Introduction to Lefschetz-thimble path integral and its applications to physics

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1 Introduction to Lefschetz-thimble path integral and its applications to physics Yuya Tanizaki Department of Physics, The University of Tokyo Theoretical Research Division, Nishina Center, RIKEN February 20, Keio University Collaborators: Takayuki Koike (Tokyo), Takuya Kanazawa (RIKEN) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 1 / 38

2 Motivation Introduction and Motivation Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 2 / 38

3 Motivation Motivation Partition function Z = tr[exp βĥ(ˆp, ˆx)] Path integral representation Z = Infinitely multiple integration! β DxDp exp dt (ipẋ H(p, x)). 0 Mean field theory + perturbations Monte Carlo sampling Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 3 / 38

4 Motivation QCD phase diagram(?) Temperature T sqgp Critical Point Quark-Gluon Plasma Inhomogeneous ScB Hadronic Phase Liquid-Gas Nuclear Superfluid Quarkyonic Matter? 0 2SC usc dsc CFL Color Superconductors CFL-K, Crystalline CSC Meson supercurrent Baryon Chemical Potential mb Gluonic phase, Mixed phase (Fukushima, Hatsuda, 2010) Z(β, µ) = DA det(γ i D i (µ, A)) exp 1 β d 4 xtrf 2 2g 2 Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 4 / 38

5 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 Feb 20, Keio U. 5 / 38

6 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 Feb 20, Keio U. 6 / 38

7 Motivation Contents How can we circumvent such oscillatory integrations? Lefschetz-thimble integrations [Witten, arxiv: , ] Applications of this new technique for path integrals Phase transition of the matrix model [YT, Phys. Rev. D91, (2015); Kanazawa, YT, arxiv: ] Quantum tunneling in the real-time path integral [YT, Koike, Ann. Physics 351 (2014) 250] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 7 / 38

8 Lefschetz-thimble integrations Introduction to Lefschetz-thimble integrations Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 8 / 38

9 Lefschetz-thimble integrations Again 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 Feb 20, Keio U. 9 / 38

10 Lefschetz-thimble integrations Steepest descent contours What is the most appropriate contour for our purpose? Re[iS(x, a)] should be made as small as possible. The contour should be perpendicular to Re[iS(x, a)] = cosnt. Steepest descent ones J must satisfy Im[iS(x, a)] = const. because of the holomorphy. Figure : Contour plots for Re[iS(x, a)] with a = exp ±i0.1. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 10 / 38

11 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 Feb 20, Keio U. 11 / 38

12 Lefschetz-thimble integrations 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 Feb 20, Keio U. 12 / 38

13 Lefschetz-thimble integrations Generalization to multiple integrals Model integral: Z = dx 1 dx n exp I(x i ). R n What properties are required for Lefschetz thimbles J? 1 J should be a n-dimensional object in C n. 2 Im[I] should be constant on J. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 13 / 38

14 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} = a=1,2 [ f g g ] f. x a p a x a p a Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 14 / 38

15 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} = a=1,2 [ f g g ] f. x a p a x a p a Hamilton equation with the Hamiltonian H = Im[I(z a )]: ( df(x, p) = {H, f} dz ( ) ) a I dt dt = z a d This is Morse s flow equation: Re[I] 0. dt We can find n good directions for J around saddle points! [Witten, 2010] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 14 / 38

16 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 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 Feb 20, Keio U. 15 / 38

17 Lefschetz-thimble integrations Harmonic oscillator Example of QM: Harmonic oscillator The action functional T I[z] = i dt 0 [ 1 2 ( ) ] 2 dz 1 dt 2 z2 Saddle-point condition (=Euler Lagrange eq.): d 2 z cl (t) dt 2 = z cl (t). The classical solution: z cl (t) = x f x i cos T sin T sin t + x i cos t. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 16 / 38

18 Lefschetz-thimble integrations Harmonic oscillator Example of QM: Flow equation Flow equation ( ) 2 z(t; u) = i u t + 1 z(t; u), 2 Boundary conditions: z(t; ) = 0, and z(0; u) = z(t, u) = 0. The set of solutions z are spanned by [( ( e iπ/4 exp πn ) ) ] 2 T 1 u sin nπt, (nπ > T ), T z n (t; u) = [( e iπ/4 exp 1 ( ) ) ] πn 2 u sin nπt, (nπ < T ). T T Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 17 / 38

19 Lefschetz-thimble integrations Harmonic oscillator Example of QM: Lefschetz thimble Lefschetz thimble J ( R ): { ν z cl (t) + e iπ/4 a l sin πlt T l=1 + eiπ/4 l=ν+1 a l sin πlt T } a l R ν: the maximal non-negative integer smaller than T/π. Integration measure on J becomes J Dz = N ν e iπ 4 dan n=1 m=ν+1 e iπ 4 dam = e iπν/2 N l=1 idal. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 18 / 38

20 Lefschetz-thimble integrations Harmonic oscillator Example of QM: Path integral Feynman kernel for the harmonic oscillator: K h.o. (x f, x i, T ) ( I[zcl ] = exp ( I[zcl ] = exp = i πν 2 1 2πi sin T exp ) N l i πν ) 1 2 ( I[zcl ] ( idal ( ) 2 πl exp 1 T 1 1 (T/πl) 2 2πi T l=1 i πν 2 ). Appearance of the Maslov Morse index ν is quite manifest. [YT, Koike, Ann. Physics 351 (2014) 250] a2 l ) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 19 / 38

21 Lefschetz-thimble integrations Harmonic oscillator Proposed Applications of Lefschetz-thimble methods Analytic properties of partition function for CS theories [Witten] Real-time description of tunneling [YT, Koike; Cherman, Unsal] Sign problem of the lattice Monte Carlo simulation [M. Cristoforetti, et al., Kikukawa, et al.] Geometrization of resurgence transseries [Basar, Dunne, Unsal; Cherman, Dorigoni, Unsal] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 20 / 38

22 Gross Neveu-like model Phase transition of a matrix model Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 21 / 38

23 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 The Hubbard Stratonovich transformation gives N Z N (G, m) = dσ e NS(σ), πg R with S(σ) σ2 G log[p2 + (σ + m) 2 ]. For simplicity, we put m = 0 in the following. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 22 / 38

24 Gross Neveu-like model Properties of S(σ) S has three saddle points: 0 = S(z) z = 2z G 2z p 2 + z 2 = z = 0, ± G p 2. Figures for G = 0.7e 0.1i and p 2 = 1 [Kanazawa, YT, arxiv: ]: Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 23 / 38

25 Gross Neveu-like model Properties of S(σ) Figures for G = 1.1e 0.1i and p 2 = 1 [Kanazawa, YT, arxiv: ]: From these figures, we learn that, for real G, 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 Feb 20, Keio U. 24 / 38

26 Gross Neveu-like model Stokes phenomenon The difference of the way of contribution can be described by Stokes phenomenon. At some special values of coupling, several critical points are connected by the flow. [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 Feb 20, Keio U. 25 / 38

27 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. 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 Feb 20, Keio U. 26 / 38

28 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 Feb 20, Keio U. 27 / 38

29 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 thimble decomposition and Lee Yang zeros is indicated. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 28 / 38

30 Resergence theory Resurgence theory of the perturbation series Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 29 / 38

31 Resergence theory Perturbation theory Let us consider the integral [ Z(λ) = dx exp 1 ( )] x 4 λ 4 x2. 2 Gaussian approximation around x = ±1 gives R Z(λ) e 1/4λ λ a n = 2 π n! a n λ n n=0 n ((2m 1) 2 1/4) m=1 (a n is growing factorially in terms of n. ) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 30 / 38

32 Resergence theory Non-convergent series M-test says the convergence radius λ c is λ c a n a n+1 = n + 1 (2n + 1) 2 1/4 0. Perturbation series = Asymptotic series (Non-convergent!) Can we make sense of it? Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 31 / 38

33 Resergence theory Define Borel summation BP (t) = n=0 a n n! tn. This is convergent at least for small t. Since 0 dtt n e t/λ = n!λ n+1, we might be able to consider the following series B(λ) = e1/4λ dtbp (t)e t/λ λ However, BP (t) can have a pole on R +. This integral becomes ambiguous. 0 Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 32 / 38

34 Resergence theory Interpreting Borel ambiguity via Lefschetz thimbles y O 1 2 x 1 2 Ambiguous imaginary part of the Borel sum J 0 is missed!. (Resurgence trans-series theory: Écalle, Voros, etc. ) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 33 / 38

35 Resergence theory Lesson & application to QFT Lesson: Large order perturbation theory can become ambiguous even after resummation of the series. Such ambiguity provides information on contributions of non-trivial classical solutions. Application to QM & QFT This provides an exact semiclassical treatment of QM & QFT. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 34 / 38

36 Resergence theory Example of nontrivial solutions: Gauge theory (KvBLL caloron) Diakonov et. al., Phys. Rev. D70, (2004) These are related to dynamics of QCD! (Ünsal, Schaefer, Poppitz, Dünne, etc.) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 35 / 38

37 Resergence theory Example of nontrivial solution: Double-well QM Imaginary-time instantons as exact solutions (YT, Koike, Ann. Phys. 351 (2014) 250): t Re z Im z 1.0 Re z Im z S classical 4 (n + m) 3 Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 36 / 38

38 Resergence theory Example of nontrivial solution: Double-well QM Real-time instanton solutions (YT, Koike, Ann. Phys. 351 (2014) 250; Cherman, Ünsal, arxiv: ): t t 5 10 Rez Rez 10 Imz 20 Imz Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 37 / 38

39 Summary Summary Lefschetz-thimble integral is a useful tool to treat multiple integrals. Semiclassical expansion of QM can be reformulated in a neat way with this technique. Perturbation theory in a strong coupling regime (?) Sign problem of Monte Carlo study (?) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral Feb 20, Keio U. 38 / 38