Tensor Networks

During the last two decades, tensor networks have emerged as a powerful new language for encoding the wave functions of quantum many-body states, and the operators acting on them, in terms of contractions of tensors. Insights from quantum information theory have led to highly efficient and accurate tensor network representations for a variety of situations, particularly for one- and two-dimensional systems. For these, tensor network-based approaches rank among the most accurate and reliable numerical methods currently available.

This module offers an introduction to tensor network-based numerical methods. It starts with a brief motivation, then introduces basic notions of tensor network methodology (tensor diagrams, matrix product states and operators, unitary and isometric transformations, singular value decomposition, projective truncations, canonical forms for 1D tensor networks, computing expectation values and correlation functions). This is followed by detailed discussions of several well-established tensor-network-based algorithms. These include the iterative diagonalization of small systems; the numerical renormalization group (NRG) for treating quantum impurity models; the density matrix renormalization group (DMRG) and the time-evolving block decimation (iTEBD) scheme for treating one-dimensional systems; the time-dependent variational principle (TDVP) for computing time evolution via a tangent-space projection; projected entangled pair states (PEPS) for treating two-dimensional quantum lattice models; the tensor renormalization group (TRG), the tensor network renormalization (TNR) approach and the multi-scale entanglement renormalization ansatz (MERA) for coarse-graining various types of lattice models. Along the way, various technical issued will be covered in detail, such as the proper treatment of fermionic signs, exploiting gauge freedom, and implementing abelian and non-abelian symmetries.

After completing the Module the student is able to:

1. Translate tensor network formulas into tensor network diagrams and vice versa.

2. Bring a matrix product state into one of the standard canonical forms, and convert one canonical form into another.

3. Work out the computational cost involved in contracting a given tensor network diagram.

4. Understand and explain various schemes available for truncating a tensor network.

5. Explain the key ideas underlying each of the algorithms covered in the module (NRG, DMRG, TEBD, TDVP, PEPS, TRG, TNR, MERA).

6. Write a working code for each of these methods, and use the code to compute physical quantities of interest for various standard models (tight-binding chain, classical and quantum Ising model, Heisenberg model, Kondo model, Anderson model, AKLT model, Kitaev’s toric code, 1D and 2D Hubbard model, t-J model). 

7. Keep track of fermionic signs where relevant.

8. Implement abelian and non-abelian symmetries in a tensor-network code.

No prerequisites beyond the requirements for the Master’s program in Quantum Science and Technology. Familiarity with quantum mechanics is assumed, at the level of an introductory module from a Bachelor’s degree in physics.

The module consists of a lecture series (4 SWS) and exercise classes (2 SWS), comprising two lecture sessions and one exercise session per week. 

During lectures, the teaching material is presented by beamer, supplemented by handouts.

Weakly homework assignments guide students towards achieving an in-depth understanding of the topics covered in the preceding lectures. Each week students have to write MATLAB codes implementing the corresponding numerical algorithms discussed in lecture. In the subsequent exercise class, solution codes written by experts will be provided, illustrating how to write compact and efficient codes. By studying these codes in detail and adapting them to solve concrete physics problems, students will gain practical, hands-on working knowledge of tensor network coding. Moreover, each week, some students will present their codes in a short beamer presentation of 15 minutes (including discussion), explaining the central concepts and ingredients underlying their code.

Participation in the exercise classes is strongly recommended, since the exercises offer thorough hands-on training in core conceptual and computational skills, thereby greatly aiding preparation as well as practicing presentation skills.

Presentations (slides), electronic board (with handouts).

There is no suitable textbook yet for this module. Introductory topics will be covered following various review articles, advanced topics by discussing the original literature. Relevant review articles include:

• The density-matrix renormalization group in the age of matrix product states, U. Schollwöck, Annals of Physics, 326, 96-192 (2011).

• Tensor networks and the numerical renormalization group, A. Weichselbaum, Phys. Rev. B 86, 243124 (2012).

• Tangent-space metods for uniform matrix product states, L. Vanderstraeten, J. Haegeman, F. Verstraete, SciPost Phys. Lect. Notes 7 (2019).

• A practical introduction to tensor networks: Matrix product states and projected entangled pair states, R. Orús, Annals of Physics, 349, 117-158 (2014).

• Algorithms for tensor network renormalization, G. Evenbly, Phys. Rev. B 95, 045117 (2017).

The exam will consist of a graded report supplemented by a presentation. Each student will be assigned a different take-home computational problem, to be completed within about 10 days. Each problem will be based on one of the methods covered in the module. This method has to be coded and used to reproduce established numerical results in a specified publication in the literature.The solution has to be submitted in the form of a report consisting of a well-commented tensor network code (typically between 500 and 1500 lines) and plots of the numerical results obtained therewith. Subsequently, each student gives a beamer presentation of about 15 minutes (including discussion). The goal of the presentation is to explain the central concepts and ingredients underlying the code, to present the numerical results obtained therewith, and to demonstrate the ability for verbal communication of technical material. The final grade will be based both on the quality and performance of the submitted code (70%) and the quality of the oral presentation (30%).

The exam evaluates the acquired level of competence by using conceptual questions and computational tasks.

Typical exam questions:

• Use NRG to compute the energy-level flow diagram of the 2-channel Kondo mode, while exploiting available abelian and non-abelian symmetries.
• Use tDMRG or TEBD to compute the impurity spectral function of the Anderson model.
• Use DMRG to compute the entanglement entropy of a spin-1 bilinear-biquadratic chain.
• Use PEPS, in combination with simple update and TRG, to compute the ground state energy of the spin-1/2 Heisenberg model on a Kagome lattice.