Tensor renormalization group approach tofour-dimensional complex 𝜙 4 theory at finite density
Shinichiro Akiyama (Univ. of Tsukuba)
in collaboration with
D. Kadoh, Y. Kuramashi, T. Yamashita, Y. Yoshimura
Based on arXiv:2005.04645 [hep-lat]
熱場の量子論とその応用 (TQFT2020)
2020.8.24

Tensor renormalization group approach
Tensor Renormalization Group (TRG)-> A real-space renormalization group to coarse grain tensor networks
Tensor Network (TN)-> Contractions of the tensors locating on a real spaceProcedures
TN rep. for 𝑋# 𝐨𝐟 𝐭𝐞𝐧𝐬𝐨𝐫𝐬 𝐢𝐧 𝐓𝐍 = (# 𝐨𝐟 𝐥𝐚𝐭𝐭𝐢𝐜𝐞 𝐬𝐢𝐭𝐞𝐬)

𝑋 → Σ𝑎𝑏𝑐𝑑⋯ 𝑇𝑎𝑖𝑤⋯ 𝑇𝑏𝑗𝑥⋯ 𝑇𝑐𝑘𝑦⋯ 𝑇𝑑𝑙𝑧⋯
TRG : Block-spin trans. for 𝑇≈ Σ𝑎′ 𝑏′ 𝑐 ′ 𝑑′⋯ 𝑇′𝑎′ 𝑖 ′𝑤 ′⋯ 𝑇′𝑏′ 𝑗 ′𝑥 ′⋯ 𝑇′𝑐 ′ 𝑘 ′𝑦′⋯ 𝑇′𝑑′ 𝑙′𝑧 ′⋯1) Construct the TN representation for the target function defined on latticeex. Partition function, Path integral, 𝑛-th moment
2) Approximately perform the tensor contraction with TRG

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Advantage of TRG approach
Tensor Renormalization Group (TRG) is a deterministic numerical method.
・No sign problem
・The computational cost scales logarithmically w. r. t. the system size
・Direct evaluation of the Grassmann integrals (w/o introducing pseudo-fermions)・Direct evaluation of the partition functions

TRG has been successfully applied to various 2d or 3d models w/ or w/o thesign problem.

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Successful applications
3d Ising model

2d complex 𝜙 4 theory
at finite density

Xie et al, PRB86(2012)045139

Kadoh et al, JHEP02(2020)161
𝑚 = 0.1, 𝜆 = 1-> Good agreement with
the Monte Carlo results-> the Silver Blaze phenomenon
is successfully confirmed

Today’s message
TRG is an effective approach not only in 2d or 3d but also in 4d !

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4d complex 𝜙 4 theory at finite density✔ a typical system with the sign problem✔ the Silver Blaze phenomenon-> thermodynamic observables at zero temperature are independent of 𝜇 up to 𝜇𝑐
・Complex Langevin method
Aarts, PRL102(2009)131601

・Thimble approach
Cristoforetti et al, PRD88(2013)051501
Fujii et al, JHEP10(2013)147

・World-line representation
Gattringer-Kloiber, NPB869(2013)56-73

etc …, and ・ Tensor renormalization group

This work is the first application of TRG to 4d QFT!!!

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TRG in 4d system
・Higher-Order TRG (HOTRG) Xie et al, PRB86(2012)045139✔ Applicable to any dimensional lattice✔ Not so economic in 4d lattice-> 4d Ising model on 𝑉 = 10244 (with parallel computation)
SA et al, PRD100(2019)054510

・Anisotropic TRG (ATRG) Adachi et al, arXiv:1906.02007✔ Also applicable to any dimensional lattice✔ Accuracy with the fixed computational time is improved compared withthe HOTRG-> 4d Ising model on 𝑉 = 10244 (with parallel computation)
SA et al, PoS(LAT2019)363

We employ the ATRG algorithm in this work

5/14Anisotropic TRG with parallel computation
ATRG is a coarse-graining (direct truncation) method based on SVD
4d ATRG

4d HOTRG

Memory𝑂(𝐷5 )𝑂(𝐷8 )Time𝑂(𝐷 9 )𝑂(𝐷15 )

𝐷: bond dimension (singular value matrix is truncated by 𝐷)

𝑂 𝐷9 calculations in 4d ATRG -> SVD and tensor contraction
Our implementation
SVD

contraction

Strategy

Randomized SVD

Parallel computing

Time𝑂(𝐷 7 )𝑂(𝐷 8 )-> Parallel computation reduces the computational cost from 𝑶 𝑫𝟗 to 𝑶 𝑫𝟖6/14

Tensor network representation (1/2)✔ 𝜙𝑛 = 𝑟𝑛 ei𝜋𝑠𝑛 : continuous d. o. f.
✔ 𝜇 : chemical potential

𝑆lat = Σ𝑛∈Γ 8 + 𝑚2 𝑟𝑛2 + 𝜆𝑟𝑛4 − 2Σ𝜈 𝑟𝑛 𝑟𝑛+ෝ𝜈 cos(𝜋𝑠𝑛+ෝ𝜈 − 𝜋𝑠𝑛 + i𝜇𝛿𝜈,4 )To derive a finite dimensional tensor, we need to discretize 𝑟𝑛 and 𝑠𝑛 :Continuous
d. o. f.

Discreted. o. f.

Quadrature rule𝑟𝑛 ∈ [0, ∞]𝛼𝑛 ∈ ℤ

Gauss-Laguerre : ‫׬‬0 d𝑟𝑛 e−𝑟𝑛 𝑓 𝑟𝑛 ≈ σ𝐾
𝛼𝑛 =0 𝑤𝛼𝑛 𝑓(𝑟𝛼𝑛 )𝑠𝑛 ∈ [−1,1]𝛽𝑛 ∈ ℤ

Gauss-Legendre:∞1‫׬‬−1 d𝑠𝑛 𝑓 𝑠𝑛 ≈ σ𝐾
𝛽𝑛 =0 𝑢𝛽𝑛 𝑓(𝑠𝛽𝑛 )-> The partition function 𝑍 is approximated by 𝑍(𝐾)
4𝑍(𝐾) = Σ{𝛼,𝛽} Π𝜈=1𝑀𝛼𝜈𝑛 𝛽𝑛,𝛼𝑛+𝜈ෝ 𝛽𝑛+𝜈ෝ

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Tensor network representation (2/2)
SVD separates 𝑛-site d. o. f. from (𝑛 + 𝜈)-siteƸ
d. o. f. :෩≔𝑈 𝜎𝑈𝑉෨ ∗ ≔ 𝑉 ∗ 𝜎𝜈𝜈∗𝜈∗𝐾2 ෩ 𝜈𝐷 ෩ 𝜈
𝑀𝛼𝑛 𝛽𝑛 ,𝛼𝑛+𝜈ෝ 𝛽𝑛+𝜈ෝ = Σ𝑙=1
𝑈𝛼𝑛 𝛽𝑛 ,𝑙 𝑉෨𝛼𝑛+𝜈ෝ 𝛽𝑛+𝜈ෝ ,𝑙 ≈ Σ𝑙=1
𝑈𝛼𝑛 𝛽𝑛 ,𝑙 𝑉෨𝛼𝑛+𝜈ෝ 𝛽𝑛+𝜈ෝ ,𝑙𝜈4( 𝑍(𝐾) = Σ{𝛼,𝛽} Π𝜈=1𝑀𝛼𝑛 𝛽𝑛 ,𝛼𝑛+𝜈ෝ 𝛽𝑛+𝜈ෝ )Tensor network representation: 𝑍 𝐾 ≈ Tr[Π𝑛 𝑇𝑥𝑛𝑦𝑛𝑧𝑛𝑡𝑛𝑥𝑛′𝑦𝑛′𝑧𝑛′𝑡𝑛′ ]
෩1
෩2
෩3
෩4
෨ 1∗
෨ 2∗
෨ 3∗
෨ 4∗
( 𝑇𝑥𝑛 𝑦𝑛 𝑧𝑛 𝑡𝑛 𝑥𝑛′𝑦𝑛′ 𝑧𝑛′𝑡𝑛′ = Σ𝛼𝐾𝑛 =1 Σ𝛽𝐾𝑛=1 𝑈
𝛼𝑛 𝛽𝑛 ,𝑥𝑛 𝑈𝛼𝑛𝛽𝑛,𝑦𝑛 𝑈𝛼𝑛𝛽𝑛,𝑧𝑛 𝑈𝛼𝑛𝛽𝑛,𝑡𝑛 𝑉𝛼 𝛽 ,𝑥 ′ 𝑉𝛼 𝛽 ,𝑦 ′ 𝑉𝛼 𝛽 ,𝑧 ′ 𝑉𝛼 𝛽 ,𝑡 ′ )𝑛 𝑛 𝑛𝑛 𝑛 𝑛𝑛 𝑛 𝑛𝑛 𝑛 𝑛✔ Tensor 𝑇 locates on each lattice site 𝑛𝑍 𝐾

✔ Tensor contraction is approximately=𝑧𝑡𝑦𝑥

done by TRG( Tensor network is coarse-grained )8/14

Numerical Results

Algorithmic-parameters dependencewith 𝑚 = 0.1, 𝜆 = 1, 𝜇 = 0.6, 𝐿 = 1024

Polynomial orderin the Gauss quadrature

Bond dimension in ATRG

𝐷 = 45

𝐾 = 64little 𝐾 dependence beyond 𝐾 ∼ 30

converging around 𝐷 ∼ 40

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Average phase factor ei𝜃 pq
with 𝑚 = 0.1, 𝜆 = 1, 𝐾 = 64, 𝐷 = 45

Reweighting in MC
𝑍pq = න[d𝜙] e−Re 𝑆
with e−𝑆 = e−Re 𝑆 ei𝜃
𝑂 =𝑂ei𝜃 pqei𝜃 pqei𝜃 pq = 𝑍/𝑍pqei𝜃 pq quickly falls off from 1 to 0 beyond 𝜇 ∼ 0.05-> difficult to perform a MC simulation on large volume

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Particle number density (1/2)
with 𝑚 = 0.1, 𝜆 = 1, 𝐾 = 64, 𝐷 = 45

Resulting 𝑛 is qualitatively not bad even in the region with ei𝜃 pq ∼ 0.
𝑛 stays around 0 up to 𝜇 ≈ 0.65 and shows the rapid increase with 𝜇 ≳ 0.65-> The Silver Blaze phenomenon is confirmed

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Particle number density (2/2)
with 𝑚 = 0.1, 𝜆 = 1, 𝐾 = 64, 𝐷 = 45, 𝐿 = 1024

𝑛 vs 𝑛 pq

The Silver Blaze phenomenon is attributed to the imaginary part of 𝑆

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𝜙 2 : a discussion of the validity of the numerical resultswith 𝑚 = 0.1, 𝜆 = 1, 𝐾 = 64, 𝐷 = 45

Mean-field estimation
MF𝜇4 sinh2 𝑐2= 𝑚2 + 4𝜆 𝜙 2 𝜇=0Aarts, JHEP05(2009)052↓
𝝁𝐌𝐅𝒄 ≈ 𝟎. 𝟕𝟎

𝜙 2 ≈ 0.125 over 0 ≲ 𝜇 ≲ 0.6
Location of 𝝁𝒄 in the current ATRG calculations seems reasonable

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Summary• This is the first application of TRG approach to 4d QFT• The Silver Blaze phenomenon (thermodynamic observables at zerotemperature are independent of 𝜇 up to 𝜇𝑐 ) is clearly observed for 𝑛and 𝜙 2• The location of 𝜇𝑐 seems reasonable compared with the mean-fieldvalue 𝜇𝑐MF
• TRG approach does not suffer from the sign problem and nicely works
to evaluate the observables on almost thermodynamic lattice• TRG will be an effective numerical approach to other 4d QFTs

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