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

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 1

Plan of Talk � Current Status for TRG Studies in Particle Physics � Application to Scalar Field Theory - 2D Real φ 4 Theory ⇒ Spontaneous Symmetry Breaking - 2D Complex φ 4 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 L D system size ∝ D � log(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

Application of TRGs to Particle Physics (1) 2D models Ising model � Levin-Nave, PRL99(2007)120601 X-Y model � Meurice+, PRE89(2014)013308 CP(1)+θ � Kawauchi-Takeda, PRD93(2016)114503 Real φ 4 theory(scalar field) � Shimizu, Mod.Phys.Lett.A27(2012)1250035, Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, JHEP1905(2019)184 Complex φ 4 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 5

Application of TRGs to Particle Physics (2) 3D models Ising � Xie+, PRB86(2012)045139 Potts model � Wan+, CPL31(2014)070503 Free Wilson fermion(fermion field) � Sakai-Takeda-Yoshimura, PTEP2017(2017)063B07, Yoshimura-YK-Nakamura-Takeda-Sakai, PRD97(2018)054511 Z 2 gauge theory at finite temperature(Z 2 gauge field) � YK-Yoshimura, arXiv:1808.08025[hep-lat] 4D models Ising � Akiyama-YK-Yamashita-Yoshimura, arXiv:1906.06060[hep-lat] ⇒ Poster by Akiyama 6

Selected Topics on Scalar Field Theories 1. 2D real φ 4 theory Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, JHEP1905(2019)184 How to treat continuous d. o. f.? Spontaneous breaking of Z 2 symmetry Continuum limit of critical coupling ⇒ comparison w/ MC results 2. 2D complex φ 4 theory at finite density Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, in preparation Complex action with finite chemical potential μ Sign problem is really solved?

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. 8

2D Real φ 4 Theory Kadoh+, JHEP1905(2019)184 Continuum action of 2D real φ 4 theory 2 ( ∂ ρ φ ( x )) 2 + µ 2 � � 1 � 2 φ ( x ) 2 + λ 4 φ ( x ) 4 d 2 x 0 S cont . = Lattice action ⎧ ⎫ 2 ρ − φ n ) 2 + µ 2 1 n + λ ⎨ ⎬ � � � 2 φ 2 0 4 φ 4 Z = D φ exp( − S ) S = ( φ n +ˆ n 2 ⎩ ⎭ n ∈ Γ L ρ =1 Introduce a constant external field h to investigate spontaneous breaking of Z 2 symmetry � S h = S − h φ n , n ∈ Γ L Boltzmann weight is expressed as 2 e − S h = � � f ( φ n , φ n +ˆ ρ ) n ∈ Γ L ρ =1 2 ( φ 1 − φ 2 ) 2 − µ 2 � − 1 � − λ + h 0 φ 2 1 + φ 2 φ 4 1 + φ 4 � � � � f ( φ 1 , φ 2 ) = exp 4 ( φ 1 + φ 2 ) 2 2 8 16 ⇒ Need to discretize the continuous d. o. f.

Tensor Network Representation Kadoh+, JHEP1905(2019)184 Use of Gauss-Hermite quadrature � ∞ K d ye − y 2 g ( y ) ≈ � w α g ( y α ) −∞ α =1 Discretized version of partition function 2 � � � y 2 � � � � Z ( K ) = w α n exp f y α n , y α n +ˆ α n ρ { α } n ∈ Γ L ρ =1 SVD for f(φ,φ) K U α i σ i V † � f ( y α , y β ) = i β , i =1 Partition function with initial tensor � � Z ( K ) = T ( K ) x n t n x n − ˆ 1 t n − ˆ 2 n ∈ Γ L { x,t } � � � K T ( K ) ijkl = √ σ i σ j σ k σ l w α e y 2 α U α i U α j V † k α V † � l α . α =1

K dependence of <φ> Kadoh+, JHEP1905(2019)184 Expectation value of φ is calculated w/ insertion of an impurity tensor K T ( K ) ijkl = √ σ i σ j σ k σ l y α w α e y 2 α U α i U α j V † k α V † ˜ � l α , α =1 K dependence of <φ> near μ 0,c 2.2e-07 D =32 D =40 2e-07 D =48 1.8e-07 1.6e-07 1.4e-07 < φ > 1.2e-07 1e-07 λ=0.05, h=10 −12 , L=1024 8e-08 Symm. Phase near μ 0,c 6e-08 4e-08 2e-08 10 100 K little K dependence beyond K � 10 11

Determination of Critical Point Kadoh+, JHEP1905(2019)184 Critical point is determined from scaling property of susceptibility ⟨ φ ⟩ h,L − ⟨ φ ⟩ 0 ,L � � µ 2 0 , c − µ 2 � � − γ χ = A χ = lim h → 0 lim , 0 h L →∞ Scaling property of susceptibility h dependence of <φ> h,∞ /h near μ 0,c 1e+10 6e-06 Symm. Phase near μ 0,c 1e+09 5e-06 1e+08 1e+07 4e-06 1e+06 χ -1/1.75 < φ >/h 3e-06 1e+05 1e+04 2e-06 1e+03 λ=0.05, D=32, K=256 1e-06 λ=0.05, D=32, K=256, L≥10 6 1e+02 1e+01 0 1e-12 1e-10 1e-08 1e-06 1e-04 1e-02 -0.1006180 -0.1006176 -0.1006172 -0.1006168 2 h µ 0 Scaling property is well described by 2D Ising universality class (γ=1.75) Consider dimensionless quantity λ/(μ c ) 2 to take the continuum limit 12

Continuum Limit of Critical Coupling Kadoh+, JHEP1905(2019)184 Comparison w/ recent Monte Carlo studies Schaich and Loinaz: cluster (2009) 11.6 Wozar and Wipf: with SLAC derivative (2012) Bosetti et al.: worm (2015) Bronzin et al.: worm with gradient flow (2018) This work 11.4 11.2 2 λ / µ c 11 10.8 10.6 10.4 0 0.02 0.04 0.06 0.08 0.1 λ λ/(μ c ) 2 =10.913(56) in the continuum limit (λ→0) Consistent with recent Monte Carlo results 13

2D Complex φ 4 Theory at Finite Density Kadoh+, in preparation Continuum action of 2D complex φ 4 theory at finite μ | ∂ ρ φ | 2 + ( m 2 − µ 2 ) | φ | 2 + µ ( φ ∗ ∂ 2 φ − ∂ 2 φ ∗ φ ) + λ | φ | 4 � d 2 x � � S cont = Introduction of finite chemical potential ⇒ complex action Lattice action � Z (original) = D φ 1 D φ 2 exp( − S ) ⎡ ⎤ 2 ⎣ (4 + m 2 ) | φ n | 2 + λ | φ n | 4 − � � e µ δ ρ , 2 φ ∗ ρ + e − µ δ ρ , 2 φ ∗ S = n φ n +ˆ ρ φ n � � ⎢ ⎥ n +ˆ ⎦ n ρ =1 TN representation is constructed in the same way as for the real φ 4 case Bose condensation is expected to occur at sufficiently large μ

Simple(st) Test Bed for Sign Problem Mori-Kashiwa-Onishi, PTEP(2018)023B04 Previous study with path optimization method (Monte Carlo) 1 1 L =4 L =6 Average Phase Factor 0.8 0.8 L =8 Re < e i θ > pq Re < e i θ > pq 0.6 0.6 ⟨ e i θ ⟩ = Z/Z pq 0.4 0.4 � Z pq = D φ 1 D φ 2 exp( − Re( S )) 0.2 0.2 L =4 L =6 0 0 L =8 0 0.5 1 1.5 2 0 0.5 1 1.5 2 µ µ 16 <e iθ > becomes close to 1 L =4 14 L =6 12 Still, it seems difficult to perform L =8 m 2 =1, λ=1 10 Mean Field approx. Re < n > a MC simulation on large L 8 6 4 2 0 -2 0 0.5 1 1.5 2 µ The authors claim ”We show that the average phase factor is significantly enhanced after the optimization and then we can safely perform the hybrid Monte Carlo method.”

Sign-Problem-Free Representation Endres, PoS(LAT2006)133 Mathematical tools Polar coordinate • φ n = ( φ n, 1 , φ n, 2 ) → ( r n cos θ n , r n sin θ n ) Character expansion • ∞ exp( x cos z ) = k = −∞ I k ( x ) exp( ikz ) � x ∈ R, z ∈ C Partition function can be expressed in a sign-problem-free form Z (positive) ⎛ ⎞ 2 � ∞ ∞ �� � � ρ =1 e − 1 ρ ) − λ 4 (4+ m 2 )( r 2 n + r 2 4 ( r 4 n + r 4 ρ ) = d r n n 2 π r n � � � n +ˆ n +ˆ ⎜ ⎟ ⎝ ⎠ 0 n n k n, 1 ,k n, 2 = −∞ ρ ) e k n, ρ µ δ ρ , 2 δ ( k n, 1 + k n, 2 − k n − ˆ · I k n, ρ (2 r n r n +ˆ 1 , 1 − k n − ˆ 2 , 2 ) , 0 TRG should work for Z(positive) Consistency check btw the results for Z(original) and Z(positive)

Results for Z(original) with TRG Kadoh+, in preparation Bose condensation at finite μ Parameters: m 2 =0.01, λ=1, K 2 =4096, D cut =64 3.5 1.1 V=2 2 × 2 2 =4 × 4 V=2 2 × 2 2 =4 × 4 V=2 4 × 2 4 =16 × 16 V=2 4 × 2 4 =16 × 16 1 3 V=2 6 × 2 6 =64 × 64 V=2 6 × 2 6 =64 × 64 0.9 2.5 V=2 8 × 2 8 =256 × 256 V=2 8 × 2 8 =256 × 256 0.8 2 <| φ | 2 > 0.7 1.5 n 0.6 1 0.5 0.5 0.4 0 0.3 -0.5 0.2 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 µ µ Average phase factor � ⟨ e i θ ⟩ = Z/Z pq Z pq = D φ 1 D φ 2 exp( − Re( S )) 1 0.9 0.8 0.7 0.6 Z / Z pq 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 µ Clear signal even in the <e iθ > � 0 region