Inhaltsbereich
Tensor Networks 2020 – Lecture notes
- Overview
- Lecture notes
- Tutorials
- References
- Videos
The course schedule (Stoffplan) can be downloaded here.
(Some minor reshuffling of topics might occur during the course of the semester.)
| Date | Notes | Lecture | Pages | Topic |
|---|---|---|---|---|
| 28.07.20 | Tutorial: MPS-based machine learning (optional) | |||
| 23.07.20 | ML.1 ML.2 |
Machine learning 1. Neural networks 2. Supervised learning with tensor networks |
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| 22.07.20 | 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 |
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| 21.07.19 | Tutorial: GILT, FET | |||
| 16.07.20 | CanF.1 CanF.2 CanF.3 CanF.4 |
2D 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 |
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| 15.07.20 | 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 |
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| 14.07.20 | Tutorial: TRG, simple update | |||
| 09.07.20 | 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 |
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| 08.07.20 | 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) |
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| 07.07.20 | Tutorial: Finite PEPS | |||
| 02.07.20 | 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) |
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| 01.07.20 | 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 |
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| 30.06.20 | Tutorial: NRG II | |||
| 25.06.20 | 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 |
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| 24.06.20 | NRG-III.1 NRG-III.2 NRG-III.3 NRG-III.4 |
NRG III: Thermal and dynamical quantities 1. Thermodynamic observables 2. Wilson ration. 3. Lehmann representation of spectral functions 4. Single-shell and patching schemes |
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| 23.06.20 | Tutorial: NRG I | |||
| 18.06.20 | 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 |
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| 28.05.20 | 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 |
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| 16.06.20 | Tutorial: tDMRG. Tangent space methods | |||
| 11.06.20 | Corpus Christi | |||
| 10.06.20 | TS.1 TS.2 TS.3 TS.4 |
Tangent space methods (TDVP) 1. MPS and canonical forms. 2. Tangent space. 3. Tangent space projector. 4. Time evolution. |
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| 09.06.20 | Tutorial: iTEBD | |||
| DMRG-III.1-4 | DMRG III: tDMRG, finite temperature (purification, XTRG) 1. tDMRG 2. Error analysis 2. Purification 4. Exponential tensor renormalization group (XTRG) |
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| 04.06.20 | DMRG-II.1 DMRG-II.2 DMRG-II.3 |
DMRG II: Traditional DMRG, tDMRG, purification 1. Relation to traditional DMRG. 2. tDMRG. 3. Finite temperature: purification. |
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| 03.06.20 | pdf |
pdf |
MPS-VI.1 iTEBD.1 iTEBD.2 iTEBD.3 iTEBD.4 |
MPS VI: 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 |
| 02.06.20 | Tutorial: DMRG | |||
| 28.05.20 | 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 |
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| 27.05.20 | 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 |
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| 26.05.20 | Tutorial: Symmetries & QSpace II | |||
| 21.05.20 | Ascension Day (if needed: Symmetries & QSpace IV) | |||
| 20.05.20 | 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 contractions and X-symbols |
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| 19.05.20 | Tutorial: Symmetries & QSpace I | |||
| 14.05.20 | 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. 1. Motivation, SU(2) basics 2. Tensor product decomposition 3. Tensor operators 4. A-matrix factorizes 5. Example: two spin 1/2's 6. Example: three spin 1/2's 7. Bookkeeping for unit matrices |
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| 13.05.20 | 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 |
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| 12.05.20 | Tutorial: Iterative diagonalization, AKLT Model | |||
| 07.05.20 | MPS-IV.1 MPS-IV.2 MPS-IV.3 MPS-IV.4 MPS-IV.5 MPS-IV.6 MPS-IV.7 |
MPS IV: Translationally invariant MPS, AKLT model 1. Transfer matrix 2. Correlation functions 3. AKLT model - general remarks 4. Construction of Hamiltonian 5. AKLT ground state 6. Transfer operator 7. String order parameter |
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| 06.05.20 | MPS-III.1 MPS-III.2 MPS-III.3 MPS-III.4 |
MPS III: Diagonalization, fermionic signs 1. Basis transformation 2. Iterative diagonalization 3. Spinless fermions 4. Spinful fermions |
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| 05.05.20 | Tutorial: MPS I | |||
| 30.04.20 | MPS-II.1 MPS-II.2 |
MPS II: Matrix Product States 1. Matrix elements, expectation values 2. Schmidt decomposition |
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| 29.04.20 | MPS-I.1 MPS-I.2 |
MPS I: Matrix Product States 1. Overlap and normalization. 2. Canonical MPS forms (left, right, site, bond) |
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| 28.04.20 | Tutorial: Tensor network basics | |||
| 23.04.19 | Tutorial: MATLAB basics | |||
| 22.04.20 | TNB-II.1 TNB-II.2 TNB-II.3 |
Tensor Network basics (TNB) II: 1. Entanglement entropy and area laws 2. Tensor network diagrams 3. Singular-value decomposition (SVD) |
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| 21.04.20 | TNB-I.1 TNB-I.2 TNB-I.3 |
Tensor Network basics (TNB) I: 1. Why study tensor networks? 2. Iterative diagonalization 3. Covariant index notation |