Inhaltsbereich
Tensor Networks for Many-Body Physics 2025 – Lecture notes
- Overview
- Lecture notes
- Tutorials
- References
- Videos
Lecture Notes: (Klick on the small grey triangles for dropdown lists of subtopics.)
| Lecture | Date | Notes | Pages | Topic |
| Changkai Zhang (21.05.26) and Kiyeon Kim (22.05.26) gave guest presentations on symmetry-enabled tensor network libraries developed by them, inspired by the symmetry-enabled QSpace tensor network library of Andreas Weichselbaum. Their presentations were not part of the 2025 Tensor Network course but are included here because tensor network enthusiasts may find them useful. For video recordings of these presentations, see Videos: Changkai Zhang, Kiyeon Kim. | ||||
| guest talk |
22.05.26 | Telum.jl docs CGT.jl |
Telum.jl Telum_docs LuCGT.jl |
Julia-based symmetry-enabled tensor network library, by Kiyeon Kim |
| guest talk |
21.05.26 | Nicole Yuzuha Alice |
Nicole Yuzuha Alice |
Python-based symmetry-enabled tensor network library, by Changkai Zhang |
| Lectures 22-25 on NRG are copy-pasted from lectures 13, 14, 17, 18 of the 2023 Tensor Network Course. The corresponding videos can be found here (as lectures 13, 14, 17, 18). | ||||
| L25 | 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 |
| L24 | 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 |
| L23 |
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 |
| L22 |
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 |
| L21 | 24.06.25 |
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 |
|
| L20 | 16.07.25 | 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 |
|
| T07b | 11.07.23 | Tutorial: NRG II: Finite-size spectra, static correlations (continued) | ||
| L19 | 10.07.25 | TRGTRG-I.1TRG-I.2 TRG-I.3 TRG-I.4-7 |
Term projects + 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) |
|
| T07a | 04.07.23 | Tutorial: NRG II: Finite-size spectra, static correlations | ||
| L21 | 24.06.25 | 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 |
|
| L18 | 03.07.25 | TCI.14-17TCI.14TCI.15 TCI.16 TCI.17 |
Quantics TCI14. Computing integrals and functions15. Quantics representation of functions (QTCI) 16. Numerical examples of QTCI 17. Quantics Fourier transform (QFT) |
|
| L17 | 02.07.25 | TCI.10-13TCI.10TCI.11 TCI.12 TCI.13 |
TCI: Unfolding tensor to TT (continued)10. Proof of nesting properties11. CI-canonicalization 12. High-level TCI algorithms 13. Relation to machine learning |
|
| L16 | 26.06.25 | TCI.7-9TCI.7TCI.8 TCI.9 |
TCI: Unfolding tensor to TT7. Ingredients of TCI form8. Nesting conditions 9. TCI unfolding algorithms (2-site, 1-site, 0-site) |
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| L15 | 25.06.25 | TCI.4-6TCI.4TCI.5 TCI.6 |
TCI: Pivoting, Schur complement, prrLU4. Finding new pivots5. Properties of Schur complement 6. Partial rank-revealing LU decomposition (prrLU) |
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| L14 | 18.06.25 | TCI.1-3TCI.1TCI.2 TCI.3 |
Tensor Cross Interpolation (TCI) I1. Motivation2. Integration of multi-variate functions 3. Matrix Cross Interpolation (CI) |
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| L13 | 12.06.25 | rSVD.1rSVD.1 |
Randomized SVD (rSVD) |
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| L12 | 11.06.25 | CBE.1-4CBE.1CBE.2 CBE.3 CBE.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 | ||
| L11 | 05.06.25 | TDVP.1-3TDVP.1TDVP.2 TDVP.3 |
Tangent space methods II: TDVP, energy variance1. 1-site TDVP2. 2-site projectors 3. 2-site TDVP |
|
| L10 | 04.06.25 | TS-I-5TS-I.1TS-I.2 TS-I.3 TS-I.4 TS-I.5 |
Tangent space1. 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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| 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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| L09 | 28.05.25 | 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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| L08 | 21.05.25 | iTEBD.1-2iTEBD.1iTEBD.2 |
iTEBD: Infinite Time-Evolving Block Decimation1. Vidal's Gamma-Lambda notation2. Basic iTEBD algorithm |
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| T04a | 23.05.23 | Tutorial: DMRG I: 1-site ground state search |
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| L07 | 15.05.25 | DMRG-IDMRG-I.1DMRG-I.2 DMRG-I.3 DMRG-I.4 |
DMRG I.1-4: 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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| L06 | 14.05.25 | MPS-IVMPS.9MPS.10 MPS.11 MPS.12 |
MPS.9-12: Matrix product operators9. Applying MPO to MPS yields MPS10. MPO representation of Heisenberg Hamiltonian 11. Applying MPO to mixed-canonical state 12. MPS representation of Fermi sea |
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| T03a | 09.05.23 | Tutorial: SVD, MPO, Diagonalization |
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| L05 | 08.05.25 | MPS-IIIMPS.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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| L04 | 07.05.25 | MPS-IIMPS.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.) | ||
| L03 | 30.04.25 | MPS-IMPS.0MPS.1 MPS.2 |
Matrix Produc States (MPS).0-2 Basics0. Reshaping generic tensor into MPS form.1. Iterative diagonalization 2. Overlaps, matrix elements. |
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| T02a | 25.04.23 | Tutorial: Tensor network basics | ||
| T01 | 20.04.23 | Tutorial: MATLAB 101 |
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| L02 | 24.04.25 | TNB-IITNB.4TNB.5 TNB.6 |
Tensor Network basics II1. Unitaries and isometries5. Singular value decomposition 6. Schmidt decomposition |
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| L01 | 23.04.25 | TNB-ITNB.1TNB.2 TNB.3 |
Tensor Network basics (TNB) I1. Why matrix product states?2. Arrow conventionsTensor network diagrams 3. Tensor network diagrams |