[PPT] - Application of Tensor Network Scheme to Particle Physics CCS, Univ. PowerPoint Presentation

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Application of Tensor Network Scheme to Particle Physics

CCS, Univ. of Tsukuba Yoshinobu Kuramashi TNSAA 2018-2019@R-CCS, Kobe Japan, Dec. 3, 2018

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Plan of Talk

1. Introduction to Particle Physics and Lattice Gauge Theory
2. An Overview of Current Status for application of TN Scheme

to QFTs relevant to Particle Physics

3. Application to QFTs
2D φ4 Theory
One-Flavor Wilson Schwinger Model w/ and w/o θ term
3D Finite Temperature Z2 Gauge Theory
4. Summary

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Questions in history of mankind

What is the smallest component of matter?
What is the most fundamental interaction?

Introduction to Particle Physics

K.-I. Ishikawa@Hiroshima U.

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e µ t ne nµ nt

−e lepton

4

u c t d s b

+2/3e −1/3e quark red,blue,green

Elementary Particles Known to Date

Higgs particle (LHC@CERN in 2012) electric charge electric charge

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Force Strength Gauge boson Theory

Only four fundamental interactions is known so far

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Fundamental Interactions

Strong EM Weak Gravity 1 0.01 0.00001 10−40 Gluon Photon Weak Boson Graviton QCD QED Weinberg-Salam Superstring(?)

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Unification of Fundamental Forces

10−35m 10−18m 10−32m 10−10m

Maxwell/Eqs/ Classical/ Mechanics/ General/ Rela;vity/ QED/ QCD/ SuperGravity/ Grand/Unified/Theory/ SuperString/ ElectroWeak/ Special/Rela;vity/ Quantum/Theory/ SuperSymmetry/ NonMAbelian/Gauge/ Theory/ Spontaneous/ Symmetry/Breaking/ Standard/Model/ / Strong Weak ElectroM/ Magne;c /Gravity

Higgs particle was the last piece of SM Beyond SM (BSM)

Standard model consists of gauge theories ⇩ Lattice QCD

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Motivation of Lattice QCD

Nonperturbative Phenomena in strong Interaction btw quraks

Confinement : quark can never be retrieved by itself
Asymptotic freedom closer to each other, arbitrarily weaker
Hierarchy : 3 quarks ⇒ protonneutron ⇒ nuclear
Finite temperature and finite density (finite T-μ): phase transition

quark proton nucleus neutron finite T-μ

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Difficulties in lattice gauge theory

Partition function of lattice gauge theory after analytic integration of fermion field Expectation value of physical quantity Monte Carlo should work for detDe−S>0 with importance sampling In case that P is negative or a complex value, Importance sampling fails = statistical error becomes uncontrollable Another problem is computational cost for fermion system Direct treatment of Grassmann numbers is practically impossible

Z =

DU det D({U}) e−Sg({U})

⟨O⟩ =

DU O({U, D−1}) det D({U}) e−Sg({U})

U: gauge field D: Dirac matrix

Sign problem / Complex action problem

P = 1 Z det D({U}) e−Sg({U})

det D½U ¼ Y

n;α

Z dψn;αd ¯ ψn;α

e ¯

ψD½Uψ

Z

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TN Scheme for Particle Physics

Advantage

Free from sign problem and complex action 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 (D≤3) dimensional models

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Important Ingredients in Particle Physics

Quantum field theory
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 important to investigate various lower dimensional models which contains the above ingredients

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How to treat QFT w/ TN Scheme (1)

Path-integral formalism Scalar field(non-compact) How to discretize continuous d. o. f.? ⇒ Expansion w/ orthogonal functions. Shimizu Mod.Phys.Lett.A27(2012)1250035 Gauss-Hermite quadrature Kadoh et al. JHEP1803(2018)141, arXiv:1811.12376 Gauge field(compactZN,U(1),SU(2),SU(3) gauge groups) Character expansionpreserve gauge symmetry, good convergence Liu et al. PRD88(2013)056005 (w/o numerical demonstration) First successful calculation of 3D Z2 gauge theory YK-Yoshimura arXiv:1808.08025[hep-lat]

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How to treat QFT w/ TN Scheme (2)

Path-integral formalism Fermion field Nilpotency of Grassmann variables ⇒ finite numbers of terms in the expansion Coarse-graining procedure for Grassmann variables Shimizu-YK PRD90(2014)014508, Takeda-Yoshimura PTEP2015(2015)043B01 We have developed necessary tools to analyze QFT in path-integral formalism w/ tensor network scheme

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Application of TN Scheme to Particle Physics (1)

2D models Ising modelLevin-Nave PRL99(2007)120601 X-Y modelMeurice et al. PRE89(2014)013308 CP(1)+θKawauchi-Takeda PRD93(2016)114503 φ4 theory(scalar field) Shimizu Mod.Phys.Lett.A27(2012)1250035, Kadoh et al. arXiv:1811.12376 QED, QED+θ(fermion+U(1) gauge fields) Shimizu-YK PRD90(2014)014508, PRD90(2014)074503, PRD97(2018)034502 Gross-Neveu model+μ(fermion) Takeda-Yoshimura PTEP2015(2015)043B01 N=1 Wess-Zumino model(fermion+scalar fields) Kadoh et al. JHEP1803(2018)141

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Application of TN Scheme to Particle Physics (2)

3D models IsingXie et al. PRB86(2012)045139 Potts modelWan et al. CPL31(2014)070503 Free Wilson fermion(fermion field) Sakai-Takeda-Yoshimura PTEP2017(2017)063B07, Yoshimura et al. PRD97(2018)054511 Z2 gauge theory(Z2 gauge field) YK-Yoshimura arXiv:1808.08025[hep-lat] 4D models Ising(φ4 theory) Akiyama et al. work in progress (parallel computation)

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Selected Our Recent Work

1. 2D φ4 theory

Kadoh et al., arXiv:1811.12376 Scalar field, spontaneous breaking of Z2 symmetry

2. One-Flavor Wilson Schwinger model w/ and w/o θ term

Shimizu-YK, PRD90(2014)014508, PRD90(2014)074503, PRD97(2018)034502 Fermion+U(1) gauge fields, sign problem, complex action

3. 3D Finite Temperature Z2 Gauge Theory

YK-Yoshimura, arXiv:1808.08025[hep-lat] Gauge field, simplest gauge theory in 3D

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Collaborators

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

Chulalongkorn U./ Keio U.

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TN Representation of 2D φ4 theory (1)

Continuum action of 2D 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 et al. arXiv:1811.12376

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)

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TN Representation of 2D φ4 theory (2)

Use of Gauss-Hermite quadrature Discretized version of partition function SVD for f(φ,φ) Partition function with initial tensor Kadoh et al. arXiv:1811.12376

∞

−∞

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.

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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 K=256 is large enough Kadoh et al. arXiv:1811.12376

˜ 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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Susceptibility of <φ>

Critical point is determined from scaling property of susceptibility Kadoh et al. arXiv:1811.12376

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+01 1e+02 1e+03 1e+04 1e+05 1e+06 1e+07 <φ>/h L h=10-12 h=10-10 h=10-8 h=10-6 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

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

Symm. Phase near μ0,c

λ=0.05, D=32, K=256

Symm. Phase near μ0,c

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Determination of Critical Coupling

Kadoh et al. arXiv:1811.12376

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 D dependence of λ/(μc)2

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

λ=0.05, D=32, K=256 λ=0.05

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Continuum Limit of Critical Coupling

Kadoh et al. arXiv:1811.12376

Comparison w/ recent Monte Carlo studies

λ/(μc)2 =10.913(56) in the continuum limit (λ→0) It may be possible to obtain more precise results w/ TNR or loop-TNR

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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Schwinger Model in Tensor Network Formalism

Path-integral for Wilson fermion action and U(1) plaquette gauge action Fermion part is expanded in terms of Ψ, Ψ Gauge part is expressed by character expansion Integration of link variables gives Tn;i,j,k,l includes Grassmann numbers Ψ, Ψ, dΨ, dΨ ⇒ Grassmann TRG Liu et al. PRD88(2013)056005

¯ ψD½Uψ ¼ 1 2κ X

n;α

¯ ψn;αψn;α − 1 2 X

n;μ;α;β

¯ ψn;αfð1 − γμÞα;βUn;μψnþˆ

μ;β

þ ð1 þ γμÞα;βU†

n−ˆ μ;μψn−ˆ μ;βg;

Z =

DψD ¯

ψDU e− ¯

ψD[U]ψ−Sg[U] Sg ¼ −β X

p

cos φp; φp ¼ φn;1 þ φnþˆ

1;2 − φnþˆ 2;1 − φn;2;

φn;1; φnþˆ

1;2; φnþˆ 2;1; φn;2 ∈ ½−π; π;

Z ¼ Z X

i;j;k;

Tn;i;j;k;lTnþˆ

1;m;o;i;pTnþˆ 2;q;r;s;j

eβ cos φp ¼ X

∞ mb¼−∞

eimbφpImbðβÞ; ≃ X

Nce mb¼−Nce

eimbφpImbðβÞ;

Imb: modified Bessel function Nce: truncation parameter

_ _ _ Shimizu-YK PRD90(2014)014508

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β-κ Phase Diagram with One Flavor Wilson Fermion

det D < 0 is allowed near κc ⇒ test bed for sign problem

Free fermion at β=∞, κc=0.25 ⇒ Ising universality class (α=0, ν=1)
Strong coupling limit at β=0 ⇒ Ising universality class (α=0, ν=1)

Gausterer-Lang NPB455(1995)785, Wenger PRD80(2009)071503 Natural to expect the Ising universality class at finite β However, previous studies were controversial (1) Microcanonical fermionic average approach Different universality class (α=1, ν=2/3) Azcoiti et al. PRD53(1996)5069 (2) Weak coupling expansion No phase transition Kenna et al. arXiv:hep-lat/9812004 ⇒ Investigation with Grassmann TRG in TN scheme

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κ=1/2(m+2) β=1/g2

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Numerical Results

Similar results are obtained at β=5.0 Our results show the Ising universality class (α=0, ν=1) even at finite β β=10.0, Dcut=96, Nce=15 Fisher zero analysis in complex κ plane Shimizu-YK PRD90(2014)014508 χ(L)max∝log(L) ⇒ α≃0 ν=0.995(19) Peak height of chiral susceptibility

χ(L) = 1 L2 ∂2 ln Z ∂(1/2κ)2

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Phase Structure in (m,g) plane

Shimizu-YK PRD97(2018)034502 Further investigation of phase diagram in wider range of (m,g) plane − Analysis of central charge − Existence of Berezinskii-Kosterlitz-Thouless (BKT) transition κ=1/2(m+2) β=1/g2

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Schwinger Model with θ term

Inclusion of θ term ⇒ complex action Character expansion including the θ term Shimizu-YK PRD90(2014)074503

Sg ¼ −βP

p

cos φp − iθQ; φp ¼ φn;1 þ φnþ^

1;2 − φnþ^ 2;1 − φn;2;

φn;1; φnþ^

1;2; φnþ^ 2;1; φn;2 ∈ ½−π; π;

Q ¼ 1 2π X

p

qp; qp ¼ φp mod 2π exp

β cos φp þ i θ

2π qp

¼

X

∞ m¼−∞

eimφp X

∞ l¼−∞

IlðβÞ 2 sinðθþ2πðm−lÞ

2

Þ θ þ 2πðm − lÞ ≃ X

Nce m¼−Nce

eimφp X

N0

ce

l¼−N0

ce

IlðβÞ 2 sinðθþ2πðm−lÞ

2

Þ θ þ 2πðm − lÞ ;

Hassan et al. PTP93(1995)161

Nce, N’ce: truncation parameters Q is integer even on the lattice

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Phase structure in θ-m/g plane

Expected phase diagram Phase transition at θ=π ⇒ θ term is relevant Coleman Ann.Phys.(NY)101(1976)239

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Numerical Results

Our results successfully reproduce the expected phase diagram β=10.0, Dcut=160, Nce=20, N’ce=120 Lee-Yang zero analysis in complex θ plane around critical endpoint Shimizu-YK PRD90(2014)074503

Imθ(∞)≠0 Crossover y=1.869(10) 2nd order phase transition Ising universality class (y=2-β/ν=1.875) y=2.009(12) 1st order phase transition (y=2)

m heavier lighter

Imθ(∞)+aL−y

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3D finite temperature Z2 gauge theory

Partition function Plaquette action with spin variable assigned at link n=(n0,n1,n2) 0≤n0<Nτ: temporal(temperature) direction 0≤n1,2<Nσspatial direction At fixed NT, it is expected to belong to 2D Ising universality class (Svetitsky-Yaffe conjecture) YK-Yoshimura arXiv:1808.08025 Z = 2−3V X

{σ=±1}

Y

n,µ>ν

e−βσn,µν σn,µν = σn,µσn+ˆ

µ,νσn+ˆ ν,µσn,ν,

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TN Representation for Z2 gauge theory

A-, B-tensors introduced by Liu et al. B-tensor is assigned at plaquette A-tensor is assigned at site Similar representation is obtained by character expansion for other gauge theories (U(1),SU(2),SU(3)) No numerical result is presented so far Liu et al. PRD88(2013)056005

Z = (cosh β)3V X

{p,q}

Y

n,µ>ν

B(n,µν) Y

n,µ

A(n,µ). Apqrs = δ mod (p+q+r+s,2)=0. Bpqrs = (tanh β)(p+q+r+s)/4δp,qδq,rδr,s.

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Reconstruction of TN Representation (1)

Graphical representation of our reconstruction Temporal gauge is employed to reduce redundant degrees of freedom − does not change the value of Z − A(n,0)=0 at n0≠0 YK-Yoshimura arXiv:1808.08025

Decomposition of A(n,0)=0 at n0=0 Decomposition of B(n,21)=0

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Reconstruction of TN Representation (2)

Final form of reconstructed tensors Tensors are assigned at only even site (not plaquette any more) Spatial size of reconstructed network is half of original one YK-Yoshimura arXiv:1808.08025 Z = X

{t,x,y}

Y

n;n0>0

T (n) ! Y

n;n0=0

S(n) !

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Coarse-Graining Procedure

Step 1: coarse-graining in temporal direction w/ HOTRG Step 2: coarse-graining in spatial direction w/ HOTRG YK-Yoshimura arXiv:1808.08025

Trace HOTRG HOTRG HOTRG

Truncation parameter D1 Truncation parameter D2

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Numerical Study (1)

Finite size scaling analysis of specific heat Peak position and Cmax is determined by fitting internal energy avoiding uncertainties associated w/ numerical derivative YK-Yoshimura arXiv:1808.08025 Clear peak structure Nσ∈[32,4096]

1.40 1.41 1.42

1/β

4.0 6.0 8.0 10.0 12.0

Nσ=32 Nσ=64 Nσ=128 Nσ=256 Nσ=512 Nσ=1024 Nσ=2048 Nσ=4096

C(Nσ)

C(Nσ) ≡ β2 ∂2 ln Z V ∂β2 . E = ∂ ln Z V ∂β = P + Cmax(Nσ) β + R 3 ✓ 1 β 1 βc(Nσ) ◆3

C(Nσ) ⇠ Cmax(Nσ) + R ✓ 1 β 1 βc(Nσ) ◆2

Nτ=3, D2=128

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Numerical Study (2)

Finite size scaling (FFS) analysis of specific heat Cmax shows clear logarithmic dependence on Nσ (α=0 in 2D Ising universality class) FFS with Nσ∈[512,4096] gives ν=1 ⇒ consistent w/ Svetitsky-Yaffe conjecture YK-Yoshimura arXiv:1808.08025

10 100 1000 10000

Nσ

4.0 6.0 8.0 10.0 12.0

Cmax(Nσ)

0.00 0.01 0.02 0.03 0.04

1/Nσ

0.707 0.708 0.709 0.710 0.711 0.712

βc(Nσ)

This work

Ref. [19]

Nτ Nσ βc(∞) ν B χ2/d.o.f. Nσ βc(∞) ν 2 [512, 4096] 0.656097(1) 1.00(1) 0.116(6) 0.086 4, 8, 16, 32 0.65608(5) 1.012(21) 3 [512, 4096] 0.711150(4) 0.99(4) 0.10(3) 0.047 24 0.71102(8) 5 [512, 4096] 0.740730(3) 0.96(5) 0.08(3) 0.012 40 0.74057(3)

Cmax(Nσ) / N α/ν

σ

, βc(Nσ) βc(1) / N −1/ν

σ

.

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Summary

What we have achieved since 2014 We have developed necessary tools to analyze QFT in path-integral formalism w/ tensor network scheme Three selected topics in today’s talk

2D φ4 theory

Construction of TN representation for scalar field

One-Flavor Wilson Schwinger model w/ and w/o θ term

Free from sign problem and complex action problem

3D Finite Temperature Z2 Gauge Theory