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TVI/TMP: Tensor Networks – Lecture notes

Course Outline
Lecture Notes:
Lecture Date Notes Lecture Pages Topic
26 26.07.17 pdf pdf
ML1.a-c
ML2.a-g
Machine learning
1. Neural networks
2. Supervised learning with tensor networks
25 24.07.17 pdf pdf

MERA1.a-g
MERA2.a-l
MERA3.a-e
Multiscale Entanglement Renormalization Ansatz (MERA)
1. Entanglement renormalization
2. Technial details
3. Tensor network renormalization (TNR)
24 19.07.17 pdf pdf

TRG1.a-d
TRG2.a-d
TRG3.a-j
Tensor renormalization group (TRG)
1. TRG for classical spin models
2. TRG for quantum ground states
3. Second TRG (SRG)
23 17.07.17 pdf pdf

PEPS9.a-e
PEPS10.a-k
PEPS V: corner transfer matrix (CTM) approach
9. Environmental gauge fixing
10. Fermionic PEPS
22 12.07.17 pdf pdf

PEPS8.a-w
PEPS IV: corner transfer matrix (CTM) approach
8. PEPS via CTM
21 10.07.17 pdf pdf

PEPS6.a-e
PEPS7.a-h
PEPS III: contractions via MPS techniques
6. PEPS via MPO-MPS
7. iPEPS via iMPO-iMPS (iTEBD)
20 05.07.17 pdf pdf

PEPS5.a-f
PEPS II: Tensor-entanglement renormalization group (TERG)
5. TERG
20 05.07.17 pdf pdf
PEPS3.a-h
PEPS4.a-c
PEPS I (continued): Examples
3. Kitaev's toric code (continued)
4. General remarks about PEPS
19 03.07.17 pdf pdf

PEPS1.a-fj
PEPS2.a-d
PEPS I: Projected entangled-pair states
1. Definition: PEPS
2. Example: RVB state
18 28.06.17 pdf pdf
Sym4.a-j
Sym5.a-d
Sym6.a-b
Non-Abelian symmetries:
4. SU(2) basics
5. Example: Heisenberg chain
6. QSpace bookkeeping for unit matrices
17 26.06.17 pdf pdf
Sym1.a-j
Sym2.a-d
Sym3.a-b
Abelian symmetries:
1. Example: spin 1/2 XXZ-chain
2. Iterative diagonalization
3. QSpace bookkeeping for unit matrices
16 21.06.17 pdf pdf

DMRG5.a-i
DMRG6.a-c
DMRG III: time-dependent DMRG:
5. Time evolution
6. Finite temperatures
15 19.06.17 pdf

iTEBD1.a-g
iTEBD2.a-c
iTEBD3.a-d
revised iTEBD: same material, revised, much clearer notation: o=odd, e=even instead of 1,2, A,B
15 19.06.17 pdf pdf

iTEBD1.a-g
iTEBD2.a-c

iTEBD3.a-d
iTEBD: Infinite Time-Evolving Block Decimation
1. Basic iTEBD algorithm
2. Hastings' method (no division by singular values)
3. Mixed-canonical form
14 14.06.17 pdf pdf

DMRG4.a-g
DMRG II: Relation to traditional DMRG
Infinite-size DMRG, finite-size DMRG, iDMRG in Gamma-Lambda notation
14 14.06.17 pdf pdf MPS8.a-f MPS Technology V: Vidal's Gamma-Lambda notation
13 12.06.17 pdf pdf

DMRG2.a-h
DMRG3.a-f
DMRG4.a-c
DMRG ground state search
(continued)

2. Lancos Method
3. DMRG: excited states
4. Two-site optimization
12 07.06.17 pdf pdf
DMRG1.a-d
DMRG ground state search
1. Single-site optimization
11 31.05.17 pdf pdf

MPS7.a-d
MPS Technology IV: Matrix product operators
Definition of MPO; applying MPO to MPS; need for truncation
10 29.05.17 pdf pdf
AKLTa-f
AKLT Model
Model, spin projectors, ground state
09 29.05.17 pdf pdf

MPS6.a-f
MPS Technology III: Translationally invariant MPS
6. Transfer operators, normalization, correlation functions
08 24.05.17 Tutorial (!!) on fdmNRG
08 22.05.17 pdf pdf

NRG9.a-m
Numerical Renormalization group (NRG) IV:
9. Spectral functions (fdmNRG)
07 17.05.17 pdf pdf

NRG7.a-l
NRG8.a-d
Numerical Renormalization group (NRG) III:
7. RG flow of Kondo model, SIAM
8. Thermodynamic observables
06 15.05.17 pdf pdf

NRG5.a-j
NRG6.a-f
Numerical Renormalization group (NRG) II:
5. Kondo Model
6. Basic RG ideas
05 10.05.17 pdf pdf

NRG1.a-d
NRG2.a-c
NRG3.a-b
NRG4.a-3
Numerical Renormalization group (NRG) I:
1. Single-impurity Anderson model
2. Logarithmic discretization
3. Wilson chain
4. Iterative diagonalization
04 08.05.17 pdf pdf
MPS4.a-g

MPS5.a-m
MPS Technology II:
4. Iterative diagonalization of short spin chain
5. Fermionic chain, Jordan-Wigner transformation
03 03.05.17 Starting from lecture 3 onward, I will use my laptop and beamer, writing on the screen instead of the blackboard, in order to record the lecture (in response to a request by some students). What I write will be a condensed version of the lecture notes posted below. You may consider printing them out ahead of time, then taking notes will be much simpler.
03 03.05.17 pdf pdf
MPS1.a-e
MPS2.a-j
MPS3.a-3
MPS Technology I:
1. Overlaps and normalization
2. Various canonical MPS forms
3. Matrix elements, expectation values
02 02.05.17 17:40 For the material presented in lecture 02 on 26.04.17, I relabelled the section numbers and made some additional minor changes. Please download it again and discard the previous version.
02 26.04.17 pdf --
TNB5.a-e
TNB6.a-d
TNB7.a-g
TNB8.a-d
TNB9.a-c
Tensor Network basics (TNB) II:
5. Singular-value decomposition (SVD)
6. Schmidt decomposition
7. Left-canonical MPS
8. Right-canonical MPS
01 24.04.17 pdf --
TNB1.a-c
TNB2.a-d
TNB3.a-c
TNB4.a-e
Tensor Network basics (TNB) I:
1. Why tensor networks?
2. Iterative diagonalization
3. Entanglement entropy and area laws
4. Tensor network diagrams