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Tensor Networks – Lecture notes

Course Outline
Lecture Notes:
Lecture Date Notes Pages Topic
L23 24.07.19 pdf
ML.1
ML.2
Machine learning
1. Neural networks
2. Supervised learning with tensor networks
L22 22.07.19 pdf
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
L21 17.07.19 pdf
CanF.1

CanF.2
CanF.3
2D Canonical Forms, Isometric PEPS
1. Canonical form for bond in 2D tensor network
2. Full environment truncation
3. Isometric PEPS
L20 15.07.19 pdf
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
T12 11.07.19 Tutorial: TERG, TRG
L19 10.07.19 pdf
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
L18 08.07.19 pdf
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)
T11 04.07.19 Tutorial: PEPS I - exact contraction on a strip
L17 03.07.19 pdf
PEPS-II.1
PEPS-II.2
PEPS II: contractions via MPS techniques
1. PEPS via finite-size MPS
2. Infinite-size PEPS (iPEPS)
L16 01.07.19 pdf
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
T10 27.06.19 Tutorial: Symmetries & QSpace (continued)
T09 26.06.19 Tutorial: Symmetries & QSpace
L15 24.06.19 pdf
Sym-II.1
Sym-II.2
Sym-II.3
Sym-II.4
Sym-II.5
Sym-II.6
Symmetries II: Non-Abelian.
1. Motivation, SU(2) basics
2. Tensor product decomposition
3. Tensor operators
4. Example: direct product of two spin 1/2's
5. Example: direct product of three spin 1/2's
6. Bookkeeping for unit matrices
20.06.19 Corpus Christi
L14 19.06.19 pdf
Sym-I.1
Sym-I.1
Sym-I.1
Symmetries I: Abelian
1. Example: spin 1/2 XXZ-chain
2. Iterative diagonalization
3. QSpace bookkeeping for unit matrices
T08 17.06.19 Tutorial: tDMRG, purification, tangent space methods
L13 13.06.19 pdf
TS.1
TS.2
TS.3
Tangent space methods
1. MPS and canonical forms.
2. Tangent space.
3. Tangent space projector.
4. Time evolution.
L12 12.06.19 pdf
DMRG-II.1
DMRG-II.2
DMRG-II.3
DMRG II
1. Relation to traditional DMRG.
2. tDMRG.
3. Finite temperature: purification.
10.06.19 Pentecost Monday
T07 06.06.19 Tutorial: iTEBD
L11 05.06.19 pdf
pdf
MPS-V.1


iTEBD.1
iTEBD.2
iTEBD.3
iTEBD.4
MPS V: Vidal's Gamma-Lambda notation
iTEBD: Infinite Time-Evolving Block Decimation
1. Basic iTEBD algorithm
2. iTEBD in Gamma-Lambda notation
3. iTEBD: Hastings' method
4. Orthogonalization
T06 03.06.19 Tutorial: DMRG
30.05.19 Ascension Day
L10 29.06.19 pdf

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 optimization
L09 27.05.19 pdf
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
T05 23.05.19 Tutorial: AKLT Model
L08 22.05.19 pdf
AKLT.1
AKLT.2
AKLT.3
AKLT.4
AKLT.5
AKLT Model
1. General remarks
2. Construction of AKLT Hamiltonian
3. AKLT ground state
4. Transfer operator
5. String order parameter
L07 20.05.19 pdf
MPS-III.1
MPS-III.2
MPS-III.3
MPS III: Translationally invariant MPS
1. Transfer matrix
2. Eigenvalues of transfer matrix
3. Correlation functions
T04 16.05.19 Tutorial: NRG II/III
L06 20.05.19 6:15 The notes for the entire lecture 06 have been thoroughly revised. Please discard previous version and use this one.
L06 15.05.19 pdf

NRG-III.1
NRG-III.2
NRG-III.3
NRG-III.4
NRG-III.5
NRG-III.6
NRG-III.7
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
6. Complete many-body basis
7. Full density matrix NRG (fdmNRG)
L05 13.05.19 pdf

NRG-II.1
NRG-II.2
NRG-II.3
NRG-II.4
NRG-II.5
NRG-II.6
NRG-II.7
Numerical Renormalization group (NRG) II: RG flow
1. Kondo model: low-order perturbation theory
2. Kondo model: poor man's scaling
3. General RG concepts
4. NRG iteration scheme from RG perspective
5. Uncoupled bath Hamiltonian: fixed points
6. Kondo model: fixed points and RG flow
7. Anderson model: fixed points and RG flow
T03 09.05.19 Tutorial: NRG I
L04 08.05.19 pdf

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
L03 06.05.19 pdf
MPS-II.1
MPS-II.2
MPS-II.3
MPS II: Diagonalization, fermionic signs
1. Iterative diagonalization of short spin chain
2. Spinless fermions
3. Spinful fermions
T02 02.05.19 Tutorial: MPS I
L02 02.05.19 9:10 In Section 3, the discussion of operator matrix elements has been revised. In section 4, handwritten text has been converted to typed text.
01.05.19 Labor day
L02 29.04.19 pdf
MPS-I.1
MPS-I.2
MPS-I.3
MPS-I.4
MPS 1: Matrix Product States
1. Overlap and normalization.
2. Canonical MPS forms (left, right, site, bond)
3. Matrix elements, expectation values
4. Schmidt decomposition
L01 30.04.19 12:38 In Section 3, Eqs. (12), (27) and (28), the order of the lower indices of A-dagger has been reversed.
L01 23.04.19 23:27 In Sections 1 to 4, important notational changes were made regarding the definition of tensor product spaces.
T01 25.04.19 Tutorial: MATLAB basics
L01 24.04.19 pdf
TNB1
TNB2
TNB3
TNB4
TNB5
TNB6
Tensor Network basics (TNB):
1. Why study tensor networks?
2. Iterative diagonalization
3. Covariant index notation
4. Entanglement entropy and area laws
5. Tensor network diagrams
6. Singular-value decomposition (SVD)
22.04.19 Easter Monday