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Tensor Networks for Many-Body Physics 2025 – Lecture notes

Lecture Notes: (Klick on the small grey triangles for dropdown lists of subtopics.)
Lecture Date Notes Pages Topic
Lectures 21-24 on NRG are copy-pasted from lectures 13, 14, 17, 18 of the 2023 Tensor Network Course. The corresponding videos can be found here.
L24 15.06.23 pdf
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
L23 14.06.23 pdf
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
L22
07.06.23 pdf
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
L21
31.05.23 pdf
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
L21 24.06.25 pdf
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
L20 16.07.25 pdf
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
T07b 11.07.23 Tutorial: NRG II: Finite-size spectra, static correlations (continued)
L19 10.07.25 pdf
TRG TRG-I.1
TRG-I.2
TRG-I.3
TRG-I.4-7
Term projects + Tensor renormalization group (TRG) 1. TRG for 2D classical Ising model
2. 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 pdf
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
L18 03.07.25 pdf
TCI.14-17 TCI.14
TCI.15
TCI.16
TCI.17
Quantics TCI 14. Computing integrals and functions
15. Quantics representation of functions (QTCI)
16. Numerical examples of QTCI
17. Quantics Fourier transform (QFT)
L17 02.07.25 pdf
TCI.10-13 TCI.10
TCI.11
TCI.12
TCI.13
TCI: Unfolding tensor to TT (continued) 10. Proof of nesting properties
11. CI-canonicalization
12. High-level TCI algorithms
13. Relation to machine learning
L16 26.06.25 pdf
TCI.7-9 TCI.7
TCI.8
TCI.9
TCI: Unfolding tensor to TT 7. Ingredients of TCI form
8. Nesting conditions
9. TCI unfolding algorithms (2-site, 1-site, 0-site)
L15 25.06.25 pdf
TCI.4-6 TCI.4
TCI.5
TCI.6
TCI: Pivoting, Schur complement, prrLU 4. Finding new pivots
5. Properties of Schur complement
6. Partial rank-revealing LU decomposition (prrLU)
L14 18.06.25 pdf
TCI.1-3 TCI.1
TCI.2
TCI.3
Tensor Cross Interpolation (TCI) I 1. Motivation
2. Integration of multi-variate functions
3. Matrix Cross Interpolation (CI)
L13 12.06.25 pdf
rSVD.1 rSVD.1
Randomized SVD (rSVD)
L12 11.06.25 pdf
CBE.1-4CBE.1
CBE.2
CBE.3
CBE.4
Energy variance. Controlled bond expansion (CBE) 1. Energy variance.
2. CBE-DMRG.
3. Shrewd selection.
4. CBE-TDVP.
T06b 27.06.23 Tutorial: Wilson chain, TDVP, Variance (continued)
T06a 22.06.23 Tutorial: Wilson chain, TDVP, Variance II
L11 05.06.25 pdf
TDVP.1-3TDVP.1
TDVP.2
TDVP.3
Tangent space methods II: TDVP, energy variance 1. 1-site TDVP
2. 2-site projectors
3. 2-site TDVP 
L10 04.06.25 pdf
TS-I-5 TS-I.1
TS-I.2
TS-I.3
TS-I.4
TS-I.5
Tangent space  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
T05a 06.06.23 Tutorial: DMRG II: 2-site ground state search
T04b 01.06.23 Tutorial: DMRG I: 1-site ground state search (cont.)
L09 28.05.25 pdf
DMRG-II DMRG-II.1
DMRG-II.2
DMRG-II.3
DMRG-II.4
DMRG II: Original DMRG, tDMRG, finite temperature (XTRG, purification) 1. Original DMRG
2. tDMRG
3. Exponential tensor renormalization group (XTRG)
4. Purification
L08 21.05.25 pdf
iTEBD.1-2 iTEBD.1
iTEBD.2 
iTEBD: Infinite Time-Evolving Block Decimation 1. Vidal's Gamma-Lambda notation
2. Basic iTEBD algorithm 
T04a 23.05.23 Tutorial: DMRG I: 1-site ground state search
L07 15.05.25 pdf
DMRG-I DMRG-I.1
DMRG-I.2
DMRG-I.3
DMRG-I.4
DMRG I.1-4: Density Matrix Renormalization Group - ground state search 1. Single-site optimization
2. Lancos Method
3. Excited states
4. Two-site update
T03b 16.05.23 Tutorial: SVD, MPO, Diagonalization (cont.)
L06 14.05.25 pdf
MPS-IV MPS.9
MPS.10 
MPS.11 
MPS.12
MPS.9-12: Matrix product operators
9. Applying MPO to MPS yields MPS
10. MPO representation of Heisenberg Hamiltonian
11. Applying MPO to mixed-canonical state
12. MPS representation of Fermi sea
T03a 09.05.23 Tutorial: SVD, MPO, Diagonalization
L05 08.05.25 pdf
MPS-III MPS.6
MPS.7
MPS.8
MPS.6-8: Diagonalization, fermionic signs 6. Iterative diagonalization of short spin chain.
7. Spinless fermions.
8. Spinful fermions.
L04 07.05.25 pdf
MPS-II MPS.3
MPS.4
MPS.5
MPS.3-5: Canonical forms 3. Left- and right-normalized states.
4. Canonical MPS forms (left, right, site, bond).
5. Basis change, projectors
T02b 02.05.23 Tutorial: Tensor network basics (cont.)
L03 30.04.25 pdf
MPS-IMPS.0
MPS.1
MPS.2
Matrix Produc States (MPS).0-2 Basics 0. Reshaping generic tensor into MPS form.
1. Iterative diagonalization
2. Overlaps, matrix elements. 
T02a 25.04.23 Tutorial: Tensor network basics
T01 20.04.23 Tutorial: MATLAB 101
L02 24.04.25 pdf
TNB-II TNB.4
TNB.5
TNB.6
Tensor Network basics II 1. Unitaries and isometries
5. Singular value decomposition
6. Schmidt decomposition
L01 23.04.25 pdf
TNB-I TNB.1
TNB.2
TNB.3
Tensor Network basics (TNB) I 1. Why matrix product states?
2. Arrow conventionsTensor network diagrams
3. Tensor network diagrams