Tensor network methods for low-dimensional systems 1100-TNM

First block: Theoretical foundations
- Entanglement in many-body systems, 1D free fermions, criticality, relation to conformal field theory, gapped phases
- Matrix Product State as an ansatz for the wavefunction (an exact ground state)
- Bethe ansatz as an MPS
- Variational problems, SVD
- Dynamics: Trotterization, quantum circuits

Second part: Hands-on experience
- Quspin, Julia Introductions, setting Itensor and learning basic usage of libraries, methods in practice
- Workshops: study of particular 1D quantum spin chains, electrons, qubits
- Workshops on static models: Ising and XXZ model in temperature T=0, phase diagrams, CFT spectra from numerics
- Workshops on time-dependent problems: quenches with Neel state, polarised state

Third part: Modern research
- Invited lecture on selected tensor network related topic from condensed matter (solid state or cold atoms), metrology
- Presentations of the projects

Kierunek podstawowy MISMaP

fizyka

Koordynatorzy przedmiotu

Miłosz Panfil
Jędrzej Wardyn

Założenia (opisowo)

Quantum Mechanics (bra-ket notation, spin operators, many-body states). Linear Algebra (Matrix diagonalisation). Basic programming proficiency in Python (Julia is to be learned during the course, as it is very similar to python).

Efekty kształcenia

By the end of this course, students will be able to:
Understand the Tensor network and specifically MPS representation of quantum eigenstates and how bond dimension (entanglement) limits its usage.
Recognise the numerical methods associated with handling the static ground and excited state computation (ED, DMRG) as well as time-dependent computation (TEBD, TDVP) for computing local and global quenches.
Understand the scope of possible applications for low-dimensional quantum systems.
Write their own code in python and julia to utilise methods from point 2.
Create new operator functions to do measurements on the obtained eigenstate MPSs.

Kryteria oceniania

- Midterm and final exam verifying theoretical as well as numerical knowledge,

- Final Project: Students choose an advanced topic to study with the help of tensor networks methods and present the project's outcome in the class

Literatura

- Julia: https://julialang.org/
- Itensor: https://docs.itensor.org/Overview/
- Quspin: [might be superseded by better ED package] https://quspin.github.io/QuSpin/

For introduction to Tensor networks:
- Density-matrix renormalization group: a pedagogical introduction G. Catarina, Bruno Murta https://arxiv.org/abs/2304.13395
- Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks Lecture Notes Jacob C. Bridgeman1, Christopher T. Chubb https://arxiv.org/pdf/1603.03039
- https://www.tensors.net/intro
- https://tensornetwork.org/
- https://archive.int.washington.edu/talks/WorkShops/int_21_1c/People/Evenbly_G/Evenbly.pdf
https://arxiv.org/pdf/1306.2164

For entanglement measures and significance in Tensor networks:
- “Introduction to quantum entanglement in many-body systems” Anubhav Kumar Srivastava, Guillem Müller-Rigat, Maciej Lewenstein and Grzegorz Rajchel-Mieldzioć https://arxiv.org/pdf/2402.09523
- Quantum Entanglement in Condensed Matter systems https://arxiv.org/pdf/1512.03388
- Entanglement entropy and quantum field theory Pasquale Calabrese and John Cardy https://arxiv.org/abs/hep-th/0405152
- Entanglement in Many-Body Systems by Frank Pollmann https://www.cond-mat.de/events/correl20/manuscripts/pollmann.pdf
- Entanglement in Many-Body Systems Luigi Amico, Rosario Fazio, Andreas Osterloh, Vlatko Vedral https://users.physics.ox.ac.uk/~vedral/old/articles/rmp.pdf