Tensor Networks, Fall 2022

Basic info

Instructor: Prof. Seung-Sup Lee, Department of Physics and Astronomy, SNU.

Dates, time, and place: Tuesdays and Thursdays, 14:00–15:15, Room 56-219. A Zoom room is also available as a virtual classroom.

Course outline: Please check the course outline (compiled on Dec. 28, 2022) for the details of the course (goal and objective, target and requirements, course style, evaluation criteria, references, and schedule).

Student forum: In this KakaoTalk Chat (closed), students can ask questions, answer them, and discuss any topic. Please join with your full name in English. Activities in this Chat will also count towards the evaluation (20% of total)

Lecture materials

Lecture materials will be uploaded in advance: by 16:00 on the preceding Tuesdays for Thursday classes; and by 23:00 on the preceding Saturdays for Tuesday classes. Students are asked to submit their pre-class exercise solutions to Seung-Sup Lee by e-mail, no later than 11:00 am on the class day, to be considered for a credit for the exercises (responsible for 20% of the grade). The solutions need not be fully correct; each submission will get full credit as long as meaningful contributions are made.

When submitting codes as exercise and exam solutions, please include and specify the main script (.m or .mlx) that generates all relevant results upon its execution. Also, please submit pen-and-paper solutions in .pdf, .jpg, or .png format.

Tutorial materials including coding and pen-and-paper exercises are available from the GitHub repository for this course. Please read readme.pdf therein to learn how to use the materials.

During the course, I will refer to books, research articles, and reviews by keywords, e.g., Bruus2004, Schollwoeck2005. The detailed information of references indicated by these keywords can be found in this .bib file (updated on Nov. 23, 2022), which can be opened by a text editor or a .bib management program such as JabRef.

The list below provides the links to lecture notes and videos. It also enlists the topics of relevant tutorial materials (coding and pen-and-paper) for completeness. The lecture notes and the associated videos are numbered as Lxx.y (e.g., L01.2), where xx indexes a class within the semester and y indexes a topic within the class. The tutorial materials are similarly numbered as Txx.y (e.g., T02.1). The pre-class exercises within the tutorials will be specified by trailing (a), (b), (c), ... after the tutorial numbers [e.g., T02.1(a)]. The rest of the exercises are in-class.

Please report if you find any mistakes or typos. Such reports will also count towards the evaluation!

[Class 26] Dec. 01 (Th)

• Q&A: on any topics covered in this lecture course

• No exercises for this last class!

[Class 25] Nov. 29 (Tu)

• [L25.1] Machine learning with MPS (lecture note) (video)

• [T25.1] Machine learning with MPS
  (a) Complete the function for the MPS-based machine learning method

• Q&A: on any topics covered in this lecture course

• Pre-class exercises: T25.1(a) (No in-class exercise!)

[Class 24] Nov. 24 (Th)

• [L24.1] Graph-independent local truncations (GILT) (lecture note) (video)

• [T24.1] Graph-independent local truncations (GILT)
  (a) Complete the function for the TRG and the GILT for a square lattice

• Pre-class exercises: T24.1(a) (only the TRG part, without implementing GILT)

[Class 23] Nov. 22 (Tu)

• [L23.1] Tensor renormalization group (TRG) (lecture note) (video)

• [L23.2] Simple update & full update (lecture note) (video)

• [T23.1] Simple update and tensor renormalization group
  (a) Complete the function for the simple update method on a honeycomb lattice
  (b) Complete the function for the TRG on a honeycomb lattice

• Pre-class exercises: T23.1(a)

[Class 22] Nov. 17 (Th)

• [L22.1] Infinite PEPS (iPEPS) & corner transfer matrix (CTM) (lecture note) (video)

• [T22.1] Symmetric CTMRG
  (a) Complete the function for the symmetric CTMRG

• Pre-class exercises: Solve one of the "complete-the-function" exercises in T08–T21 from scratch, i.e., by filling out an empty .m file. Take an exercise from the classes other than those for which you gave a presentation or you chose to solve as the pre-class exercises of Classes 17 and 20.

[Class 21] Nov. 15 (Tu)

• [L21.1] Projected entangled-pair states (PEPS) (lecture note) (video)

• [L21.2] Resonating valence bond state (lecture note) (video)

• [L21.3] Kitaev’s toric code (lecture note) (video)

• [T21.1] Contraction of finite PEPS
  (a) Complete the function for the MPO-MPS method of PEPS contraction
  (b) Spin-spin correlation function of the RVB state with different lattice sizes
  (c) Ground state of Kitaev's toric code

• Pre-class exercises: Compose a question on any topic within the TS module (Classes 18–20).

[Class 20] Nov. 10 (Th)

• [L20.1] Controlled bond expansion (CBE) (lecture note) (video)

• [T20.1] Controlled bond expansion (CBE) DMRG for ground state search
  (a) Complete the function for CBE-DMRG
  (b) Error analysis

• Pre-class exercises: Solve one of the "complete-the-function" exercises in T08–T19 from scratch, i.e., by filling out an empty .m file. Take an exercise from the classes other than those for which you gave a presentation or you chose to solve as the pre-class exercise of Class 17.

[Class 19] Nov. 08 (Tu)

• [L19.1] 2-site TDVP (lecture note) (video)

• [L19.2] Energy variance (lecture note) (video)

• [T19.1] Time-dependent variational principle (TDVP): 2-site variant
  (a) Complete the function for 2-site TDVP
  (b) Compare the accuracy between 1-site TDVP and tDMRG

• [T19.2] Two-site energy variance
  (a) Complete the function for the two-site energy variance
  (b) Extrapolate the ground-state energy as a function of the two-site variance
  (c) Hubbard chain

• Pre-class exercises: T19.2(a)

[Class 18] Nov. 03 (Th)

• [L18.1] Tangent space (lecture note) (video)

• [L18.2] One-site time-dependent variational principle (TDVP) (lecture note) (video)

• [T17.1] Time-dependent variational principle (TDVP): 1-site variant
  (a) Complete the function for 1-site TDVP
  (b) Compare the accuracy between 1-site TDVP and tDMRG

• Pre-class exercises: Compose a question on any topic within the NRG module (Classes 14–17).

[Class 17] Nov. 01 (Tu)

• [L17.1] Dynamical mean-field theory (DMFT) (lecture note) (video)

• [T17.1] NRG: Bulla's self-energy trick
  (a) Complete the function for the Kramers–Kronig (KK) relation
  (b) Complete the function for Bulla's self-energy trick
  (c) Dynamical susceptibilities

• [T17.2] DMFT+NRG
  (a) Complete the DMFT+NRG function
  (b) Obtain the DMFT+NRG solutions starting from the U=0 result
  (c) Obtain the DMFT+NRG solutions starting from the U=3.5D result

• Pre-class exercises: Solve one of the "complete-the-function" exercises in T08–T16 from scratch, i.e., by filling out an empty .m file. For those who already gave mid-term presentations: choose an exercises which was not your presentation topic.

[Class 16] Oct. 27 (Th)

• [L16.1] Lehmann representation (lecture note) (video)

• [L16.2] Full density matrix NRG (lecture note; revised on Oct. 27) (video)

• [T16.1] NRG: Impurity spectral function
  (a) Complete the function for fdmNRG
  (b) Temperature dependence of the impurity spectral function
  (c) Dynamical susceptibilities

• [T16.2] Lehmann representation for a non-interacting system (pen-and-paper)
  (a) Evaluate the spectral function
  (b) Explain why the local spectral function can be interpreted as the local density of states

• Pre-class exercises: T16.2 (a)–(b)

[Class 15] Oct. 25 (Tu)

• [L15.1] NRG: Anders–Schiller basis (lecture note) (video)

• [L15.2] NRG: Full density matrix (lecture note) (video)

• [L15.3] NRG: Operator expansion (local operators) (lecture note; revised on Oct. 27) (video)

• [T15.1] NRG: Full density matrix
  (a) Complete the function for constructing the full density matrix

• Pre-class exercises: No pre-class exercise, as my upload was late. Sorry for the inconvenience!

[Class 14] Oct. 18 (Tu)

• [L14.1] NRG: Intro (lecture note) (video)

• [L14.2] NRG: Logarithmic discretization, iterative diagonalization (lecture note; revised on Oct. 22) (video 1, video 2)

• [L14.3] NRG: Energy flow (lecture note) (video)

• [L14.4] NRG: Thermodynamic properties (lecture note) (video)

• [T14.1] NRG: Iterative diagonalization and energy flow
  (a) Complete the functions for the logarithmic discretization and the iterative diagonalization
  (b) Reproduce lowest-lying energies in the strong-coupling regime by fixed-point Hamiltonians
  (c) Single-impurity Kondo model

• [T14.2] NRG: Impurity contribution to thermodynamic properties
  (a) Complete the function for computing thermodynamic properties
  (b) Single-impurity Kondo model

• Pre-class exercises: T14.1(a)

[Class 13] Oct. 13 (Th)

• [L13.1] Finite temperatures: Purification (lecture note) (video)

• [L13.2] Exponential tensor renormalization group (XTRG) (lecture note) (video)

• [T13.1] Finite temperatures: Purification and XTRG
  (a) Complete the function for variational multiplication of MPOs
  (b) Complete the function for the purification method
  (c) Complete the function for the XTRG

• Pre-class exercises: Compose a question on any topic covered so far.

[Class 12] Oct. 11 (Tu)

• [L12.1] Time-dependent DMRG (tDMRG) (lecture note) (video)
  1. Bond-by-bond (local) time evolution and truncation
  2. Variational (global) application of time evolution
  3. Errors in tDMRG

• [T12.1] Time-dependent DMRG (tDMRG): Error analysis
  (a) Complete the function for tDMRG
  (b) Change Nkeep and dt
  (c) Longer time evolution
  (d) Different initial state where only one spin is up

• [T12.2] MPO representation of time evolution operators
  (a) MPO for the first-order Trotterization
  (b) MPO for the first-order Taylor expansion

• Pre-class exercises: T12.2(a)–(b)

[Class 11] Oct. 06 (Th)

• [L11.1] Hastings’ modified iTEBD (lecture note) (video)

• [L11.2] Orthonormalization (lecture note) (video)

• [T11.1] iTEBD: Hastings' method, orthonormalization
  (a) Complete the function for Hastings' version of iTEBD
  (b) Complete the function for the orthonormalization
  (c) Correlation length of the spin-1 Heisenberg model

• Pre-class exercise: T11.1(a)

[Class 10] Oct. 04 (Tu)

• [L10.1] Vidal’s Gamma-Lambda notation (lecture note) (video)

• [L10.2] iTEBD (lecture note) (video)

• [T10.1] iTEBD: Ground state search
  (a) Complete the function for Vidal's original iTEBD
  (b) Check the energy convergence

• Pre-class exercise: Compose a question on any topic covered so far.

[Class 09] Sep. 29 (Th)

• [L09.1] DMRG: excited state search (lecture note) (video)

• [L09.2] DMRG: two-site update (lecture note) (video)

• [T09.1] DMRG: Single-site update for excited state search
  (a) Complete the single-site DMRG for first excited state search

• [T09.2] DMRG: Two-site update for ground state search
  (a) Complete the two-site DMRG function
  (b) Majumdar–Ghosh model

• Pre-class exercise: T09.2(a)

[Class 08] Sep. 27 (Tu)

• [L08.1] Lanczos method (lecture note) (video)

• [L08.2] Density-matrix renormalization group (DMRG): single-site update for ground state search (lecture note) (video)

• [T08.1] DMRG: Single-site update for ground state search
  (a) Initialize MPS with the iterative diagonalization result
  (b) Complete the single-site DMRG function

• Pre-class exercise: T08.1(a)

[Class 07] Sep. 22 (Th)

• [L07.1] Matrix product operators (MPOs) (lecture note) (video)
  1. Applying an MPO to an MPS yields another MPS; 2. Applying an MPO Hamiltonian to a site-canonical MPS

• [L07.2] MPO representation of Hamiltonians (lecture note) (video)

• [T07.1] Applying an MPO onto an MPS
  (a) MPO representation of the AKLT Hamiltonian
  (b) Confirm whether the AKLT states are the eigenstates of the AKLT Hamiltonian

• Pre-class exercise: Compose a question on any topic covered so far.

[Class 06] Sep. 20 (Tu)

• [L06.1] Translationally invariant MPS (lecture note) (video)
  1. Transfer operator; 2. Correlation functions

• [T06.2] AKLT state (lecture note; typo fixed on Mar. 3, 2024) (video)
  1. AKLT model; 2. AKLT state

• [T06.1] AKLT state (pen-and-paper)
  (a) Left- and right-normalization
  (b), (c) Transfer operators
  (d) Correlation function and the string order parameter

• [T06.2] Expectation values in the AKLT state
  (a) Magnetization
  (b) Spin-spin correlation

• Pre-class exercises: T06.1(a)–(d)

[Class 05] Sep. 15 (Th)

• [L05.1] Iterative diagonalization (lecture note) (video)

• [L05.2] Canonical forms of MPSs (lecture note) (video)
  1. Left-, right-, bond-, and site-canonical forms; 2. Right-to-left sweep for bringing MPS into a canonical form

• [T05.1] Iterative diagonalization
  (a) Non-interacting tight-binding chain

• [T05.2] Canonical forms of MPS
  (a) Complete the function that transforms MPSs into canonical forms
  (b) Truncate bond dimensions

• Pre-class exercise: T05.1(a)

[Class 04] Sep. 13 (Tu)

• [L04.1] Local operators (lecture note) (video)
  1. Spins; 2. Spinless fermions, fermionic sign operator; 3. Spinfull fermions

• [L04.2] Matrix elements, overlaps, and expectation values (lecture note) (video)

• [L04.3] Symmetries: Abelian and non-Abelian (lecture note; revised on Sep. 13; typo fixed on Mar. 3, 2024) (video)
  (a) Abelian symmetry example: U(1) spin symmetry
  (b) Non-Abelian symmetry example: SU(2) spin symmetry

• [T04.1] Diagonalize many-body Hamiltonians
  (a) Spin-1/2 Heisenberg triangle (pen-and-paper)
  (b) Spin-1/2 Heisenberg triangle (coding)
  (c) Non-interacting tight-binding chain

• Pre-class exercise: T04.1(a)

[Class 03] Sep. 08 (Th)

• [L03.1] Unitaries and isometries (lecture note) (video)
  1. Identities; 2. Left- and right-normalized tensors (left and right isometries)

• [L03.2] Decompose tensors into MPS (lecture note; revised on Sep. 7) (video)
  1. Tensor decomposition with QR; 2. Tensor decomposition with SVD

• [L03.3] Area law of entanglement (lecture note) (video)

• [T03.1] Tensor decomposition and entanglement entropy
  (a) Check the integrity of the tensor decomposition
  (b) Entanglement entropies for different bipartitions
  (c) Use the SVD for the tensor decomposition and compute the entanglement entropy

• Pre-class exercise: T03.1(a)

[Class 02] Sep. 06 (Tu)

• [L02.1] Tensor network diagrams (lecture note; revised on Sep. 6) (video)

• [L02.2] Tensor contractions (lecture note) (video)

• [L02.3] Schmidt decomposition (lecture note) (video)

• [T02.1] W state
  (a) Tensor representation of the W state

• [T02.2] Tensor contractions
  (a) First contract A and C, and then contract AC and B

• Pre-class exercise: T02.1(a)

[Class 01] Sep. 01 (Th)

• [L01.1] Quantum many-body theory basics (lecture note; revised on Sep. 6) (video)
  1. First quantization; 2. Second quantization; 3. Example: non-interacting fermions

• [L01.2] Singular value decomposition (lecture note) (video)

• [T01.1] MATLAB 101

• [T01.2] Non-interacting fermions on a tight-binding chain
  (a) Compute the energy and degeneracy of ground states

• [T01.3] SVD Example: Image compression
  (a) Understanding singular vectors

• No pre-class exercises for this very first class!