinite MPS
    5. Examples
    6. Properties of MPS: Norms, expectation values, correlations
    7. Translational invariant infinite MPS: Construction and correlations
  3. Simulations with MPS
    1. Ground states: The DMRG method and beyond
      For the basic DMRG algorithm described in part (a), you can check this code from this course - this is Lecture 5 from that course, and the instructions for the code are given in Exercise Sheet 2 (Problem 3+4).
      For a DMRG algorithm based on an MPO encoding of the Hamiltonian, as discussed in part (c) of the current course, see this code, which will be also discussed in the lecture. (Note that "ncon.m" is a library routine for tensor contraction and not part of the code.)
    2. Other MPS-based algorithms
  4. Solvable models and the classification of phases
    1. The AKLT model
    2. Solvable MPS models and classification of phases
  5. Two dimensions: Projected Entangled Pair States
    1. Projected Entangled Pair States (PEPS)
    2. Examples

Exercise sheets


Further references for the individual parts of the lecture will be posted with the lecture notes.

Organisatorial issues

The lecture (course 260070) takes place Monday 10:30-13:00. The lecture will be held via zoom until further notice. The zoom link is available on the lecture's moodle page. If you are interested in participating in the lecture but cannot register formally, please get in touch with me. 

Passing and grade will be based on (i) regular presence, (ii) presentation of at least one exercise problem, and (iii) active participation.