Tensor Networks – Lecture notes

Course Outline
Lecture Notes:

Lecture	Date	Notes	Pages	Topic
L23	24.07.19	pdf	ML.1 ML.2	Machine learning 1. Neural networks 2. Supervised learning with tensor networks
L22	22.07.19	pdf	F-PEPS.1 F-PEPS.2 F-PEPS.3 F-PEPS.4	Fermionic PEPS 1. Parity conservation 2. Fermionic signs 3. Jump move 4. Examples
L21	17.07.19	pdf	CanF.1 CanF.2 CanF.3	2D Canonical Forms, Isometric PEPS 1. Canonical form for bond in 2D tensor network 2. Full environment truncation 3. Isometric PEPS
L20	15.07.19	pdf	TNR.1 TNR.2 TNR.3 TNR.4 TNR.5 TNR.6	TNR: Tensor network renormalization 1. Motivation 2. TNR idea 3. Projective truncation 4. TNR details 5. TNR results in MERA 6. TNR benchmark results
T12	11.07.19			Tutorial: TERG, TRG
L19	10.07.19	pdf	TRG-II.1 TRG-II.2 TRG-II.3 TRG-II.4 TRG-II.5 TRG-II.6	TRG-II: Graph-independent local truncations (Gilt) 1. Motivation 2. Why is TRG insufficient? 3. Environment spectrum 4. Gilt: Graph-independent local truncations 5. Gilt-TNR 6. Benchmark results
L18	08.07.19	pdf	TRG-I.1 TRG-I.2 TRG-I.3 TRG-I.4	Tensor renormalization group (TRG) 1. TRG for 2D classical lattice models 2. TRG for quantum lattice models 3. Second renormalization (SRG) of tensor network states 4. Core tensor renormalization group (CTRG)
T11	04.07.19			Tutorial: PEPS I - exact contraction on a strip
L17	03.07.19	pdf	PEPS-II.1 PEPS-II.2	PEPS II: contractions via MPS techniques 1. PEPS via finite-size MPS 2. Infinite-size PEPS (iPEPS)
L16	01.07.19	pdf	PEPS-I.1 PEPS-I.2 PEPS-I.3 PEPS-I.4	PEPS I: Projected entangled-pair states 1. Motivation and Definition 2. Example: RVB state 3. Example: Kitaev's Toric Code 4. Example: Resonating AKLT state
T10	27.06.19			Tutorial: Symmetries & QSpace (continued)
T09	26.06.19			Tutorial: Symmetries & QSpace
L15	24.06.19	pdf	Sym-II.1 Sym-II.2 Sym-II.3 Sym-II.4 Sym-II.5 Sym-II.6	Symmetries II: Non-Abelian. 1. Motivation, SU(2) basics 2. Tensor product decomposition 3. Tensor operators 4. Example: direct product of two spin 1/2's 5. Example: direct product of three spin 1/2's 6. Bookkeeping for unit matrices
	20.06.19			Corpus Christi
L14	19.06.19	pdf	Sym-I.1 Sym-I.1 Sym-I.1	Symmetries I: Abelian 1. Example: spin 1/2 XXZ-chain 2. Iterative diagonalization 3. QSpace bookkeeping for unit matrices
T08	17.06.19			Tutorial: tDMRG, purification, tangent space methods
L13	13.06.19	pdf	TS.1 TS.2 TS.3	Tangent space methods 1. MPS and canonical forms. 2. Tangent space. 3. Tangent space projector. 4. Time evolution.
L12	12.06.19	pdf	DMRG-II.1 DMRG-II.2 DMRG-II.3	DMRG II 1. Relation to traditional DMRG. 2. tDMRG. 3. Finite temperature: purification.
	10.06.19			Pentecost Monday
T07	06.06.19			Tutorial: iTEBD
L11	05.06.19	pdf pdf	MPS-V.1 iTEBD.1 iTEBD.2 iTEBD.3 iTEBD.4	MPS V: Vidal's Gamma-Lambda notation iTEBD: Infinite Time-Evolving Block Decimation 1. Basic iTEBD algorithm 2. iTEBD in Gamma-Lambda notation 3. iTEBD: Hastings' method 4. Orthogonalization
T06	03.06.19			Tutorial: DMRG
	30.05.19			Ascension Day
L10	29.06.19	pdf	DMRG-I.1 DMRG-I.2 DMRG-I.3 DMRG-I.4	DMRG I: Density Matrix Renormalization Group - ground state search 1. Single-site optimization 2. Lancos Method 3. Excited states 4. Two-site optimization
L09	27.05.19	pdf	MPS-IV.1 MPS-IV.2 MPS-IV.3	MPS IV: Matrix product operators 1. Applying MPO to MPS yields MPS 2. MPO representation of Heisenberg Hamiltonian 3. Applying MPO to mixed-canonical state
T05	23.05.19			Tutorial: AKLT Model
L08	22.05.19	pdf	AKLT.1 AKLT.2 AKLT.3 AKLT.4 AKLT.5	AKLT Model 1. General remarks 2. Construction of AKLT Hamiltonian 3. AKLT ground state 4. Transfer operator 5. String order parameter
L07	20.05.19	pdf	MPS-III.1 MPS-III.2 MPS-III.3	MPS III: Translationally invariant MPS 1. Transfer matrix 2. Eigenvalues of transfer matrix 3. Correlation functions
T04	16.05.19			Tutorial: NRG II/III
L06	20.05.19 6:15	The notes for the entire lecture 06 have been thoroughly revised. Please discard previous version and use this one.
L06	15.05.19	pdf	NRG-III.1 NRG-III.2 NRG-III.3 NRG-III.4 NRG-III.5 NRG-III.6 NRG-III.7	NRG III: Thermal and dynamical quantities 1. Thermodynamic observables 2. Lehmann representation of spectral functions 3. Single-shell and patching schemes 4. Graphical notation for basis change 5. MPS notation for discarded/kept states 6. Complete many-body basis 7. Full density matrix NRG (fdmNRG)
L05	13.05.19	pdf	NRG-II.1 NRG-II.2 NRG-II.3 NRG-II.4 NRG-II.5 NRG-II.6 NRG-II.7	Numerical Renormalization group (NRG) II: RG flow 1. Kondo model: low-order perturbation theory 2. Kondo model: poor man's scaling 3. General RG concepts 4. NRG iteration scheme from RG perspective 5. Uncoupled bath Hamiltonian: fixed points 6. Kondo model: fixed points and RG flow 7. Anderson model: fixed points and RG flow
T03	09.05.19			Tutorial: NRG I
L04	08.05.19	pdf	NRG-I.1 NRG-I.2 NRG-I.3 NRG-I.4	NRG I: Numerical Renormalization group - Wilson chain 1. Single-impurity Anderson model 2. Logarithmic discretization 3. Wilson chain 4. Iterative diagonalization
L03	06.05.19	pdf	MPS-II.1 MPS-II.2 MPS-II.3	MPS II: Diagonalization, fermionic signs 1. Iterative diagonalization of short spin chain 2. Spinless fermions 3. Spinful fermions
T02	02.05.19			Tutorial: MPS I
L02	02.05.19 9:10	In Section 3, the discussion of operator matrix elements has been revised. In section 4, handwritten text has been converted to typed text.
	01.05.19			Labor day
L02	29.04.19	pdf	MPS-I.1 MPS-I.2 MPS-I.3 MPS-I.4	MPS 1: Matrix Product States 1. Overlap and normalization. 2. Canonical MPS forms (left, right, site, bond) 3. Matrix elements, expectation values 4. Schmidt decomposition
L01	30.04.19 12:38	In Section 3, Eqs. (12), (27) and (28), the order of the lower indices of A-dagger has been reversed.
L01	23.04.19 23:27	In Sections 1 to 4, important notational changes were made regarding the definition of tensor product spaces.
T01	25.04.19			Tutorial: MATLAB basics
L01	24.04.19	pdf	TNB1 TNB2 TNB3 TNB4 TNB5 TNB6	Tensor Network basics (TNB): 1. Why study tensor networks? 2. Iterative diagonalization 3. Covariant index notation 4. Entanglement entropy and area laws 5. Tensor network diagrams 6. Singular-value decomposition (SVD)
	22.04.19			Easter Monday