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 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!
Q&A: on any topics covered in this lecture course
No exercises for this last class!
[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!)
[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)
[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)
[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.
[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).
[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.
[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)
[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).
[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.
[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)
[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!
[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)
[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.
[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)
[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)
[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.
[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)
[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)
[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.
[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)
[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)
[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)
[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)
[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)
[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!