Tensor Renormalization Group Approach to Scalar Field Theories in - - PowerPoint PPT Presentation

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Tensor Renormalization Group Approach to Scalar Field Theories in Particle Physics

CCS, Univ. of Tsukuba Yoshinobu Kuramashi CAQMP 2019@ISSP, Kashiwa Japan, July 22, 2019

Plan of Talk

Current Status for TRG Studies in Particle Physics Application to Scalar Field Theory

• 2D Real φ4 Theory ⇒ Spontaneous Symmetry Breaking
• 2D Complex φ4 Theory at Finite Density ⇒ Sign Problem

Summary

Ingredients in Particle Physics

• 4D relativisitic quantum field theory in path-integral formalism
• Gauge symmetry (U(1), SU(2), SU(3) etc.)
• Fermion(quark, lepton), gauge boson(photon, gluon, weak

boson), scalar particle(Higgs)

• Spontaneous symmetry breaking

It is often useful or important to investigate various lower (≤3) dimensional models which contains the above ingredients

TRG for Path-Integral Formalism

Advantage

• Free from sign problem in Monte Carlo method
• Computational cost for LD system size ∝ Dlog(L)
• Direct treatment of Grassmann numbers
• Direct measurement of Z itself

Possible applications in particle physics Light quark dynamics in QED/QCD, Finite density QCD, Strong CP problem, Chiral gauge theories, Lattice SUSY etc. Disadvantage Computational cost increases for higher dimensions ⇒ better to start with lower (≤3) dimensional models

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Application of TRGs to Particle Physics (1)

2D models Ising modelLevin-Nave, PRL99(2007)120601 X-Y modelMeurice+, PRE89(2014)013308 CP(1)+θKawauchi-Takeda, PRD93(2016)114503 Real φ4 theory(scalar field) Shimizu, Mod.Phys.Lett.A27(2012)1250035, Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, JHEP1905(2019)184 Complex φ4 theory at finite density(scalar field) Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, in preparation QED, QED+θ(fermion+U(1) gauge fields) Shimizu-YK, PRD90(2014)014508, PRD90(2014)074503, PRD97(2018)034502 Gross-Neveu model at finite density(fermion field) Takeda-Yoshimura, PTEP2015(2015)043B01 N=1 Wess-Zumino model(fermion+scalar fields) Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, JHEP1803(2018)141

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Application of TRGs to Particle Physics (2)

3D models IsingXie+, PRB86(2012)045139 Potts modelWan+, CPL31(2014)070503 Free Wilson fermion(fermion field) Sakai-Takeda-Yoshimura, PTEP2017(2017)063B07, Yoshimura-YK-Nakamura-Takeda-Sakai, PRD97(2018)054511 Z2 gauge theory at finite temperature(Z2 gauge field) YK-Yoshimura, arXiv:1808.08025[hep-lat] 4D models IsingAkiyama-YK-Yamashita-Yoshimura, arXiv:1906.06060[hep-lat] ⇒ Poster by Akiyama

Selected Topics on Scalar Field Theories

• 1. 2D real φ4 theory

Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, JHEP1905(2019)184 How to treat continuous d. o. f.? Spontaneous breaking of Z2 symmetry Continuum limit of critical coupling ⇒ comparison w/ MC results

• 2. 2D complex φ4 theory at finite density

Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, in preparation Complex action with finite chemical potential μ Sign problem is really solved?

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Collaborators

• Y. Kuramashi, Y. Yoshimura U. Tsukuba
• S. Akiyama
• Y. Nakamura R-CCS
• S. Takeda, R. Sakai Kanazawa U.
• D. Kadoh

Chulalongkorn U./ Keio U.

2D Real φ4 Theory

Continuum action of 2D real φ4 theory Lattice action Introduce a constant external field h to investigate spontaneous breaking of Z2 symmetry Boltzmann weight is expressed as ⇒ Need to discretize the continuous d. o. f. Kadoh+, JHEP1905(2019)184

• Scont. =
• d2x

1 2 (∂ρφ (x))2 + µ2 2 φ (x)2 + λ 4φ (x)4

• S =
• n∈ΓL

⎧ ⎨ ⎩ 1 2

2

• ρ=1

(φn+ˆ

ρ − φn)2 + µ2

2 φ2

n + λ

4φ4

n

⎫ ⎬ ⎭

Sh = S − h

• n∈ΓL

φn, e−Sh =

• n∈ΓL

2

• ρ=1

f (φn, φn+ˆ

ρ)

f (φ1, φ2) = exp

• −1

2 (φ1 − φ2)2 − µ2 8

• φ2

1 + φ2 2

• − λ

16

• φ4

1 + φ4 2

• + h

4 (φ1 + φ2)

• Z =
• Dφ exp(−S)

Tensor Network Representation

Use of Gauss-Hermite quadrature Discretized version of partition function SVD for f(φ,φ) Partition function with initial tensor Kadoh+, JHEP1905(2019)184

∞

−∞

dye−y2g (y) ≈

K

• α=1

wαg (yα) Z (K) =

• {α}
• n∈ΓL

wαn exp

• y2

αn

• 2
• ρ=1

f

• yαn, yαn+ˆ

ρ

• f (yα, yβ) =

K

• i=1

UαiσiV †

iβ,

Z (K) =

• {x,t}
• n∈ΓL

T (K)xntnxn−ˆ

1tn−ˆ 2

• T (K)ijkl = √σiσjσkσl

K

• α=1

wαey2

αUαiUαjV †

kαV † lα.

2e-08 4e-08 6e-08 8e-08 1e-07 1.2e-07 1.4e-07 1.6e-07 1.8e-07 2e-07 2.2e-07 10 100 <φ> K D=32 D=40 D=48

K dependence of <φ>

Expectation value of φ is calculated w/ insertion of an impurity tensor little K dependence beyond K10 Kadoh+, JHEP1905(2019)184

˜ T (K)ijkl = √σiσjσkσl

K

• α=1

yαwαey2

αUαiUαjV †

kαV † lα,

K dependence of <φ> near μ0,c

λ=0.05, h=10−12, L=1024

• Symm. Phase near μ0,c

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Determination of Critical Point

Critical point is determined from scaling property of susceptibility Kadoh+, JHEP1905(2019)184

χ = A

• µ2

0,c − µ2

• −γ

χ = lim

h→0 lim L→∞

⟨φ⟩h,L − ⟨φ⟩0,L h ,

1e+01 1e+02 1e+03 1e+04 1e+05 1e+06 1e+07 1e+08 1e+09 1e+10 1e-12 1e-10 1e-08 1e-06 1e-04 1e-02 <φ>/h h

h dependence of <φ>h,∞/h near μ0,c λ=0.05, D=32, K=256, L≥106

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• Symm. Phase near μ0,c

1e-06 2e-06 3e-06 4e-06 5e-06 6e-06

• 0.1006180
• 0.1006176
• 0.1006172
• 0.1006168

χ-1/1.75 µ0

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Scaling property of susceptibility λ=0.05, D=32, K=256

Scaling property is well described by 2D Ising universality class (γ=1.75) Consider dimensionless quantity λ/(μc)2 to take the continuum limit

Continuum Limit of Critical Coupling

Kadoh+, JHEP1905(2019)184

Comparison w/ recent Monte Carlo studies

λ/(μc)2 =10.913(56) in the continuum limit (λ→0) Consistent with recent Monte Carlo results

10.4 10.6 10.8 11 11.2 11.4 11.6 0.02 0.04 0.06 0.08 0.1

λ/µc

2

λ

Schaich and Loinaz: cluster (2009) Wozar and Wipf: with SLAC derivative (2012) Bosetti et al.: worm (2015) Bronzin et al.: worm with gradient flow (2018) This work

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2D Complex φ4 Theory at Finite Density

Continuum action of 2D complex φ4 theory at finite μ Introduction of finite chemical potential ⇒ complex action Lattice action TN representation is constructed in the same way as for the real φ4 case Bose condensation is expected to occur at sufficiently large μ Kadoh+, in preparation

Scont =

• d2x
• |∂ρφ|2 + (m2 − µ2)|φ|2 + µ(φ∗∂2φ − ∂2φ∗φ) + λ|φ|4
• Z(original) =
• Dφ1Dφ2 exp(−S)

S =

• n

⎡ ⎢ ⎣(4 + m2)|φn|2 + λ|φn|4 −

2

• ρ=1
• eµδρ,2φ∗

nφn+ˆ ρ + e−µδρ,2φ∗ n+ˆ ρφn

• ⎤

⎥ ⎦

Simple(st) Test Bed for Sign Problem

Previous study with path optimization method (Monte Carlo) The authors claim ”We show that the average phase factor is significantly enhanced after the

• ptimization and then we can safely perform the hybrid Monte Carlo method.”

Mori-Kashiwa-Onishi, PTEP(2018)023B04

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 Re <eiθ>pq µ L=8 L=6 L=4 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 Re <eiθ>pq µ L=8 L=6 L=4

• 2

2 4 6 8 10 12 14 16 0.5 1 1.5 2 Re <n> µ Mean Field approx. L=8 L=6 L=4

Average Phase Factor

⟨eiθ⟩ = Z/Zpq Zpq =

• Dφ1Dφ2 exp(−Re(S))

<eiθ> becomes close to 1 Still, it seems difficult to perform a MC simulation on large L m2=1, λ=1

Sign-Problem-Free Representation

Mathematical tools

• Polar coordinate
• Character expansion

Partition function can be expressed in a sign-problem-free form TRG should work for Z(positive) Consistency check btw the results for Z(original) and Z(positive) Endres, PoS(LAT2006)133

φn = (φn,1, φn,2) → (rn cos θn, rn sin θn) exp(x cos z) =

∞

• k=−∞ Ik(x) exp(ikz)

x ∈ R, z ∈ C Z(positive) =

⎛ ⎜ ⎝

• n

∞

• kn,1,kn,2=−∞

⎞ ⎟ ⎠

• n

∞

drn

n 2πrn 2

• ρ=1 e−1

4(4+m2)(r2 n+r2 n+ˆ ρ)−λ 4(r4 n+r4 n+ˆ ρ)

·Ikn,ρ(2rnrn+ˆ

ρ)ekn,ρµδρ,2δ(kn,1+kn,2−kn−ˆ

1,1−kn−ˆ 2,2),0

Results for Z(original) with TRG

Bose condensation at finite μ Parameters: m2=0.01, λ=1, K2=4096, Dcut=64 Average phase factor Clear signal even in the <eiθ> 0 region Kadoh+, in preparation

⟨eiθ⟩ = Z/Zpq Zpq =

• Dφ1Dφ2 exp(−Re(S))
• 0.5

0.5 1 1.5 2 2.5 3 3.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 n µ V=22×22=4×4 V=24×24=16×16 V=26×26=64×64 V=28×28=256×256 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 <|φ|2> µ V=22×22=4×4 V=24×24=16×16 V=26×26=64×64 V=28×28=256×256

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 Z/Zpq µ

Comparison btw Z(original) and Z(positive)

Parameters: m2=0.01, V=8x128, λ=1, K2=4096, Dcut=64, NCE=128 Good agreement in large μ region (severe sign problem in MC) ⇒ Free from sign problem Kadoh+, in preparation

0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.8 0.9 1 1.1 1.2 1.3

<|φ|2> µ

naive TRG Endres’s form w/ TRG

Summary

We have been developing necessary tools to analyze QFT in path-integral formalism w/ TRGs in tensor network scheme Focused on

• 2D real φ4 theory

Construction of TN representation for scalar field

• 2D complex φ4 theory at finite density

Free from sign problem ⇒ Next step is an extension to 4D models [Poster by Akiyama]