Inhaltsbereich
Tensor Networks 2021 – Lecture notes
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Lecture notes
The course schedule (Stoffplan) can be downloaded here.
(Some minor reshuffling of topics might occur during the course of the semester.)
| Lecture | Date | Notes | Pages | Topic |
| T16 | 20.07.21 | Tutorial: MPS-based machine learning (optional) | ||
| L26 | 15.07.21 | ML ML.1 ML.2 |
Machine learning 1. Neural networks 2. Supervised learning with tensor networks |
|
| L25 | 14.07.21 | 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 |
|
| T15 | 13.07.21 | Tutorial: GILT, FET | ||
| L24 | 08.07.21 | CanF 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 |
|
| L23 | 07.07.21 | TNR 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 |
|
| T14 | 06.07.21 | Tutorial: TRG, simple update | ||
| L22 | 01.07.21 | 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 res |
|
| L21 | 30.07.21 | 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) |
|
| T13 | 29.07.21 | Tutorial: Finite PEPS | ||
| L20 | 24.07.21 | 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) |
|
| L19 | 23.07.21 | 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 |
|
| T12 | 22.06.21 | Tutorial: NRG II | ||
| L18 | 17.06.21 | 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 |
|
| L17 | 18.06.21 | NRG-III 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 |
|
| T11 | 15.06.21 | Tutorial: NRG I | ||
| L16 | 10.06.21 | 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 |
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| L15 | 09.06.21 | 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 |
|
| T10 | 08.06.21 | Tutorial: Finite temperature. TDVP. QSpace tDMRG (optional) | ||
| L14 | 02.06.21 | TS 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. |
|
| T09 | 01.06.21 | Tutorial: tDMRG, subspace expansion, error estimates | ||
| L13 | 27.05.21 | DMRG-III.1-4 | DMRG III: tDMRG, finite temperature (purification, XTRG) 1. tDMRG 2. Error analysis 2. Purification 4. Exponential tensor renormalization group (XTRG) |
|
| L12 | 26.05.21 | DMRG-II DMRG-II.1 DMRG-II.2 DMRG-II.3 DMRG-II.4 |
DMRG II: original, subspace expansion, error estimates 1. Relation to original DMRG. 2. One-site update with subspace expansion. 3. Error estimates |
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| T08 | 25.05.21 | Tutorial: iTEBD. QSpace DMRG (optional) | ||
| L11 | 20.05.21 | 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 |
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| T07 | 19.05.21 | Tutorial: DMRG | ||
| T06 | 18.05.21 | Tutorial: Symmetries & QSpace II | ||
| L10 | 12.05.21 | 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 |
|
| L09 | 12.05.21 | Sym-III.1 Sym-III.2 Sym-III.3 Sym-III.4 |
Symmetries III: Outer multiplicity, X-symbols. (optional; was omitted in 2021) 1. Motivation 2. Outer multiplicity 3. Arrow inversion 4. Pairwise contractions and X-symbols |
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| T05 | 12.05.21 | Tutorial: Symmetries & QSpace I | ||
| L08 | 06.05.21 | 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. 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 |
|
| L07 | 05.05.21 | Sym-I Sym-I.1 Sym-I.1 Sym-I.1 |
Symmetries I: Abelian (revised and improved version) 1. Example: spin 1/2 XXZ-chain 2. Iterative diagonalization 3. QSpace bookkeeping for unit matrices |
|
| T04 | 04.05.21 | Tutorial: AKLT Model, MPO methodology | ||
| L06 | 29.04.21 | MPS-IV 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 4. MPS representation of Fermi sea |
|
| L05 | 28.04.21 | MPS-III MPS-III.1 MPS-III.2 MPS-III.3 MPS-III.4 MPS-III.5 MPS-III.6 MPS-III.7 |
MPS III: Translationally invariant MPS, AKLT model 1. Transfer matrix 2. AKLT model 3. AKLT ground state 4. Transfer operator and string order parameter |
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| T02 | 27.04.21 | Tutorial: MPS, iterative diagonalization | ||
| L04 | 22.04.21 | MPS-II MPS-II.1 MPS-II.2 MPS-II.3 |
MPS II: Diagonalization, fermionic signs 1. Basis change. 2. Iterative diagonalization of short spin chain. 3. Spinless fermions 4. Spinful fermions |
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| L03 | 21.04.21 | MPS-I MPS-I.1 MPS-I.2 MPS-I.3 MPS-I.4 |
MPS I: Basic properties 1. Overlaps, matrix elements. 2. Left- and right-normalized states. 3. Matrix elements of local operators. 4. Canonical MPS forms (left, right, site, bond). |
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| T02 | 20.04.21 | Tutorial: Tensor network basics | ||
| T01 | 15.04.21 | Tutorial: MATLAB 101 | ||
| L02 | 14.04.21 | TNB-II TNB-II.1 TNB-II.2 TNB-II.3 TNB-II.4 |
Tensor Network basics II 1. Singular-value decomposition (SVD) 2. Schmidt decomposition 3. Iterative diagonalization 4. Rewriting any tensor as MPS |
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| L01 | 13.04.21 | TNB-I TNB-I.1 TNB-I.2 TNB-I.3 TNB-I.4 |
Tensor Network basics (TNB) I 1. Notation for generic quantum lattice systems. 2. Entanglement entropy and area laws 3. Tensor network diagrams 4. Covariant index notation |
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