Lefschetz-thimble path integral for solving the mean-field sign problem

Transcription

1 Lefschetz-thimble path integral for solving the mean-field sign problem Yuya Tanizaki Department of Physics, The University of Tokyo Theoretical Research Division, Nishina Center, RIKEN Jul 15, LATTICE 2015 Collaborators: Kouji Kashiwa (YITP, Kyoto), Hiromichi Nishimura (Bielefeld) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral July 15, Kobe 1 / 17

2 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 July 15, Kobe 2 / 17

3 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 July 15, Kobe 3 / 17

4 Lefschetz-thimble integrations First thing first Lefschetz decomposition formula 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[iS] is constant on it: dz i dt = ( S(z) z i ). n σ : intersection numbers of duals K σ and R n. [Witten, arxiv: , ] ( Ünsal, Friday 9:00-; Scorzato, Sat. 14:15-) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral July 15, Kobe 4 / 17

5 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 July 15, Kobe 5 / 17

6 : 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, ) Question Can we make a tied connection bet. the saddle-point approximation and the mean-field approximation with complex S? Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral July 15, Kobe 6 / 17

7 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 July 15, Kobe 7 / 17

8 Charge conjugation in the Polyakov loop model Charge conjugation acts as L(θ 1, θ 2 ) L (θ 1, θ 2 ) L(θ 1, θ 2 ): 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 July 15, Kobe 8 / 17

9 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, PRD91, ) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral July 15, Kobe 9 / 17

10 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, PRD91, ) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral July 15, Kobe 10 / 17

11 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 July 15, Kobe 11 / 17

12 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 July 15, Kobe 12 / 17

13 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, PRD91, ) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral July 15, Kobe 13 / 17

14 General Theorem The partition function: Z = σ Σ 0 n σ J σ d n z exp S(z) + τ Σ + n τ Each term on the r.h.s. is real. (YT, Nishimura, Kashiwa, PRD91, ) J τ +J K τ d n z exp S(z). Consequence Charge conjugation ensures the reality of physical observables manifestly in the Lefschetz-thimble decomposition. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral July 15, Kobe 14 / 17

15 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 July 15, Kobe 15 / 17

16 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 ) = det M(µ qk, A). (First equality: γ 5 -transformation, Second one: charge conjugation) Consequence Our theorem applies to the (lattie) QCD: The thimble decomposition manifestly ensures the reality of physical observables. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral July 15, Kobe 16 / 17

17 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 July 15, Kobe 17 / 17