Inhaltsbereich
Tensor Networks 2022 – Lecture notes
- Overview
- Lecture notes
- Tutorials
- References
- Videos
Lecture notes
| Lecture | Date | Notes | Pages | Topic |
| L26 | 26.07.22 | F-PEPS 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 |
|
| L25 | 14.07.22 |
CanF CanF.1 CanF.2 CanF.3 CanF.4 |
2 D Canonical Forms, Isometric PEPS 1. Canonical form for bond in 2D tensor network 2. Full environment truncation 3. Isometric PEPS: Moses move 4. Isometric PEPS: Applications |
|
| L24 | 19.07.22 |
TNR TNR.1 TNR.2 TNR.3 TNR.4 TNR.5 TNR.6 |
TNR: Tensor network renormalization, MERA 1. Motivation 2. TNR idea 3. Projective truncation 4. TNR details 5. TNR results in MERA 6. TNR benchmark results |
|
| L23 | 14.07.22 |
TRG-II 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 |
|
| T07 | 13.07.22 | Tutorial: NRG II: Finite-size spectra, static correlations | ||
| L22 | 30.07.22 | TRG 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) |
|
| L21 | 07.07.22 |
PEPS-II PEPS-II.1 PEPS-II.2 PEPS-II.3 |
PEPS II: contractions via MPS techniques 1. PEPS via finite-size MPS 2. Infinite-size PEPS (iPEPS) 3. Corner transfer matrix (CTM) |
|
| T06 | 06.07.22 | Tutorial: Wilson chain, TDVP, Variance (continued) | ||
| L20 | 05.07.22 | PEPS-I 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 |
|
| L19 | 30.06.22 | TS TS-II.1 TS-II.2 TS-II.3 TS-II.4 |
Tangent space methods II: TDVP, energy variance 1. 1-site TDVP 2. 2-site projectors 3. 2-site TDVP 4. Energy variance |
|
| T06 | 29.06.22 | Tutorial: Wilson chain, TDVP, Variance II | ||
| L18 | 28.06.22 | TS TS-I.1 TS-I.2 TS-I.3 TS-I.4 TS-I.5 |
Tangent space methods (TDVP) 1. Motivation: why is tangent space useful? 2. MPS canonical forms 3. Kept and discarded spaces 4. Kept and discarded projectors 5. Tangent space projector |
|
| T05 | 22.06.22 | Tutorial: DMRG II: 2-site, iTEBD (continued) | ||
| L17 | 21.06.22 | NRG-IV NRG-IV.1 NRG-IV.2 NRG-IV.3 NRG-IV.4 |
NRG IV: Spectral function, fdm-NRG 1. MPS notation for discarded/kept states 2. Complete many-body basis 3. Full-density-matrix NRG (fdmNRG) 4. Spectral functions for SIAM |
|
| L16 | 16.06.22 | NRG-III NRG-III.1 NRG-III.2 NRG-III.3 NRG-III.4 NRG-III.4 |
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 |
|
| T05 | 15.06.22 | Tutorial: DMRG II: 2-site, iTEBD | ||
| L15 | 14.06.22 | NRG-II NRG-II.1 NRG-II.2 NRG-II.3 NRG-II.4 NRG-II.5 |
NRG II: RG flow, fixed points 1. General RG concepts 2. NRG iteration scheme from RG perspective 3. Uncoupled bath Hamiltonian: fixed points 4. Kondo model: fixed points and RG flow 5. Anderson model: fixed points and RG flow |
|
| L14 | 09.06.22 |
NRG-I 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 |
|
| T04 | 08.06.22 | Tutorial: DMRG I: 1-site ground state search (continued) | ||
| L13 | 07.06.22 | Sym-III Sym-III.1 Sym-III.2 Sym-III.3 Sym-III.4 |
Symmetries III: Outer multiplicity, X-symbols 1. Motivation 2. Outer multiplicity 3. Arrow inversion 4. Pairwise contraction and X-symbols |
|
| L12 | 02.06.22 |
Sym-II Sym-II.1 Sym-II.2 Sym-II.3 Sym-II.4 Sym-II.5 Sym-II.6 Sym-II.7 |
Symmetries II: Non-Abelian (optional) 1. Motivation, SU(2) basics 2. Tensor product decomposition 3. Tensor operators 4. Example: two spin 1/2's 5. Example: three spin 1/2's 6. A-matrix factorizes 7. Bookkeeping for unit matrices |
|
| T04 | 01.06.22 | Tutorial: DMRG I: 1-site ground state search | ||
| L11 | 31.05.22 |
DMRG-II DMRG-II.1 DMRG-II.2 DMRG-II.3 DMRG-II.4 |
DMRG I2I: Original DMRG, tDMRG, finite temperature (XTRG, purification) 1. Original DMRG 2. tDMRG 3. Exponential tensor renormalization group (XTRG) 4. Purification |
|
| L10 | 26.05.22 | Sym-I Sym-I.1 Sym-I.1 Sym-I.1 |
Symmetries I: Abelian (optional) 1. Example: spin 1/2 XXZ-chain 2. Iterative diagonalization 3. QSpace bookkeeping for unit matrices |
|
| T03 | 25.05.22 | Tutorial: SVD, MPO, Diagonalization (continued) | ||
| L09 | 24.05.22 | iTEBD iTEBD.1 iTEBD.2 iTEBD.3 iTEBD.4 iTEBD.5 |
iTEBD: Infinite Time-Evolving Block Decimation 1. Vidal's Gamma-Lambda notation 2. Basic iTEBD algorithm 3. iTEBD in Gamma-Lambda notation 4. Hasting's method 5. Orthogonalization |
|
| L08 | 18.05.22 | DMRG-I 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 update |
|
| L07 | 17.05.22 |
MPS-IV MPS-IV.1 MPS-IV.2 MPS-IV.3 MPS-IV.4 |
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 4. MPS representation of Fermi sea |
|
| T03 | 17.05.22 | Tutorial: SVD, MPO, Diagonalization | ||
| L06 | 12.05.22 |
MPS-III MPS-III.1 MPS-III.2 MPS-III.3 MPS-III.4 |
MPS III: Translationally invariant MPS, AKLT model 1. Transfer matrix 2. AKLT model 3. AKLT ground state 4. Transfer operator and string order parameter |
|
| L05 | 11.05.22 |
MPS-II MPS-II.1 MPS-II.2 MPS-II.3 MPS-II.4 |
MPS II: Diagonalization, fermionic signs 1. Basis change. 2. Iterative diagonalization of short spin chain. 3. Spinless fermions 4. Spinful fermions |
|
| T02 | 10.05.22 | Tutorial: Tensor network basics (cont.) | ||
| L04 | 05.05.22 |
MPS-I MPS-I.1 MPS-I.2 MPS-I.3 |
MPS I: Basic properties 1. Overlaps, matrix elements. 2. Left- and right-normalized states. 3. Canonical MPS forms (left, right, site, bond) |
|
| L03 | 04.05.22 | TNB-III TNB-III.1 TNB-III.2 TNB-III.3 TNB-III.4 |
Tensor Network basics III 1. Unitaries and isometries 2. Singular value decomposition (SVD) 3. Schmidt decomposition 4. Reshaping generic tensor into MPS form |
|
| T02 | 03.05.22 | Tutorial: Tensor network basics | ||
| T01 | 28.04.22 | Tutorial: MATLAB 101 | ||
| L02 | 27.04.22 | TNB-II TNB-II.1 TNB-II.2 TNB-II.3 |
Tensor Network basics II 1. Covariant index notation 2. Arrow conventions 3. Iterative diagonalization |
|
| L01 | 26.04.22 |
TNB-I TNB-I.1 TNB-I.2 TNB-I.3 |
Tensor Network basics (TNB) I 1. Notation for generic quantum lattice systems. 2. Entanglement entropy and area laws 3. Tensor network diagrams |