Anisotropic tensor renormalization group and BTRG The University - - PowerPoint PPT Presentation

anisotropic tensor renormalization group and btrg
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Anisotropic tensor renormalization group and BTRG The University - - PowerPoint PPT Presentation


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SLIDE 1

Anisotropic tensor renormalization group 
 and BTRG

The University of Tokyo, Tsuyoshi Okubo

  • Ref. D. Adachi, T. Okubo, and S. Todo, arXiv:1906.02007

C o m p u t a t i o n a l S c i e n c e A l l i a n c e

T h e U n i v e r s i t y o f T o k y o

東大マーク集 2

東大マーク 基本型 〈タテ〉

基本型〈タテ〉 東大マークには、 使用時の最小サイズが設定 されています。 本項で示された最小サイズ以 下で使用すると、 東大マークの再現性を著し く 損なう恐れがあり、 表示を正確に伝達するこ とができなく なります。 この最小使用サイズは 、 東大マークの印刷物における再生上の規定 です。 使用する媒体の特性やスペース等を十 分に検討し、 最適のサイズで使用してくださ い。 また、 印刷方式、 媒体の条件などによって もマークの再現性が異なることについても 注意が必要です。 最小サイズ

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SLIDE 2

Collaborators

Department of Physics, The University of Tokyo Daiki Adachi Synge Todo

東大マーク集 2

東大マーク 基本型 〈タテ〉

基本型〈タテ〉 東大マークには、 使用時の最小サイズが設定 されています。 本項で示された最小サイズ以 下で使用すると、 東大マークの再現性を著し く 損なう恐れがあり、 表示を正確に伝達するこ とができなく なります。 この最小使用サイズは 、 東大マークの印刷物における再生上の規定 です。 使用する媒体の特性やスペース等を十 分に検討し、 最適のサイズで使用してくださ い。 また、 印刷方式、 媒体の条件などによって もマークの再現性が異なることについても 注意が必要です。 最小サイズ

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SLIDE 3

Contents

  • Tensor renormalization group for high dimensions
  • Anisotropic tensor renormalization group: ATRG
  • Bond-weighted TRG: BTRG....
  • Summary
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SLIDE 4

Tensor network renormalization group

Purpose: approximate contraction of tensor network 
 by using "coarse-graining" of the network

Corse-graining (Renormalization)
 into √2times longer scale.

: D×D×D×D

Approximation

: D×D×D×D

L×L (L×L)/2

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SLIDE 5

Tensor Renormalization Group (TRG) algorithm

TRG M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007)

Low rank approximation by SVD

Computation cost: Memory:

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SLIDE 6

Higher Oder Tensor Renormalization Group (HOTRG)

Anisotropic coarse-graining by using HOSVD instead of SVD

  • Z. Y. Xie et al, Phys. Rev. B 86, 045139 (2012)

HOTRG algorithm:

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Isometry is defined through HOSVD of

Computation cost: O(D7) > O(D5)

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(TRG)

Better accuracy than TRG, although,

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SLIDE 7

Application to high dimensions

TRG: Not easy to generalize to high dimensions. HOTRG: Easy to generalize to high dimensions, but its cost is

We want to perform tensor network RG for high dimensions! However,

  • 3d classical systems
  • 2d and 3d quantum systems
  • Much higher dimensions...

Interests in O(D4d−1)

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d=3 : O(D11) d=4 : O(D15)

Is it possible to construct lower cost algorithm ?

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SLIDE 8

Anisotropic TRG = ATRG

  • D. Adachi, T. Okubo, and S. Todo, arXiv:1906.02007
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SLIDE 9

Central idea of Anisotropic TRG

In ATRG, we coarse-grain tensors anisotropically as HOTRG: In order to reduce the computation cost, we decompose the local tensor into small pieces before performing coarse-graining.

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HOTRG

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ATRG

O(D7)

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O(D5)

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SLIDE 10

Recipe of ATRG

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α

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α

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β

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x0

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x1

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x2

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x3

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A

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A

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B

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B

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C

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C

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D

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D

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A

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A

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D

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D

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'

<latexit sha1_base64="PRV4SUAJ4CufmvkxYgucWKcBoY0=">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</latexit>

SVD (truncation) T

A

<latexit sha1_base64="6fFfiOA5K65pfNgux7h1VXJdczM=">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</latexit>

U

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C

<latexit sha1_base64="6fFfiOA5K65pfNgux7h1VXJdczM=">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</latexit>

U

<latexit sha1_base64="b+9697juV/g7xBK5zdmQ0cjyzh0=">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</latexit>

D ≡

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V †

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λ

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V †

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λ

<latexit sha1_base64="rtX6k4yPzVB8sjfmDc5K05ya3M=">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</latexit>

B ≡

<latexit sha1_base64="6fFfiOA5K65pfNgux7h1VXJdczM=">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</latexit>

M

<latexit sha1_base64="6fFfiOA5K65pfNgux7h1VXJdczM=">ACaXichVFNLwNBGH6vq+Wi4NF7GpODWzJSEO0sTFkVZLQtPsrsGwX/ajSTX+gJOb4EQiIn6Giz/g4CeI4mLg7fbTYQG72Rmnmfd5ZkZzDOH5jD3FpI7Oru6eG+ir39gcCiZGi57duDqvKTbhu2ua6rHDWHxki98g687LldNzeBr2v5ic3+txl1P2NaqX3d4xVR3LEtdNUnqrzJDwJRqyZlmVhjLcDJQIyoli2kzfYxBZs6AhgsOCT9iACo/aBhQwOMRV0CDOJSTCfY4jJEgbUBanDJXYfRp3aLURsRatmzW9UK3TKQZ1l5TjyLBHdste2QO7Y8/s49dajbBG0udZq2l5U516DhdfP9XZdLsY/dL9adnH9uYC70K8u6ETPMWektfOzx9Lc4XMo1JdsVeyP8le2L3dAOr9qZfr/DCBRL0AcrP524H5VxWmc7mVmbk/EL0FXGMYQJT9N6zyGMJyjRuXs4wRnOYy9SkpLo61UKRZpRvAtJPkT06uMOA=</latexit>

C B

  • D. Adachi, T. Okubo, and S. Todo, arXiv:1906.02007
slide-11
SLIDE 11

Recipe of ATRG

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α

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β

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y

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y0

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y

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y

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y1

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y

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y

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y2

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y

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x

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x0

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x

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x1

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x

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x2

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x

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x3

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A

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A

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D

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A

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A

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D

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D

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M

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C B

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(swap by SVD)

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A

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A

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D

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D

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sha1_base64="hivCqyYbH51N4cP8kosyQiYXfU=">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</latexit>

D Y X A

'

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D Y X A

Inserte projector Different from HOTRG, this step can be done with O(D5) T'

<latexit sha1_base64="6fFfiOA5K65pfNgux7h1VXJdczM=">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</latexit>
  • D. Adachi, T. Okubo, and S. Todo, arXiv:1906.02007
slide-12
SLIDE 12

Summary of 2D ATRG

Memory storage: O(D3)

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(We need only 3-leg tensors!)

Computation time: O(D5)

<latexit sha1_base64="W9Px0LdcerYpmAhHELgDSV0gOcw=">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</latexit>

* We do not explicitly create 4-leg tensor. * By using partial SVD technique, such as the Arnoldi method, we can reduce SVD cost.

T'

<latexit sha1_base64="6fFfiOA5K65pfNgux7h1VXJdczM=">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</latexit>
  • D. Adachi, T. Okubo, and S. Todo, arXiv:1906.02007

We can perform HOTRG like anisotropic coarse-graining
 with smaller cost!

slide-13
SLIDE 13

Generalization to high dimension

  • D. Adachi, T. Okubo, and S. Todo, arXiv:1906.02007

(a)

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(b)

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(c)

<latexit sha1_base64="qgMZnqs8p3nIx8FcHrdpi7CALM=">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</latexit><latexit sha1_base64="qgMZnqs8p3nIx8FcHrdpi7CALM=">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</latexit><latexit sha1_base64="qgMZnqs8p3nIx8FcHrdpi7CALM=">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</latexit><latexit sha1_base64="qgMZnqs8p3nIx8FcHrdpi7CALM=">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</latexit>

(f)

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(d) , (e)

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We can easily generalize this to high dimensions Key points: Before coarse graining, 
 we decompose a local tensor
 into two tensors.

Memory storage: Computation time:

O(Dd+1)

<latexit sha1_base64="QMRxQFcJ4WcqaF+EPNRq7nDe4Bw=">ACbXicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKRflruMRVp2gb1mrGCygb6BmAgQImwxDKUGaAgoB8geUMQwpDPkMyQylDLkMqQx5DCVAdg5DIkMxEYzGDIYMBQAxWIZqoFiRUBWJlg+laGWgQuotxSoKhWoIhEomg0k04G8aKhoHpAPMrMYrDsZaEsOEBcBdSowqBpcNVhp8NnghMFqg5cGf3CaVQ02A+SWSiCdBNGbWhDP3yUR/J2grlwgXcKQgdCF180lDGkMFmC3ZgLdXgAWAfkiGaK/rGr652CrINVqNYNFBq+B7l9ocNPgMNAHeWVfkpcGpgbNZuACRoAhenBjMsKM9AyN9YwCTZQd7KBRwcEgzaDEoAEMb3MGBwYPhgCGUKC9eQxTGOYyzGN8zyTKJMkB1HKxAjVI8yApjUAVjljNQ=</latexit>

O(D2d+1)

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<latexit sha1_base64="sIcpimnfiBQBU6yJCAa4OAl6X5c=">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</latexit>

HOTRG

Memory: O(D2d)

<latexit sha1_base64="947pJFxqOm+SNulvO6WFBYnFLw=">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</latexit>

Time: O(D4d−1)

<latexit sha1_base64="Ct/hTEJtEb9yoClEC9yUGAanYfU=">ACbnicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKRftruMRVm6ToGtZqxgsoG+gZgIECJsMQylBmgIKAfIHlDEMKQz5DMkMpQy5DKkMeQwlQHYOQyJDMRBGMxgyGDAUAMViGaqBYkVAViZYPpWhloELqLcUqCoVqCIRKJoNJNOBvGioaB6QDzKzGKw7GWhLDhAXAXUqMKgaXDVYafDZ4ITBaoOXBn9wmlUNgPklkognQTRm1oQz98lEfydoK5cIF3CkIHQhdfNJQxpDBZgt2YC3V4AFgH5Ihmiv6xq+udgqyDVajWDRQavge5faHDT4DQB3lX5KXBqYGzWbgAkaAIXpwYzLCjPQMjfWMAk2UHSygUcHBIM2gxKABDG9zBgcGD4YAhlBwiE1lmMcwn/EDkxiTLJM8RCkTI1SPMAMKYNIAOsnjQ4=</latexit>

*We may consider 
 more decompositions.

slide-14
SLIDE 14

Benchmarks

2d square lattice Ising model at Tc

10-9 10-8 10-7 10-6 10-5 10 20 50 100

δ f

χ TRG ATRG HOTRG

At the same bond dimension, accuracy of ATRG is 
 between those of TRG and HOTRG.

HOTRG is better? No so straightforward!

D

<latexit sha1_base64="gtucZmzsqVUGxXRQ1mOyFbKRsEs=">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</latexit>
  • D. Adachi, T. Okubo, and S. Todo, arXiv:1906.02007

Relative error of the free energy

slide-15
SLIDE 15

Benchmarks

The computation costs are different between ATRG and HOTRG.

10-8 10-7 10-6 10-5 106 107 108 109 1010 1011 1012

δ f τ

TRG ATRG HOTRG

TRG, ATRG: HOTRG:

Leading order computation time: Relative error of the free energy

ATRG is the best!

(ATRG is ~102 faster than HOTRG!)

O(D5)

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O(D7)

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τ ≡ ( D5 TRG and ATRG D7 HOTRG

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slide-16
SLIDE 16

Benchmarks

3d square lattice Ising model at Tc

In order to obtain same free energy value, ATRG needs 
 ~103 times smaller computation cost than HOTRG.

  • D. Adachi, T. Okubo, and S. Todo, arXiv:1906.02007

free energy density

  • 3.5085
  • 3.508
  • 3.5075
  • 3.507
  • 3.5065
  • 3.506
  • 3.5055
  • 3.505
  • 3.5045
  • 3.504

107 108 109 1010101110121013101410151016 free energy density

τ

ATRG HOTRG

τ ≡ ( D7 ATRG D11 HOTRG

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By ATRG, we can investigate
 high dimensional systems much easier!

slide-17
SLIDE 17

Bond-weighted TRG = BTRG

  • D. Adachi, T. Okubo, and S. Todo, in preparation
slide-18
SLIDE 18

Why ATRG is better than TRG?

2d square lattice Ising model at Tc

10-9 10-8 10-7 10-6 10-5 10 20 50 100

δ f

χ TRG ATRG HOTRG

Relative error of the free energy D

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ATRG is better than TRG, although the cost is same. Why?

slide-19
SLIDE 19

Key point: weight of bonds

T

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T

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T

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T

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α

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β

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x0

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x1

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x2

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x3

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A

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A

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B

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B

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C

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C

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D

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D

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A

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A

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D

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D

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'

<latexit sha1_base64="PRV4SUAJ4CufmvkxYgucWKcBoY0=">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</latexit>

SVD (truncation) T

A

<latexit sha1_base64="6fFfiOA5K65pfNgux7h1VXJdczM=">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</latexit>

U

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C

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U

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D ≡

<latexit sha1_base64="6fFfiOA5K65pfNgux7h1VXJdczM=">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</latexit>

V †

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λ

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V †

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λ

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B ≡

<latexit sha1_base64="6fFfiOA5K65pfNgux7h1VXJdczM=">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</latexit>

In ATRG, we assign singular values to only one tensor. In TRG, usually it is divided to both tensors.

slide-20
SLIDE 20

Disadvantage of local SVD truncation

i j k l

Truncation by SVD

Local SVD is not necessarily the best "D-rank" approximation
 for the whole TN (the partition functions). We need to consider the "environment" for better truncations. This is the key idea of SRG (second renormalization)

  • Z. Y. Xie, et al Phys. Rev. Lett. 103, 160601 (2008).
slide-21
SLIDE 21

Basic strategy of SRG

<latexit sha1_base64="R6iI9fLWCtKLmIFtvaDpnRYqmq4=">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</latexit>

Me: Environment tensor

T

Me

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Me =

T

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Better truncation: truncation based on SVD of

<latexit sha1_base64="1TEPgTzUD/7Gp/8sWxcNbesk2c=">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</latexit>

In order to obtain Me we need iterative calculations, 
 which increases computation cost.

  • Z. Y. Xie, et al Phys. Rev. Lett. 103, 160601 (2008).

(Z = Tr ˜ T)

<latexit sha1_base64="PRJw3D25fU9S3Tm53napuSof1cI=">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</latexit>

Disadvantage of SRG:

slide-22
SLIDE 22

Mean field environment

As an approximated Me, we might use "weight" of bonds as mean-field environment.

<latexit sha1_base64="1TEPgTzUD/7Gp/8sWxcNbesk2c=">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</latexit>

T

: Diagonal non-negative matrix How can we estimate the bond weight?

  • We can estimate it from the singular values of a local tensor.
  • 1. The weights are repeatedly improved by the singular values of .
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Mean-field SRG. H.-H. Zhao et al, PRB 81, 174411 (2010).

2. Alternatively, we can consider the renormalization of 
 extended tensor networks which contains bond weight explicitly.

Bond-weighted TRG

slide-23
SLIDE 23

Summary

  • We introduced Anisotropic TRG (ATRG)
  • By using ATRG, we can contract a d-dimensional TN with O(D2d+1)

computation cost and O(Dd+1) memory.

  • When we look accuracy per computation time, ATRG is much

more efficient than HOTRG.

  • We considered TRG of bond-weighed TN (BTRG)
  • By choosing proper exponent, BTRG shows better accuracy than

HOTRG.

  • BTRG is easily generalized for other lattices and HOTRG.