A lattice study of N =2 Landau-Ginzburg model using a Nicolai map - PowerPoint PPT Presentation

a A lattice study of N =2 Landau-Ginzburg model using a Nicolai map based on arXiv:1005.4671 Hiroki Kawai (in collaboration with Y.Kikukawa) Institute of Physics, The University of Tokyo

Outline 1.Purpose of this study 2.Lattice formulation of WZ model 3.Simulation Method 4.Numerical results 5.Summary and future plan

1 Purposeg 2d CFT ● critical phenomena of 2d statistical systems P c i = 9 = ⇒ N = 1 space-time SUSY (compactified string) ● N = 2 minimal models : ● A problem which remains unsolved is the determination of the correspondence between CFTs and systems (Lagrangians) . 2d N =2 Landau-Ginzburg model (LG model) � � � � d 2 xd 4 θ K (Φ , ¯ d 2 xd 2 θ W (Φ) + c.c. Φ … chiral superfield. S = Φ) + , At the IR fixed point, W (Φ) = λ Φ n is believed to describe the N = 2 , c = 3(1 − 2 n ) minimal model. ✕ ✁ → check for K (Φ , ¯ Φ) = ¯ ֒ ΦΦ (WZ model) ✁ λ eff → ∞ , lattice ! Why it is believed that LG models describe CFTs ? ● 2d bosonic case ’86 A.B.Zamolodchikov … φ 2 n − 3 6 (2 , 2) ∝ ∂ 2 φ (2 , 2) In the c = 1 − n ( n +1) minimal model, the fusion rule implies … φ 2 n − 3 ∝ ∂ 2 φ In the 2d bosonic LG model L = 1 2 ∂ µ φ∂ µ φ + λφ 2 n − 2 , EOM is conjecture ⇒ φ = φ (2 , 2) at the IR fixed point. Extending this idea, ...

How to check the conjecture ● early studies → ’89 Kastor, Martinec and Shenker RG flow of c -functions → ’89 Vafa and Warner → For W (Φ) = λ Φ n , catastrophe theory  → ’89 Howe and West c = 3(1 − 2 ϵ -expansion n )     → ’93 Witten Φ : ( h, ¯  h ) = ( 1 2 n , 1 elliptic genus 2 n )   Φ 2 : ( h, ¯ h ) = ( 2 2 n , 2 2 n ) ...   ：     Φ n − 2 : ( h, ¯  h ) = ( n − 2 2 n , n − 2 2 n ) ● We computed correlation functions non-perturbatively for W (Φ) ∝ Φ 3 . susceptibility of CFT: � � 1 finite volume h ∝ V 1 − h − ¯ d 2 x 〈 φ ( x ) φ ∗ (0) 〉 d 2 x h χ ≡ − → | x | 2 h +2¯ V ⇒ log χ = (1 − h − ¯ h ) log V + const. ✁ ✕ ✁ ✁ 6 = 0.666... For the present W (Φ) ∝ Φ 3 , the conjecture expects 1 − h − ¯ h = 1 − 1 6 − 1

2 Lattice Formulation of WZ modelg Relying on the existence of the Nicolai map as the guiding principle, ’83 Sakai and Sakamoto ’09 Kadoh and Suzuki � � � � φ ∗ Tφ + W ∗ (1 − a 2 ’02 Kikukawa and Nakayama W ′ ( − S 1 + iS 2 ) φ + c.c. S = 4 T ) W + � � � D + 1 + γ 3 W ′′ 1 + ˆ + 1 − γ 3 W ′′∗ 1 − ˆ γ 3 γ 3 + ¯ ψ ψ 2 2 2 2 � � D = 1 X X = 1 − a � � γ µ ( ∇ + µ − ∇ − µ ) − a ∇ + µ ∇ − √ where 1 + = T + γ 1 S 1 + γ 2 S 2 , , µ 2 2 X † X W = λ E 3 Φ 3 . 1 ✻ a � enough continuum limit : aλ → 0 modes ! λ ● λ is the unique mass parameter (besides a ) ⇒ ✶ ✏ ✏ To see CFT, L ≫ ( aλ ) − 1 is needed. 0 � ◎ one SUSY Q ← Nicolai map ● no extra fine-tunings ⇐ ◎ Z 3 R-symmetry ← overlap fermion ● This lattice model faces the sign problem | D + F | is real, but can be negative. ⇐ γ 1 ( D + F ) γ 1 = ( D + F ) ∗ � � � � � � e − S lat. = Π n dφ n dφ ∗ e − S B Π n dφ n · · · | D + F | n � �� � real, but can be negative.

3 Simulation Methodg Idea ’91 Curci et al. We utilized the Nicolai map : η = W ′ + ( φ − a 2 W ′ ) T + ( φ ∗ − a 2 W ∗′ )( S 1 + iS 2 ) . � � � e − S lat. Π n dφ n · · · � D φ D φ ∗ ≡ Π n dφ n dφ ∗ D φ D φ ∗ | D + F | e − S B , = n � � �� � η − W ′ − ( φ − a 2 W ′ ) T − ( φ ∗ − a � D φ D φ ∗ D η D η ∗ δ 2 W ∗′ )( S 1 + iS 2 ) | D + F | e − S B = N ( η ) � � � � � � φ − φ i ( η ) δ � D φ D φ ∗ D η D η ∗ | D + F | e − S B = || D + F || i =1 � N ( η ) � � � n | η n | 2 . D η D η ∗ e − P = sgn | D + F ( φ i ) | i =1 � � N ( η ) � � D η D η ∗ X e − P i =1 O ( φ i ) sgn | D + F ( φ i ) | n | η n | 2 η ⇒ 〈O〉 = where 〈 X 〉 η ≡ , . � � N ( η ) � � n | η n | 2 D η D η ∗ e − P i =1 sgn | D + F ( φ i ) | η � �� � ❇ ▼ positive ❇ ❇ � d 2 x 〈 φ ( x ) φ ∗ (0) 〉 . Using this expression, we calculated the susceptibility χ =

Algorithm � � N ( η ) � i =1 O ( φ i ) sgn | D + F ( φ i ) | η 〈O〉 = � � N ( η ) � a → 0 i =1 sgn | D + F ( φ i ) | → Witten index ∆ = 2 (cubic potential) η  R � � x | η | 2 Π n dη n dη ∗ X ( η ) e − P  n 〈 X 〉 η ≡  R � �  x | η | 2 e − P Π n dη n dη ∗ where n    N ( η ) counts the solutions of the Nicolai map φ 1 , .., φ N ( η ) η = W ′ + ( φ − a 2 W ′ ) T + ( φ ∗ − a 2 W ∗′ )( S 1 + iS 2 ) 1. Assigning { η , η ∗ } as the standard normal distribution, 2. Solving the Nicolai map by the Newton-Raphson algorithm, 3. We sample the configurations of { φ , φ ∗ } . … no sign problem, no autocorrelation ● advantage … N ( η ) ● difficulty

Tests for the configurations ● 〈 � N ( η ) a → 0 i =1 sgn | D + F |〉 η − → Witten index ∆ = 2 (cubic potential) Why Witten index ? D + m (1 − a 2 Φ 2 (∆ = 1) , (Re η, Im η ) = (Re φ, Im φ ) � � → P.B.C. & For W (Φ) = m 2 D ) � �� � → correctly normalized positive ⇒ ∆=1 is correctly reproduced � � η ( x 1 ) · · · η ( x m ) η ∗ ( y 1 ) · · · η ∗ ( y n ) ● Ward identity for on the lattice Qη ∗ = From Qψ + = − η ∗ , δ δ Qψ − = − η , Qη = δψ + S lat. , δψ − S lat. , 〈 Q ( · · · ) 〉 = 0 , and the Schwinger-Dyson eq. , � � η ( x 1 ) · · · η ∗ ( y n ) � N ( η ) � i =1 sgn | D + F | 0 m ̸ = n η = � � � N ( η ) � σ Π m m = n. k =1 δ x k ,y σ ( k ) i =1 sgn | D + F | η For example, m = n = 1 provides � x 〈 η ( x ) η ∗ ( x ) 〉 = 〈 S B 〉 = L 2 . ⇒ If � N ( η ) i =1 sgn | D + F | = 2 over the η , OK.

4 Numerical Resultsg Samples with W (Φ) = λ 3 Φ 3 , aλ = 0 . 3 , L = 18 , 20 , ..., 32 (Newton iter. from 100 initial config. for each noise) × 320 noises 18 20 22 24 26 28 30 32 L test ... ❳ ② ❳ P sgn | D + F | = 2 (+ , +) 316 319 319 316 316 314 307 316 ✘ ✘ ✾ ( − , + , + , +) 3 0 1 3 4 6 10 4 ❳ ② ❳ P sgn | D + F | ̸ = 2 (+) 1 1 0 0 0 0 1 0 ✘ ✘ ✾ (+ , + , +) 0 0 0 1 0 0 2 0 ,but rare. ∆ 1.997 1.997 2 2.003 2 2 1.994 2 δ [%] 0.3 0.0 0.1 0.4 0.4 0.4 0.4 0.2 δ … 〈 S B 〉− L 2 ∆ … Witten index, (a Ward identity) L 2 ⇒ For 99% noises, � N ( η ) i =1 sgn | D + F | = 2 ⇒ Witten index ∆ = 2 and Ward identities are well reproduced.

Susceptibility: χ φ ≡ � x ≥ 3 〈 φ ( x ) φ (0) 〉 W (Φ) = λ 3 Φ 3 , aλ = 0 . 3 , L = 18 , 20 , .., 32 4.8 4.5 linear fit by least-square-method ❅ ❅ ❘ ln χ φ 4.2 3.9 6 6.5 7 ln L 2 ⇒ χ φ ∝ V 0.660 ± 0.011 ⇒ consistent with the conjecture χ φ ∝ V 0.666... ◎

5 Summary and future plan Summary � V d x 2 〈 φ ( x ) φ ∗ (0) 〉 in the cubic potential case, and got the consistent result with • We observed χ = the conjecture χ ∼ V 0 . 666 ... . • We also extracted the effective coupling constant K of the Gaussian model, 3 and obtained K = 0 . 242 ± 0 . 010 which is consistent with the N = 2 SUSY point K = 4 π = 0 . 238 .. . This implies the restoration of all supersymmetries in the IR. (see more detail in arXiv:1005.4671) Future Plan • further check of the A-D-E classification: → A 3 model ？ Φ 4 W = Φ 3 + Φ ′ 4 → E 6 = A 2 ⊗ A 3 model ？ Φ 2 + ΦΦ ′ 2 → D 3 model ？ , ... • c-function → central charge, c-theorem • 2d N = 1 LG model with W ∝ Φ 3 ( infrared → tricritical ising model) ⇒ dynamical SUSY breaking

a Appendix

Lattice formulation of WZ model ● continuum theory � � � ∂φ ∗ − W ′ ψ + − W ∗′ ψ − − Hψ − + 2 ψ + ¯ d 2 x E S cont. = Q � � � + W ∗′′ 1 − γ 3 γ µ ∂ µ + W ′′ 1 + γ 3 ∂ µ φ ∗ ∂ µ φ + | W ′ | 2 + ¯ � � d 2 x E = H -onshell. ψ ψ , 2 2 notation γ 1 = σ 3 , γ 2 = − σ 2 , γ 3 = − iγ 1 γ 2 = σ 1 , � � ψ 1 , ¯ ψ = ( ¯ ψ 1 , ¯ 2 ( ψ 1 ± ψ 2 ) , ¯ 2 ( ¯ ψ 1 ∓ ¯ 1 1 ψ 2 ) , ∂ = 1 ψ = ψ 2 ) , ψ ± = ψ ± = 2 ( ∂ 1 − i∂ 2 ) and √ √ ψ 2  Qφ ∗ = − ¯ Qφ = − ¯ Q ¯ ψ − , ψ + , ψ ± = 0 ,  �  Q 2 = 0 ∂φ ∗ + H ∗ , Qψ − = 2¯ ⇒ QS cont. = Q 2 ( · · · ) = 0 . Qψ + = 2 ∂φ + H,  QH ∗ = 2¯ QH = 2 ∂ ¯ ∂ ¯  ψ − , ψ + , symmetry N = 2 SUSY, SO(2), translation, U (1) R ( φ → e − 2 iα φ, ψ → e iαγ 3 ψ, ¯ ψ → ¯ ψe iαγ 3 ) for W = λ 3 φ 3 U (1) V ,