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Tensor networks: from basics to tangent space – Lecture notes

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
ML ML.1
ML.2
Machine learning 1. Neural networks
2. Supervised learning with tensor networks
L27 19.07.23 pdf
CanF CanF.1
CanF.2
CanF.3
CanF.4
Isometric PEPS, 2D Canonical Forms 1. Isometric PEPS: Moses move
2. Isometric PEPS: Applications
3. Canonical form for bond in 2D tensor network
4. Full environment truncation
L26 13.07.23 pdf
TNR
 TNR.1
TNR.2
TNR.3
TNR.4
TNR.5
TNR.6
TNR: Tensor network renormalization, MERA
 1. Motivation
2. TNR idea
3. Projective truncation
4. TNR details
5. TNR results in MERA
6. TNR benchmark results
L25 12.07.23 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)
L24 06.07.23 pdf
TRG TRG-I.1
TRG-I.2
TRG-I.3
TRG-I.4-7
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)
L23 05.07.23 pdf
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)
T07a 04.07.23 Tutorial: NRG II: Finite-size spectra, static correlations
L22 29.06.23 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
L21 28.06.23 pdf
TS-III TS-III.1
TS-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.
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-II TS-II.1
TS-II.2
TS-II.3
TS-II.4
Tangent space methods II: TDVP, energy variance 1. 1-site TDVP
2. 2-site projectors
3. 2-site TDVP
4. Energy variance.
L19 20.06.23 pdf
TS-I TS-I.1
TS-I.2
TS-I.3
TS-I.4
TS-I.5
Tangent space methods I: projector formalism 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
L18 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
L17 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
T05b 13.06.23 Tutorial: DMRG II: 2-site ground state search (cont.)
L16 08.06.23 pdf

Sym-III Sym-III.1
Sym-III.2
Sym-III.3
Sym-III.4
Symmetries III: Outer multiplicity, X-symbols (optional; was omitted in 2023) 1. Motivation
2. Outer multiplicity
3. Arrow inversion
4. Pairwise contractions and X-symbols
L15
 08.06.23 pdf
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 (optional) 1. Motivation, SU(2) basics
2. 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
L14
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
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.)
L13
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
30.05.23 Pentacost Tuesday (no lecture or tutorial)
L12 25.05.23 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
L11 24.05.23 pdf
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
T04a
 23.05.23 Tutorial: DMRG I: 1-site ground state search
L10 18.05.23 pdf
Sym-I Sym-I.1
Sym-I.1
Sym-I.1
Symmetries I: Abelian (optional) 1. Example: spin 1/2 XXZ-chain
2. Iterative diagonalization
3. QSpace bookkeeping for unit matrices
18.05.23 Ascension Day
L09
17.05.23 pdf
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
T03b
16.05.23

Tutorial: SVD, MPO, Diagonalization (cont.)
L08
 11.05.23 pdf
MPS-IV MPS.13
MPS.14
MPS.15
MPS.16
MPS.13-16: Matrix product operators
 13. Applying MPO to MPS yields MPS
14. MPO representation of Heisenberg Hamiltonian
15. Applying MPO to mixed-canonical state
16. MPS representation of Fermi sea
L07
 10.05.23 pdf
MPS-III MPS.9
MPS.10
MPS.11
MPS.12
MPS.9-12 Translationally invariant MPS, AKLT 9. Transfer matrix
10. AKLT model
11. AKLT ground state
12. Transfer operator and string order parameter
T03a
09.05.23

Tutorial: SVD, MPO, Diagonalization
L06
 04.05.23 pdf
MPS.6-8 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.
L05
 03.05.23 pdf
MPS.3-5 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.)
L04
27.04.23 pdf
TNB-III TNB-III.4
MPS.1-2 MPS.1
MPS.2
Tensor Network basics III (cont.)
4. Schmidt decomposition
MPS.1-2 Basics
 1. Reshaping generic tensor into MPS form.
2. Overlaps, matrix elements.
L03
 26.04.23
 pdf
TNB-III TNB-III.1
TNB-III.2
TNB-III.3
TNB-III.4
Tensor Network basics III 1. Reshaping generic tensor into MPS form
2. Unitaries and isometries
3. Singular value decomposition (SVD)
4. Schmidt decomposition
T02a
25.04.23

Tutorial: Tensor network basics
T01 20.04.23 Tutorial: MATLAB 101
L02
 19.04.23 pdf
TNB-II TNB-II.1
TNB-II.2
TNB-II.3
Tensor Network basics II 1. Covariant index notation
2. Arrow conventions
3. Iterative diagonalization
L01
 18.04.23 pdf
TNB-I TNB-I.1
TNB-I.2
TNB-I.3
Tensor Network basics (TNB) I 1. Notation for generic quantum lattice systems.
2. Entanglement entropy and area laws
3. Tensor network diagrams