Lefschetz-thimble path integral and its physical application

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1 Lefschetz-thimble path integral and its physical application Yuya Tanizaki Department of Physics, The University of Tokyo Theoretical Research Division, Nishina Center, RIKEN May 21, KEK Theory Center Collaborators: Takuya Kanazawa (RIKEN) Kouji Kashiwa (YITP, Kyoto), Hiromichi Nishimura (Bielefeld) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 1 / 41

2 Motivation Introduction and Motivation Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 2 / 41

3 Motivation Motivation Path integral with complex weights appear in many important physics: Finite-density lattice QCD, spin-imbalanced nonrelativistic fermions Gauge theories with topological θ terms Real-time quantum mechanics Oscillatory nature hides many important properties of partition functions. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 3 / 41

4 Motivation Example: Airy integral Let s consider a one-dimensional oscillatory integration: ( ) dx x 3 Ai(a) = 2π exp i 3 + ax. R RHS is well defined only if Ima = 0, though Ai(z) is holomorphic * cos(x 3 /3+x) exp(-0.5x)*cos(x 3 /3) Figure : Real parts of integrands for a = 1 ( 10) & a = 0.5i Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 4 / 41

5 Motivation Contents How can we circumvent such oscillatory integrations? Lefschetz-thimble integrations [Witten, arxiv: , ] [YT, Koike, Ann. Physics 351 (2014) 250] Applications of this new technique for path integrals Study on phase transitions of matrix models. Lefschetz-thimble method elucidates Lee-Yang zeros. [Kanazawa, YT, JHEP 1503 (2015) 044] General theorem ensuring that Z R. Sign problem in MFA can be solved. Application of the theorem to the SU(3) matrix model [YT, Nishimura, Kashiwa, arxiv: [hep-th], to appear in PRD] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 5 / 41

6 Lefschetz-thimble integrations Introduction to Lefschetz-thimble integrations Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 6 / 41

7 Lefschetz-thimble integrations Introduction to Lefschetz-thimble integrations Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 7 / 41

8 Lefschetz-thimble integrations Lefschetz-thimble method = Steepest descent integration Oscillatory integrals with many variables can be evaluated using the steepest descent cycles J σ : d n x e is(x) = n σ d R n σ J n z e is(z). σ J σ are called Lefschetz thimbles, and Im[iS] is constant on it. n σ : intersection numbers of duals K σ and R n. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 8 / 41

9 Lefschetz-thimble integrations Example: Airy integral Airy integral: Ai(a) = R ( ) dx x 3 2π exp i 3 + ax. The integrand is holomorphic w.r.t x The contour can be deformed continuously without changing the value of the integration! Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 9 / 41

10 Lefschetz-thimble integrations Rewrite the Airy integral There exists two Lefschetz thimbles J σ (σ = 1, 2) for the Airy integral: Ai(a) = ( ) dz z 3 n σ σ J σ 2π exp i 3 + az. n σ : intersection number of the steepest ascent contour K σ and R. Figure : Lefschetz thimbles J and duals K (a = exp(0.1i), exp(πi)) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 10 / 41

11 Lefschetz-thimble integrations Tips: Airy integral & Airy equation Let us consider the equation of motion : dz d /3+az) 2π dz ei(z3 = 0. This is nothing but the Airy equation: ( ) d 2 da a Ai(a) = 0. 2 Two possible integration contours = Two linearly independent solutions of eom. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 11 / 41

12 Lefschetz-thimble integrations Generalization to multiple integrals Model integral: Z = dx 1 dx n exp S(x i ). R n What properties are required for Lefschetz thimbles J? 1 J should be an n-dimensional object in C n. 2 Im[S] should be constant on J. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 12 / 41

13 Lefschetz-thimble integrations Short note on technical aspects Complexified variables (a = 1,..., n): z a = x a + ip a. Regard x a as coordinates and p a as momenta, so that Poisson bracket is given by {f, g} = [ f g g ] f. x a p a x a p a a=1,2 Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 13 / 41

14 Lefschetz-thimble integrations Short note on technical aspects Hamilton equation with the Hamiltonian H = Im[S(z a )]: ( df(x, p) = {H, f} dz ( ) ) a S dt dt = z a This is Morse s flow equation (= gradient flow): d ( ) 2 dt Re[S] = S z 0 We can find n good directions for J around saddle points! (This is because a π/2-rot. of coord. around a saddle point multiplies ( 1) to an eigenvalue. ) [Witten, 2010] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 14 / 41

15 Lefschetz-thimble integrations Multiple integral: Lefschetz-thimble method Oscillatory integrals with many variables can be evaluated using the steepest descent cycles J σ : d n x e S(x) = n σ d R n σ J n z e S(z). σ J σ are called Lefschetz thimbles, and Im[S] is constant on it. n σ : intersection numbers of duals K σ and R n. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 15 / 41

16 Gross Neveu-like model Phase transition associated with symmetry Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 16 / 41

17 Gross Neveu-like model 0-dim. Gross Neveu-like model The partition function of our model study is the following: { Z N (G, m) = d ψdψ N exp ψ a (i/p + m)ψ a + G ( N } ) 2 ψ a ψ a. 4N a=1 a=1 ψ, ψ: 2-component Grassmannian variables with N flavors. /p = p 1 σ 1 + p 2 σ 2. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 17 / 41

18 Gross Neveu-like model Hubbard Stratonovich transformation Bosonization (σ ψψ ): with Z N (G, m) = For simplicity, we put m = 0 in the following. S has three saddle points: N dσ e NS(σ), πg R S(σ) σ2 G log[p2 + (σ + m) 2 ]. 0 = S(z) z = 2z G 2z p 2 + z 2 = z = 0, ± G p 2. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 18 / 41

19 Gross Neveu-like model Behaviors of the flow Figures for G = 0.7e 0.1i, 1.1e 0.1i at p 2 = 1 [Kanazawa, YT, arxiv: ]: z = 0 is the unique critical point contributing to Z if G < p 2. All three critical points contribute to Z if G > p 2. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 19 / 41

20 Gross Neveu-like model Stokes phenomenon The difference of the way of contribution can be described by Stokes phenomenon. [Witten, arxiv: , ] Figures of G-plane for ImS(0) = ImS(z ± ) [Kanazawa, YT, arxiv: ]: I G R G Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 20 / 41

21 Gross Neveu-like model Dominance of contribution The Stokes phenomenon tells us the number of Lefschetz thimbles contributing to Z N. However, it does not tell which thimbles give main contribution. Z N # exp( NS(0)) + # exp( NS(z ± )) In order to obtain σ 0 in the large-n limit, z ± should dominate z = 0. ReS(z ± ) ReS(0) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 21 / 41

22 Gross Neveu-like model Connection with Lee Yang zero I G R G Blue line: ImS(z ± ) = ImS(0). Green line: ReS(z ± ) = ReS(0). Red points: Lee-Yang zeros at N = 40. [Kanazawa, YT, arxiv: ] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 22 / 41

23 Gross Neveu-like model Conclusions for studies with GN-like models 1 Decomposition of the integration path in terms of Lefschetz thimbles is useful to visualize different phases. 2 The possible link between Lefschetz-thimble decomposition and Lee Yang zeros is indicated. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 23 / 41

24 Gross Neveu-like model Preliminary comments on a recent paper [Nishimura, Shimasaki, arxiv: ] from the Lefschetz-thimble viewpoint Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 24 / 41

25 Gross Neveu-like model Singularity of the drift term Model: Z = dx(x + iα) p e x2 /2. Drift term in the complex Langevin equation: S z = z p z + iα. The singularity of the drift term breaks the formal proof for the correctness of CLE at the stage of integration by parts. [Nishimura, Shimasaki, arxiv: ] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 25 / 41

26 Gross Neveu-like model Expectation values of x 2 for various α CLE breaks down for α 14 at p = 50. [Nishimura, Shimasaki, arxiv: ] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 26 / 41

27 Gross Neveu-like model Structure of Lefschetz thimbles The Stokes phenomenon happens at α = 4p. y y x (a) α < 2 p x (b) α > 2 p Is this phenomenon related to the breakdown of CLE? Looks interesting. No one can answer yet, though... Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 27 / 41

28 Sign problem in MFA Formal discussion Sign problem in MFA and its solution: Formal discussion Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 28 / 41

29 Sign problem in MFA Formal discussion Motivation At finite-density QCD (in the heavy-dense limit), the Polyakov-loop effective action looks like S eff (l) d 3 x [ e µ l(x) + e µ l(x) ] R. Even after the MFA, the effective potential becomes complex! The integration over the order parameter plays a pivotal role for reality. (Fukushima, Hidaka, PRD75, ) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 29 / 41

30 Sign problem in MFA Formal discussion General set up Consider the oscillatory multiple integration, Z = d n x exp S(x). R n To ensure Z R, suppose the existence of a linear map L, satisfying S(x) = S(L x). L 2 = 1. Let s construct a systematic computational scheme of Z with Z R. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 30 / 41

31 Sign problem in MFA Formal discussion Flow eq. and Complex conjugation Morse s downward flow: dz i dt = Complex conjugation of the flow: dz i dt = ( ( S(z) ) S(z) = z i z i ). ( S(L z) z i therefore z = L z satisfy the same flow eqation: dz i dt = ( S(z ) z i ). ), Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 31 / 41

32 Sign problem in MFA Formal discussion General Theorem: Saddle points & thimbles Let us decompose the set of saddle points into three parts, Σ = Σ 0 Σ + Σ, where Σ 0 = {σ z σ = L z σ }, Σ ± = {σ ImS(z σ ) 0}. The antilinear map gives Σ + Σ. This correspondence also applies to Lefschetz thimbles. (YT, Nishimura, Kashiwa, arxiv: [hep-th]) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 32 / 41

33 Sign problem in MFA Formal discussion General Theorem The partition function: Z = n σ d σ Σ J n z exp S(z) σ 0 + d n z exp S(z). τ Σ + n τ Each term on the r.h.s. is real. J τ +J K τ (YT, Nishimura, Kashiwa, arxiv: [hep-th]) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 33 / 41

34 Sign problem in MFA Application to QCD Sign problem in MFA and its solution: Application to QCD-like theories Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 34 / 41

35 Sign problem in MFA Application to QCD The QCD partition function at finite density The QCD partition function: Z QCD = DA detm(µ qk, A) exp S YM [A], w./ the Yang-Mills action S YM = 1 2 tr β 0 dx4 d 3 x F µν 2 (> 0), and detm(µ, A) = det [ γ ν ( ν + iga ν ) + γ 4 µ qk + m qk ]. is the quark determinant. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 35 / 41

36 Sign problem in MFA Application to QCD Charge conjugation If µ qk 0, the quark determinant takes complex values Sign problem of QCD. However, Z QCD R is ensured thanks to the charge conjugation A A t : det M(µ qk, A) = det M( µ qk, A ) (First equality: γ 5 -transformation, Second equality: charge conjugation) = det M(µ qk, A). Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 36 / 41

37 Sign problem in MFA Application to QCD Polyakov-loop effective model The Polyakov line L: L = 1 3 diag [ e i(θ 1+θ 2 ), e i( θ 1+θ 2 ), e ] 2iθ 2. Let us consider the SU(3) matrix model: Z QCD = dθ 1 dθ 2 H(θ 1, θ 2 ) exp [ V eff (θ 1, θ 2 )], where H = sin 2 θ 1 sin 2 ((θ 1 + 3θ 2 )/2) sin 2 ((θ 1 3θ 2 )/2). Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 37 / 41

38 Sign problem in MFA Application to QCD Charge conjugation in the Polyakov loop model Charge conjugation acts as L L : V eff (z 1, z 2 ) = V eff (z 1, z 2 ). Simple model for dense quarks (l := trl): V eff = h (32 1) ( ) e µ l 3 (θ 1, θ 2 ) + e µ l 2 3 (θ 1, θ 2 ) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 38 / 41

39 Sign problem in MFA Application to QCD Behaviors of the flow Figure : Flow at h = 0.1 and µ = 2 in the Re(z 1 )-Im(z 2 ) plane. The black blob: a saddle point. The red solid line: Lefschetz thimble J. The green dashed line: its dual K. (YT, Nishimura, Kashiwa, arxiv: [hep-th]) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 39 / 41

40 Sign problem in MFA Application to QCD Saddle point approximation at finite density The saddle point approximation can now be performed. Polyakov-loop phases (z 1, z 2 ) takes complex values, so that l, l R. Since Im(z 2 ) < 0 and l 1 3 (2eiz 2 cos θ 1 + e 2iz 2 ), we can confirm that l > l. (YT, Nishimura, Kashiwa, arxiv: [hep-th]) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 40 / 41

41 Sign problem in MFA Application to QCD Summary for the sign problem in MFA Lefschetz-thimble integral is a useful tool to treat multiple integrals. Saddle point approximation can be applied without violating Z R. Sign problem of effective models of QCD is (partly) explored. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 41 / 41