Inhaltsbereich
Tensor Networks for Many-Body Physics 2023 – Lecture notes
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Lecture Notes: (Klick on the small grey triangles for dropdown lists of subtopics.)
Lecture notes are corrected/revised after lecture; if revisions are substantial, this will be indicated by writing the links as pdf-r (r for revised).
| Lecture | Date | Notes | Gaps | Lecture | Pages | Topic |
| L28 | 20.07.23 | pdf |
MLML.1ML.2 |
Machine learning1. Neural networks2. Supervised learning with tensor networks |
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| L27 | 19.07.23 | pdf |
CanFCanF.1CanF.2 CanF.3 CanF.4 |
Isometric PEPS, 2D Canonical Forms1. Isometric PEPS: Moses move2. Isometric PEPS: Applications 3. Canonical form for bond in 2D tensor network 4. Full environment truncation |
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| L26 | 13.07.23 |
TNR
TNR.1
TNR.2 TNR.3 TNR.4 TNR.5 TNR.6 |
TNR: Tensor network renormalization, MERA
1. Motivation2. TNR idea 3. Projective truncation 4. TNR details 5. TNR results in MERA 6. TNR benchmark results |
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| L25 | 12.07.23 | pdf |
TRG-II>TRG-II.1TRG-II.2 TRG-II.3 TRG-II.4 TRG-II.5 TRG-II.6 |
TRG-II: Graph-independent local truncations (Gilt)1. Motivation2. Why is TRG insufficient? 3. Environment spectrum 4. Gilt: Graph-independent local truncations 5. Gilt-TNR 6. Benchmark results |
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| T07b | 11.07.23 | Tutorial: NRG II: Finite-size spectra, static correlations (continued) | ||||
| L24 | 06.07.23 | pdf |
TRGTRG-I.1TRG-I.2 TRG-I.3 TRG-I.4-7 |
Tensor renormalization group (TRG)1. TRG for 2D classical Ising model2. Variational uniform MPS (VUMPS) 3. FPCM (Fixed Point Corner Matrix) 4. Related approaches - TRG for honeycomb, TRG for quantum lattice models, Second renormalization (SRG) of tensor network states, Core tensor renormalization group (CTRG) |
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| L23 | 05.07.23 | pdf |
PEPS-IIPEPS-II.1PEPS-II.2 PEPS-II.3 |
PEPS II: contractions via MPS techniques1. PEPS via finite-size MPS2. Infinite-size PEPS (iPEPS) 3. Corner transfer matrix (CTM) |
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| T07a | 04.07.23 | Tutorial: NRG II: Finite-size spectra, static correlations | ||||
| L22 | 29.06.23 |
PEPS-IPEPS-I.1PEPS-I.2 PEPS-I.3 PEPS-I.4 |
PEPS I: Projected entangled-pair states1. Motivation and Definition2. Example: RVB state 3. Example: Kitaev's Toric Code 4. Example: Resonating AKLT state |
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| L21 | 28.06.23 |
pdf |
TS-IIITS-III.1TS-III.2 TS-III.3 TS-III.4 |
Energy variance. Controlled bond expansion (CBE)1. Energy variance.2. CBE-DMRG. 3. Shrewd selection. 4. CBE-TDVP. |
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| T06b | 27.06.23 | Tutorial: Wilson chain, TDVP, Variance (continued) | ||||
| T06a | 22.06.23 | Tutorial: Wilson chain, TDVP, Variance II | ||||
| L20 | 21.06.23 |
pdf |
TS-IITS-II.1TS-II.2 TS-II.3 TS-II.4 |
Tangent space methods II: TDVP, energy variance1. 1-site TDVP2. 2-site projectors 3. 2-site TDVP 4. Energy variance. |
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| L19 | 20.06.23 | pdf |
TS-ITS-I.1TS-I.2 TS-I.3 TS-I.4 TS-I.5 |
Tangent space methods I: projector formalism1. Motivation: why is tangent space useful?2. MPS canonical forms 3. Kept and discarded spaces 4. Kept and discarded projectors 5. Tangent space projector |
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| L18 | 15.06.23 | pdf |
NRG-IVNRG-IV.1NRG-IV.2 NRG-IV.3 NRG-IV.4 |
NRG IV: Spectral function, fdm-NRG1. MPS notation for discarded/kept states2. Complete many-body basis 3. Full-density-matrix NRG (fdmNRG) 4. Spectral functions for SIAM |
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| L17 | 14.06.23 | pdf |
NRG-IIINRG-III.1NRG-III.2 NRG-III.3 NRG-III.4 NRG-III.4 |
NRG III: Thermal and dynamical quantities1. Thermodynamic observables2. Lehmann representation of spectral functions 3. Single-shell and patching schemes 4. Graphical notation for basis change 5. MPS notation for discarded/kept states |
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| T05b | 13.06.23 | Tutorial: DMRG II: 2-site ground state search (cont.) |
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| L16 | 08.06.23 | pdf |
Sym-IIISym-III.1Sym-III.2 Sym-III.3 Sym-III.4 |
Symmetries III: Outer multiplicity, X-symbols (optional; was omitted in 2023)1. Motivation2. Outer multiplicity 3. Arrow inversion 4. Pairwise contractions and X-symbols |
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| L15 |
08.06.23 |
pdf |
Sym-IISym-II.1Sym-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) basics2. 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 |
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| L14 |
07.06.23 | pdf |
NRG-IINRG-II.1NRG-II.2 NRG-II.3 NRG-II.4 NRG-II.5 |
NRG II: RG flow, fixed points1. General RG concepts2. 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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| T05a | 06.06.23 | Tutorial: DMRG II: 2-site ground state search |
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| T04b | 01.06.23 | Tutorial: DMRG I: 1-site ground state search (cont.) |
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| L13 |
31.05.23 | pdf |
NRG-INRG-I.1NRG-I.2 NRG-I.3 NRG-I.4 |
NRG I: Numerical Renormalization group - Wilson chain1. Single-impurity Anderson model2. Logarithmic discretization 3. Wilson chain 4. Iterative diagonalization |
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| 30.05.23 | Pentacost Tuesday (no lecture or tutorial) |
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| L12 | 25.05.23 | pdf |
DMRG-IIDMRG-II.1DMRG-II.2 DMRG-II.3 DMRG-II.4 |
DMRG II: Original DMRG, tDMRG, finite temperature (XTRG, purification)1. Original DMRG2. tDMRG 3. Exponential tensor renormalization group (XTRG) 4. Purification |
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| L11 | 24.05.23 |
iTEBDiTEBD.1iTEBD.2 iTEBD.3 iTEBD.4 iTEBD.5 |
iTEBD: Infinite Time-Evolving Block Decimation1. Vidal's Gamma-Lambda notation2. Basic iTEBD algorithm 3. iTEBD in Gamma-Lambda notation 4. Hasting's method 5. Orthogonalization |
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| T04a |
23.05.23 | Tutorial: DMRG I: 1-site ground state search |
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| L10 | 18.05.23 | pdf |
Sym-ISym-I.1Sym-I.1 Sym-I.1 |
Symmetries I: Abelian (optional)1. Example: spin 1/2 XXZ-chain2. Iterative diagonalization 3. QSpace bookkeeping for unit matrices |
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| 18.05.23 | Ascension Day | |||||
| L09 |
17.05.23 | pdf |
DMRG-IDMRG-I.1DMRG-I.2 DMRG-I.3 DMRG-I.4 |
DMRG I: Density Matrix Renormalization Group - ground state search1. Single-site optimization2. Lancos Method 3. Excited states 4. Two-site update |
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| T03b |
16.05.23 | Tutorial: SVD, MPO, Diagonalization (cont.) |
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| L08 |
11.05.23 | pdf |
MPS-IVMPS.13MPS.14 MPS.15 MPS.16 |
MPS.13-16: Matrix product operators
13. Applying MPO to MPS yields MPS14. MPO representation of Heisenberg Hamiltonian 15. Applying MPO to mixed-canonical state 16. MPS representation of Fermi sea |
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| L07 |
10.05.23 | pdf |
MPS-IIIMPS.9MPS.10 MPS.11 MPS.12 |
MPS.9-12 Translationally invariant MPS, AKLT9. Transfer matrix10. AKLT model 11. AKLT ground state 12. Transfer operator and string order parameter |
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| T03a |
09.05.23 | Tutorial: SVD, MPO, Diagonalization |
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| L06 |
04.05.23 |
pdf |
MPS.6-8MPS.6MPS.7 MPS.8 |
MPS.6-8: Diagonalization, fermionic signs6. Iterative diagonalization of short spin chain.7. Spinless fermions. 8. Spinful fermions. |
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| L05 |
03.05.23 | pdf |
MPS.3-5MPS.3MPS.4 MPS.5 |
MPS.3-5: Canonical forms3. Left- and right-normalized states.4. Canonical MPS forms (left, right, site, bond). 5. Basis change, projectors |
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| T02b |
02.05.23 | Tutorial: Tensor network basics (cont.) | ||||
| L04 |
27.04.23 | pdf |
TNB-IIITNB-III.4MPS.1-2MPS.1MPS.2 |
Tensor Network basics III (cont.)
4. Schmidt decompositionMPS.1-2 Basics
1. Reshaping generic tensor into MPS form.2. Overlaps, matrix elements. |
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| L03 |
26.04.23 |
pdf |
TNB-IIITNB-III.1TNB-III.2 TNB-III.3 TNB-III.4 |
Tensor Network basics III1. Reshaping generic tensor into MPS form2. Unitaries and isometries 3. Singular value decomposition (SVD) 4. Schmidt decomposition |
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| T02a |
25.04.23 | Tutorial: Tensor network basics | ||||
| T01 | 20.04.23 | Tutorial: MATLAB 101 |
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| L02 |
19.04.23 | pdf |
TNB-IITNB-II.1TNB-II.2 TNB-II.3 |
Tensor Network basics II1. Covariant index notation2. Arrow conventions 3. Iterative diagonalization |
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| L01 |
18.04.23 |
TNB-ITNB-I.1TNB-I.2 TNB-I.3 |
Tensor Network basics (TNB) I1. Notation for generic quantum lattice systems.2. Entanglement entropy and area laws 3. Tensor network diagrams |