SLIDE 1
Satoshi Morita (ISSP , Univ. Tokyo)
Improvement of the higher-order tensor renormalization group method
SLIDE 2 Outline
○ Introduction
➢ Real-space renormalization based on TN ➢ How to calculate a projector in HOTRG
○ Calculation of higher-order moments by HOTRG
➢ Renormalization of multi-impurity tensors ➢ Finite-scaling analysis on q-state Potts model
○ Entanglement filtering in HOTRG
➢ HOTRG + Full Environment Truncation (FET)* ➢ Benchmark on 2d Ising model*
○ Summary
2
* Unpublished results are removed in this PDF.
SLIDE 3 Tensor network methods
○ Hamiltonian mechanics
➢ Wave function of many-body systems
○ Lagrangian mechanics
➢ Partition function (Action)
𝑃 𝑒𝑂 coefficients 𝑃 𝑒𝑂 terms Tensor network representations reduce exponential computational cost to polynomial order.
- Approx. by tensor decomp.
Representation by tensor decomp.
SLIDE 4 TN representation of the partition function
Sum over states Tensor contraction
- Tensor index corresponds to spin direction.
- One tensor contains two spin.
- In higher dimensional systems, a tensor has many indices (=2𝑒).
𝑇1 𝑇2 𝑇4 𝑇3 𝑇1 𝑇2 𝑇4 𝑇3
𝐵𝑇1𝑇2𝑇3𝑇4 = 𝑓𝐿 𝑇1𝑇2+𝑇2𝑇3+𝑇3𝑇4+𝑇4𝑇1
Ising model 𝐵
SLIDE 5 TN representation of the partition function
5
Sum over states Tensor cont. Local Boltzmann factors
- Bond index is index of eigenvalue.
- Spin is already traced out.
- One tensor contains one spin.
- # of indices is 2𝑒.
𝜀 𝑦 𝑧′ 𝑧 𝑦′ 𝜏 𝜀𝑦𝑧𝑦′𝑧′
𝜏
= 𝜀𝜏𝑦𝜀𝜏𝑧𝜀𝜏𝑦′𝜀𝜏𝑧′ 𝑋 𝑋 𝑊 𝑊 𝜀 𝜀 𝑋
=
Λ 𝑉 𝑉† ED
=
𝑌 = 𝑉 Λ 𝑌† 𝑌† 𝑌 𝑌 𝜀 𝑊 𝑌† 𝑈
=
Kronecker’s delta
𝑈 𝑈 𝑈 𝑈
SLIDE 6 Real-space renormalization group
○ TRG
Tensor Renormalization Group
○ HOTRG
Higher-order Tensor Renormalization Group
Levin, Nave (2007) Xie, et al. (2012)
https://github.com/smorita/TN_animation
SLIDE 7 Real-space renormalization
○ TRG (Tensor Renormalization Group)
Levin, Nave (2007)
Decomposition & Contraction 𝑃 𝜓6 𝐵 Contraction Truncated SVD 𝑃 𝜓5
SLIDE 8 TRG-base methods
○ TNR (Tensor Network
Renormalization)
○ Loop-TNR ○ TNR+
Bal, et al. (2017) Evenbly, Vidal (2015) Yang, Gu, Wen (2017)
○ 𝑃 𝜓5 algorithms
➢ TRG + Randomized SVD
SM, Igarashi, Zhao, Kawashima (2018)
➢ Projectively Truncated TRG
Nakamura, Oba, Takeda (2019)
Only for 2-d systems
SLIDE 9 HOTRG
○ Higher-order Tensor Renormalization Group
Xie, et al. (2012)
SLIDE 10 Advantage of HOTRG
TRG HOTRG 2D Ising 𝜓 = 24
➢Higher-dimensional systems
𝑃 𝜓11
➢Accuracy
➢ Conservation of lattice structure ➢ No tensor decomposition
✓ 3-state Potts model on cubic lattice
Wang, et al., (2014) [arXiv:1405.1179]
SLIDE 11 Key parts of HOTRG
○ Renormalization
11
○ HOSVD (Higher-Order Singular Value Decomposition)
➢ truncated Tucker decomposition 𝑉 𝑉† 𝐵𝑢 𝐵𝑢 𝐵𝑢+1 =
=
CPU cost: 𝑃 𝜓7 Memory: 𝑃(𝜓4)
𝑉 𝑉′
𝑇
HOSVD
SLIDE 12 Key parts of HOTRG
○ Renormalization
12
○ HOSVD (Higher-Order Singular Value Decomposition)
➢ truncated Tucker decomposition 𝑉 𝑉† 𝐵𝑢 𝐵𝑢 𝐵𝑢+1 =
≈
𝑉 𝑊† Σ
truncated SVD
𝐵𝑢
CPU cost: 𝑃 𝜓7 Memory: 𝑃(𝜓4)
𝐵𝑢 𝐵𝑢
†
𝐵𝑢
† CPU cost: 𝑃 𝜓6 𝜓 𝜓
SLIDE 13 Optimal Projector for 2x2 Cluster
○ Problem
13 𝑄𝑀 𝑄𝑆
−
Δ = min
𝑄, 𝑅 Lower bound
=
𝑉𝑢 𝑊
𝑢 †
Σ𝑢
≈
Truncated SVD Δ ≥
𝑗>𝜓
𝜏𝑗
2
2
𝑄𝑀 𝑄𝑆
𝑉𝑢 Σt 𝑊
𝑢 †
Σt
= =
How do we obtain 𝑄𝑀 and 𝑄𝑆 satisfying the follow conditions? 𝑈
1
𝑈2 𝑈3 𝑈
4
SLIDE 14 Algorithm based on QR decomposition
14
𝐵𝑀
=
𝑅𝑀 𝑆𝑀 𝐵𝑆
=
𝑅𝑆 𝑆𝑆 𝑆𝑀 𝑆𝑆
≈
෩ 𝑉𝑢 ෨ 𝑊
𝑢 †
Σ𝑢 truncated SVD
𝑄𝑀
𝑆𝑆 ෨ 𝑊
𝑢
Σ𝑢
− Τ 1 2
= 𝑄𝑆 =
𝑆𝑀 Σ𝑢
− Τ 1 2
෩ 𝑉𝑢
†
Wang, Verstraete, arXiv:1110.4362, Corboz, Rice, Troyer, PRL 113, 046402 (2014)
QR RQ
𝑄𝑀𝑄𝑆 is an “oblique” projector
SLIDE 15 Derivation of “Oblique” Projector
15 ≡
𝐵𝑀 𝐵𝑆
𝐵𝑀𝐵𝑆 = 𝑉 Σ 𝑊† ≈ 𝑉𝑢 Σ𝑢 𝑊
𝑢 † SVD truncation
𝐵𝑀𝑄𝑀 = 𝑉𝑢 Σ𝑢 𝐵𝑀 = 𝑅𝑀𝑆𝑀 𝐵𝑆 = 𝑆𝑆𝑅𝑆
QR decomp. RQ decomp.
𝑆𝑀𝑆𝑆 = ෩ 𝑉 Σ ෨ 𝑊† = ෩ 𝑉𝑢 Σ𝑢 ෨ 𝑊
𝑢 † SVD truncation
= 𝑉Σ𝑊† 𝑊
𝑢Σ𝑢 − Τ 1 2
= 𝐵𝑀𝐵𝑆 𝑊
𝑢Σ𝑢 − Τ 1 2
= 𝐵𝑀 𝑆𝑆𝑅𝑆 𝑊
𝑢Σ𝑢 − Τ 1 2
= 𝐵𝑀 𝑆𝑆 ෨ 𝑊
𝑢Σ𝑢 − Τ 1 2
𝑉𝑢 = 𝑅𝑀 ෩ 𝑉𝑢 𝑊
𝑢 † = ෨
𝑊
𝑢 †𝑅𝑆
෨ 𝑊
𝑢 = 𝑅𝑆𝑊 𝑢
𝑄𝑀 𝑄𝑆 = Σ𝑢
− Τ 1 2 ෩
𝑉𝑢
†𝑆𝑀
෩ 𝑉𝑢
† = 𝑉𝑢 †𝑅𝑀 In the same way, we obtain We can easily prove 𝑄𝑀𝑄
𝑆 = 𝑄𝑀𝑄𝑆 2.
Comparing two SVDs, we obtain × Σ𝑊†𝑊Σ𝑢
− Τ 1 2
SLIDE 16 Computational Cost
16
𝐵𝑀
=
𝑅𝑀 𝑆𝑀 𝐵𝑆
=
𝑃 𝜓8 𝑅𝑆 𝑆𝑆 𝑆𝑀 𝑆𝑆
≈
෩ 𝑉𝑢 ෨ 𝑊
𝑢 †
Σ𝑢 truncated SVD 𝑃 𝜓6 Cost
𝑄𝑀
𝑆𝑆 ෨ 𝑊
𝑢
Σ𝑢
− Τ 1 2
= 𝑄𝑆 =
𝑆𝑀 Σ𝑢
− Τ 1 2
෩ 𝑉𝑢
†
𝑃 𝜓4
Wang, Verstraete, arXiv:1110.4362, Corboz, Rice, Troyer, PRL 113, 046402 (2014)
For HOTRG, 𝑃(𝜓8) cost is unacceptable. We can avoid the QR decomp by using ED because 𝑅𝑀 and 𝑅𝑆 is unnecessary. QR RQ
SLIDE 17 Modified Algorithm for “Oblique” Projector
17
𝑃 𝜓8 Cost
- cf. Iino, SM, Kawashima, arXiv:1905.02351, to be published in PRB
𝐵𝑀
†
𝐵𝑀 𝑉 𝑉† Λ
=
ED 𝑆𝑀
∼
𝑉 Λ Use Λ 𝑉 instead of 𝑆𝑀
HOSVD Oblique Modified HOSVD Oblique Modified
➢ Benchmark on random tensors
SLIDE 18 Boundary Tensor Renormalization Group
○ HOTRG with open boundaries
18
Iino, SM, Kawashima, arXiv:1905.02351, to be published in PRB
Index
Scaling dimension of boundary CFT
Ising model 𝑈 = 𝑈
𝑑
𝜓 = 72
Iino’s talk
July 22, 15:30-
SLIDE 19
19
1st Part:
Higher-order moments by Higher-order Tensor Renormalization Group
SLIDE 20 Finite-size scaling analysis
Binder ratio
- K. Binder: Z. Phys. B 43, 119 (1981)
✓ Dimensionless quantity ✓ Step function in 𝑂→∞ ✓ Crossing point → 𝑈
𝑑
(Ising model) 2D Ising MC 𝑢𝑀 Τ
1 𝜉
𝜉 = 1
Magnetization 2D Ising MC
SLIDE 21 Order parameter 𝒏
- 1. Derivative of free energy
➢ Error from numerical differential approximation. ➢ External field breaks symmetry of tensor.
Tr 𝑇𝑗 𝑓−𝛾𝐼 = tTr Multi-point correlations are necessary for high-order moments 𝑛𝑜 .
2D Ising model, 𝑈 = 𝑈
𝑑
SLIDE 22 Multipoint correlation functions
+ + + … + + + …
1 𝑂 1 𝑂2
𝑇1𝑇1 𝑇1𝑇2 𝑇1𝑇3 𝑇1 𝑇2 𝑇3 𝑂 terns 𝑂2 terns
We calculate the renormalized tensor of the summation of multipoint correlation functions by using HOTRG.
➢ 1st-order moment ➢ 2nd-order moment
𝐵𝑢
1
𝐵𝑢
2
≃ ≃
“the average of the local operators”
SLIDE 23
Renormalization of multi-impurity tensors
SLIDE 24
- Use the same isometry 𝑉(𝑢) for the local tensor 𝑈(𝑢)
- Generalization for multiple kinds of impurities
Renormalization of multi-impurity tensors
SLIDE 25 2D 𝑟-state classical Potts model
○ Phase transition at
➢ 𝑟 ≤ 4 : 2nd-order ➢ 𝑟 > 4 : 1st-order ➢ Correlation length 26
We consider ℎ = 0. Weakly 1st-order
SLIDE 26 TN representation
○ Local tensor
27
Eigen value of local Boltzmann factor
𝑎𝑟 symmetry
○ Order parameter
➢ Complex magnetization ➢ Impurity tensor for 𝑛𝑙𝑛∗𝑙′
Covariant with spin rotation （charge 𝑙 − 𝑙′）
𝑦 + 𝑧 − 𝑦′ − 𝑧′ ≡ 0 mod 𝑟 𝑦 + 𝑧 − 𝑦′ − 𝑧′ ≡ 𝑙 − 𝑙′ mod 𝑟
SLIDE 27 Magnetization
28
𝑂 = 240 𝜓 = 48 ➢ Ising model (𝑟 = 2) ➢ 𝑟-state Potts model
𝑈(𝑢) 𝑂 = 2𝑢 Periodic BC 𝑂 = 220 𝜓 = 32
SLIDE 28 𝜓-dependence
Ising model Τ 𝑈 𝑈
𝑑
Binder ratio
SLIDE 29 Transition temperature
Bisection search
Τ 𝑈 𝑈
𝑑 = 0.999951171
Τ 𝑈 𝑈
𝑑 = 1.000007324
Τ 𝑈 𝑈
𝑑 = 1.00000313922
Τ 𝑈 𝑈
𝑑 = 1.00000313960
- Estimated values converge to the exact value with oscillation
- Relative error seems to decay in proportion to 𝜓−3.5
𝜓-dependence
SLIDE 30 Finite-size scaling analysis
31
4-state Potts model 5-state Potts model Τ 1 𝜉 = 2 𝛾 = 0
Τ 1 𝜉 = Τ 3 2 𝛾 = Τ 1 12
Exact
✓ Fitting by Bayesian scaling (Harada, 2011) ✓ Effect of logarithmic correction is small ✓ No fitting ✓ Plateau indicate the 1st-order phase transition.
Δ𝑛 2
SLIDE 31
Slope of Binder ratio
SLIDE 32 Summary of the 1st part
○ Renormalization of tensors with multiple impurities
𝑂 = 240 𝐸 = 48 Magnetization of Potts model
SM and N. Kawashima, Comput. Phys. Comm 236, 65 (2019)
# of states
✓ Beyond the system size which the Monte-Carlo method can treat. ✓ Finite-size scaling analysis of the magnetization and the Binder ratio. ✓ Distinguish weakly first-order and continuous phase transitions
SLIDE 33 34
2nd Part:
HOTRG with entanglement filtering
Unpublished results are removed in this part.
SLIDE 34 Internal Correlations
➢ Irrelevant correlations with 𝑃(1) length scale ➢ Converge to a fictitious fixed-point tensor
- Corner Double Line structure
We need to remove internal correlations within a loop. “Entanglement Filtering”
SLIDE 35 Internal Correlations
➢ Irrelevant correlations with 𝑃(1) length scale ➢ Converge to a fictitious fixed-point tensor
- Corner Double Line structure
We need to remove internal correlations within a loop. “Entanglement Filtering”
Gu, Wen (2009)
SLIDE 36
TRG with Entanglement Filtering
○ TRG-base methods
➢ TEFT [Gu, Wen (2009)] ➢ TNR [Evenbly, Vidal (2015)] ➢ loop-TNR [Yang, Gu, Wen (2017)] ➢ TNR+ [Bal, et al, (2017)] Only for 2-d systems
SLIDE 37
Entanglement Filtering
○ TRG-base methods
➢ TEFT [Gu, Wen (2009)] ➢ TNR [Evenbly, Vidal (2015)] ➢ loop-TNR [Yang, Gu, Wen (2017)] ➢ TNR+ [Bal, et al, (2017)]
○ General methods
➢ Graph Independent Local Truncation [Hauru, Delcamp, Mizera (2017)] ➢ Tensor Network Skeletonization [Ying (2016)] ➢ Entanglement Branching [Harada (2018)] ➢ Full Environment Truncation [Evenbly (2018)] Only for 2-d systems We consider a combination with HOTRG and FET.
SLIDE 38 Full Environment Truncation (FET)
Optimize isometries 𝑣, 𝑤 and bond diagonal matrix 𝜏
𝐵𝐶 to minimize
the difference
Evenbly (2018)
✓ Simple criterion ✓ TRG + FET
TRG TRG+FET
𝜓 < 𝜓
SLIDE 39
54
Thank you for your attention!
