Improvement of the higher-order tensor renormalization group method Satoshi Morita (ISSP , Univ. Tokyo)
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Entanglement filtering in HOTRG ➢ HOTRG + Full Environment Truncation (FET)* ➢ Benchmark on 2d Ising model* ○ Summary * Unpublished results are removed in this PDF.
Tensor network methods ○ Hamiltonian mechanics ➢ Wave function of many-body systems 𝑃 𝑒 𝑂 coefficients Approx. by tensor decomp. ○ Lagrangian mechanics ➢ Partition function (Action) Representation by tensor decomp. 𝑃 𝑒 𝑂 terms Tensor network representations reduce exponential computational cost to polynomial order.
TN representation of the partition function Sum over states Tensor contraction 𝑇 4 𝑇 3 𝑇 4 𝑇 3 𝐵 𝑇 2 𝑇 1 𝑇 2 𝑇 1 Ising model 𝐵 𝑇 1 𝑇 2 𝑇 3 𝑇 4 = 𝑓 𝐿 𝑇 1 𝑇 2 +𝑇 2 𝑇 3 +𝑇 3 𝑇 4 +𝑇 4 𝑇 1 • Tensor index corresponds to spin direction. • One tensor contains two spin. • In higher dimensional systems, a tensor has many indices (= 2 𝑒 ).
5 TN representation of the partition function 𝑌 Sum over states Tensor cont. 𝑊 𝑊 = 𝑈 𝑌 † 𝜀 𝑌 𝑈 𝑈 𝑋 𝑌 † 𝑋 𝑊 𝑈 𝑈 ED 𝑉 † 𝑉 𝑋 Λ 𝜀 𝜀 = Local Boltzmann factors = 𝑌 † 𝑌 = 𝑉 Λ 𝑧 𝜏 Bond index is index of eigenvalue. • 𝜀 𝑦′ 𝑦 Spin is already traced out. • Kronecker’s delta One tensor contains one spin. • 𝜏 𝑧′ 𝜀 𝑦𝑧𝑦 ′ 𝑧 ′ = 𝜀 𝜏𝑦 𝜀 𝜏𝑧 𝜀 𝜏𝑦 ′ 𝜀 𝜏𝑧 ′ # of indices is 2𝑒 . •
Real-space renormalization group ○ TRG ○ HOTRG Tensor Renormalization Group Higher-order Tensor Renormalization Group Levin, Nave (2007) Xie, et al. (2012) https://github.com/smorita/TN_animation
Real-space renormalization ○ TRG (Tensor Renormalization Group) 𝐵 Levin, Nave (2007) Truncated SVD 𝑃 𝜓 5 Contraction 𝑃 𝜓 6 Decomposition & Contraction
TRG-base methods ○ TNR (Tensor Network ○ Loop-TNR Yang, Gu, Wen (2017) Renormalization) Evenbly, Vidal (2015) ○ 𝑃 𝜓 5 algorithms ○ TNR+ Bal, et al. (2017) ➢ TRG + Randomized SVD SM, Igarashi, Zhao, Kawashima (2018) ➢ Projectively Truncated TRG Nakamura, Oba, Takeda (2019) Only for 2-d systems
HOTRG ○ Higher-order Tensor Renormalization Group Xie, et al. (2012)
Advantage of HOTRG ➢ Accuracy ➢ Higher-dimensional systems TRG HOTRG 2D Ising 𝜓 = 24 𝑃 𝜓 11 ➢ Conservation of lattice structure ➢ No tensor decomposition ✓ 3-state Potts model on cubic lattice Wang, et al., (2014) [arXiv:1405.1179]
11 Key parts of HOTRG ○ Renormalization 𝐵 𝑢 𝑉 † 𝑉 𝐵 𝑢+1 = CPU cost: 𝑃 𝜓 7 Memory: 𝑃(𝜓 4 ) 𝐵 𝑢 ○ HOSVD (Higher-Order Singular Value Decomposition) ➢ truncated Tucker decomposition 𝑉′ 𝑉 HOSVD 𝑇 =
12 Key parts of HOTRG ○ Renormalization 𝐵 𝑢 𝑉 † 𝑉 𝐵 𝑢+1 = CPU cost: 𝑃 𝜓 7 Memory: 𝑃(𝜓 4 ) 𝐵 𝑢 ○ HOSVD (Higher-Order Singular Value Decomposition) ➢ truncated Tucker decomposition † 𝑊 † 𝐵 𝑢 𝐵 𝑢 𝑉 Σ CPU cost: 𝑃 𝜓 6 ≈ 𝜓 𝜓 truncated SVD † 𝐵 𝑢 𝐵 𝑢
13 Optimal Projector for 2x2 Cluster ○ Problem 2 𝑄 𝑀 𝑄 𝑆 𝑈 𝑈 3 1 − Δ = min 𝑄, 𝑅 𝑈 2 𝑈 4 Lower bound † 𝑉 𝑢 𝑊 𝑢 Σ 𝑢 ≈ = 2 Δ ≥ 𝜏 𝑗 Truncated 𝑗>𝜓 SVD How do we obtain 𝑄 𝑀 and 𝑄 𝑆 satisfying the follow conditions? † 𝑊 𝑉 𝑢 𝑄 𝑆 𝑄 𝑀 𝑢 Σ t Σ t = =
14 Algorithm based on QR decomposition Wang, Verstraete, arXiv:1110.4362, Corboz, Rice, Troyer, PRL 113 , 046402 (2014) 𝑅 𝑀 𝑆 𝑆 𝑅 𝑆 𝑆 𝑀 = = 𝐵 𝑀 𝐵 𝑆 QR RQ ෩ † ෨ 𝑆 𝑀 𝑆 𝑆 𝑉 𝑢 𝑊 𝑢 Σ 𝑢 ≈ truncated SVD 𝑄 𝑀 𝑆 𝑆 − Τ 1 2 𝑄 𝑆 † 𝑆 𝑀 ෨ ෩ 𝑊 Σ 𝑢 𝑉 𝑢 𝑢 − Τ 1 2 Σ 𝑢 = = 𝑄 𝑀 𝑄 𝑆 is an “oblique” projector
15 Derivation of “Oblique” Projector − Τ 1 2 𝐵 𝑀 𝑄 𝑀 = 𝑉 𝑢 Σ 𝑢 Σ𝑊 † 𝑊Σ 𝑢 × ≡ 𝐵 𝑀 𝐵 𝑆 = 𝑉Σ𝑊 † 𝑊 − Τ 1 2 𝑢 Σ 𝑢 𝐵 𝑀 𝐵 𝑆 = 𝑉 Σ 𝑊 † ≈ 𝑉 𝑢 Σ 𝑢 𝑊 † − Τ 1 2 = 𝐵 𝑀 𝐵 𝑆 𝑊 𝑢 Σ 𝑢 𝑢 SVD truncation − Τ 1 2 = 𝐵 𝑀 𝑆 𝑆 𝑅 𝑆 𝑊 𝑢 Σ 𝑢 𝐵 𝑆 = 𝑆 𝑆 𝑅 𝑆 𝐵 𝑀 = 𝑅 𝑀 𝑆 𝑀 QR decomp. RQ decomp. − Τ 1 2 = 𝐵 𝑀 𝑆 𝑆 ෨ 𝑊 𝑢 Σ 𝑢 † 𝑊 † = ෩ 𝑆 𝑀 𝑆 𝑆 = ෩ 𝑉 Σ ෨ 𝑉 𝑢 Σ 𝑢 ෨ 𝑊 𝑄 𝑀 𝑢 truncation SVD In the same way, we obtain 1 2 ෩ − Τ Comparing two SVDs, we obtain † 𝑆 𝑀 𝑄 𝑆 = Σ 𝑢 𝑉 𝑢 † = ෨ 𝑉 𝑢 = 𝑅 𝑀 ෩ † 𝑅 𝑆 𝑉 𝑢 𝑊 𝑊 𝑢 𝑢 † = 𝑉 𝑢 † 𝑅 𝑀 ෨ We can easily prove 𝑄 𝑀 𝑄 𝑆 = 𝑄 𝑀 𝑄 𝑆 2 . ෩ 𝑊 𝑢 = 𝑅 𝑆 𝑊 𝑉 𝑢 𝑢
16 Computational Cost Wang, Verstraete, arXiv:1110.4362, Corboz, Rice, Troyer, PRL 113 , 046402 (2014) 𝑅 𝑀 𝑆 𝑆 𝑅 𝑆 𝑆 𝑀 Cost = = 𝑃 𝜓 8 𝐵 𝑀 𝐵 𝑆 QR RQ ෩ † ෨ 𝑆 𝑀 𝑆 𝑆 𝑉 𝑢 𝑊 𝑢 Σ 𝑢 ≈ 𝑃 𝜓 6 truncated SVD 𝑄 𝑀 𝑆 𝑆 − Τ 1 2 𝑄 𝑆 † 𝑆 𝑀 ෨ ෩ 𝑊 Σ 𝑢 𝑉 𝑢 𝑢 − Τ 1 2 Σ 𝑢 𝑃 𝜓 4 = = For HOTRG, 𝑃(𝜓 8 ) cost is unacceptable. We can avoid the QR decomp by using ED because 𝑅 𝑀 and 𝑅 𝑆 is unnecessary.
17 Modified Algorithm for “Oblique” Projector cf. Iino, SM, Kawashima, arXiv:1905.02351, to be published in PRB Use Λ 𝑉 instead of 𝑆 𝑀 𝑉 † 𝑉 𝑉 Λ 𝑆 𝑀 Λ Cost = ∼ † 𝐵 𝑀 𝐵 𝑀 ED 𝑃 𝜓 8 ➢ Benchmark on random tensors HOSVD HOSVD Oblique Oblique Modified Modified
18 Boundary Tensor Renormalization Group Iino, SM, Kawashima, arXiv:1905.02351, to be published in PRB Iino’s talk ○ HOTRG with open boundaries July 22, 15:30- Scaling dimension of boundary CFT Ising model 𝑈 = 𝑈 𝑑 𝜓 = 72 Index
19 1 st Part: Higher-order moments by Higher-order Tensor Renormalization Group
Binder ratio K. Binder: Z. Phys. B 43 , 119 (1981) Magnetization 2D Ising MC (Ising model) ✓ Dimensionless quantity ✓ Step function in 𝑂 →∞ 2D Ising ✓ Crossing point → 𝑈 MC 𝑑 𝜉 = 1 Finite-size scaling analysis 𝑢𝑀 Τ 1 𝜉
2D Ising model, 𝑈 = 𝑈 Order parameter 𝒏 𝑑 1. Derivative of free energy ➢ Error from numerical differential approximation. ➢ External field breaks symmetry of tensor. 2. Impurity tensor Tr 𝑇 𝑗 𝑓 −𝛾𝐼 = tTr Multi-point correlations are necessary for high-order moments 𝑛 𝑜 .
Multipoint correlation functions ➢ 1 st -order moment “the average of the local operators” 𝑇 1 𝑇 2 𝑇 3 1 𝐵 𝑢 1 ≃ + + + … 𝑂 𝑂 terns ➢ 2 nd -order moment 𝑇 1 𝑇 1 𝑇 1 𝑇 2 𝑇 1 𝑇 3 2 𝐵 𝑢 1 + … ≃ + + 𝑂 2 𝑂 2 terns We calculate the renormalized tensor of the summation of multipoint correlation functions by using HOTRG.
Renormalization of multi-impurity tensors
Renormalization of multi-impurity tensors • Use the same isometry 𝑉 (𝑢) for the local tensor 𝑈 (𝑢) • Generalization for multiple kinds of impurities
26 2D 𝑟 -state classical Potts model We consider ℎ = 0 . ○ Phase transition at ➢ 𝑟 ≤ 4 : 2 nd -order ➢ 𝑟 > 4 : 1 st -order ➢ Correlation length Weakly 1 st -order
27 TN representation ○ Local tensor Eigen value of local Boltzmann factor 𝑎 𝑟 symmetry 𝑦 + 𝑧 − 𝑦 ′ − 𝑧 ′ ≡ 0 mod 𝑟 ○ Order parameter ➢ Complex magnetization ➢ Impurity tensor for 𝑛 𝑙 𝑛 ∗𝑙 ′ Covariant with spin rotation （ charge 𝑙 − 𝑙 ′ ） 𝑦 + 𝑧 − 𝑦 ′ − 𝑧 ′ ≡ 𝑙 − 𝑙′ mod 𝑟
28 Magnetization 𝑈 (𝑢) ➢ 𝑟 -state Potts model Periodic BC 𝑂 = 2 𝑢 ➢ Ising model ( 𝑟 = 2 ) 𝑂 = 2 20 𝜓 = 32 𝑂 = 2 40 𝜓 = 48
𝜓 -dependence Binder ratio Ising model Τ 𝑈 𝑈 𝑑
Transition temperature 𝜓 -dependence Bisection search 𝑈 𝑈 Τ 𝑑 = 0.999951171 Τ 𝑈 𝑈 𝑑 = 1.000007324 𝑈 𝑈 Τ 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
31 Finite-size scaling analysis 4-state Potts model 5-state Potts model Τ 1 𝜉 = 2 Exact 𝛾 = 0 1 𝜉 = Τ 3 2 Τ Δ𝑛 2 Τ 𝛾 = 1 12 ✓ Fitting by Bayesian scaling ✓ No fitting (Harada, 2011) ✓ Plateau indicate the 1 st -order ✓ Effect of logarithmic correction phase transition. is small
Slope of Binder ratio
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