On the Lefschetz thimbles structure of the Thirring model F. Di - PowerPoint PPT Presentation

On the Lefschetz thimbles structure of the Thirring model F. Di Renzo 1 K. Zambello 1 , 2 francesco.direnzo@pr.infn.it kevin.zambello@pr.infn.it Wuhan, 18/06/2019 1 University of Parma and INFN - Gruppo collegato di Parma 2 Speaker

Outline Introduction The sign problem Lefschetz thimbles regularization On the Lefschetz thimbles structure of the Thirring model The theory Lefschetz thimbles regularization Conclusions

Introduction The sign problem ◮ We would like to compute expectation values � O � ≡ 1 � D ψ O [ ψ ] e − S [ ψ ] Z by importance sampling, sampling fields configurations from P ∝ e − S [ ψ ] . ◮ However, there are cases of physical interests where S is complex → sign problem ◮ Some interesting approaches to attack the sign problem are those in which one complexifies the field variables (these are the subject of today’s and tomorrow’s parallel talks).

Introduction Lefschetz thimbles regularization ◮ One of such approaches is thimble regularization. Idea: complexify the degrees of freedom of the theory and deform the integration paths. Picard-Lefschetz theory: attached to each critical point p σ exists a manifold J σ s.t. � � dz n O ( z ) e − S ( z ) = dz n O ( z ) e − S R n σ e − iS I � σ σ C J σ σ ◮ The thimble J σ attached to a critical point p σ is the union of the steepest ascent paths leaving the critical points dt = ∂ ¯ dz i S , with i.c. z i ( −∞ ) = z σ, i ∂ ¯ z i Along the flow, the imaginary part of the action is constant. ◮ The tangent space at p σ is spanned by the Takagi vectors, which can be found by diagonalizing the Hessian at the critical point H ( p σ ) v ( i ) = λ σ v ( i ) i ¯

Introduction Lefschetz thimbles regularization ◮ A natural parametrization for a point on the thimble is z ∈ J σ ↔ (ˆ n , t ), where ˆ n defines the direction along which the path leaves the critical point and t is the integration time. ◮ Using this parametrization, the thimbles decomposition of an expectation value � O � takes the form n , t ) O e i ω (ˆ � i λ σ i n 2 � dt e − S eff (ˆ n , t ) � n 2 � σ n σ D ˆ i � O � = n , t ) e i ω (ˆ � i λ σ i n 2 � dt e − S eff (ˆ n , t ) � σ n σ D ˆ n 2 � i where V (ˆ n , t ) is the parallel transported basis, S eff (ˆ n , t ) = S R (ˆ n , t ) − log | detV (ˆ n , t ) | and ω (ˆ n , t ) = arg ( detV (ˆ n , t ))).

Introduction Lefschetz thimbles regularization ◮ Observe that, when only one thimble contributes, one can rewrite � O � = � Oe i ω � σ � e i ω � σ , having defined n Z ˆ � � f � σ = D ˆ Z σ f ˆ n n � i λ σ i n 2 � dt e − S eff (ˆ n , t ) Z σ = D ˆ n , Z ˆ n = (2 � i ) n Z ˆ 1 i n 2 n , t ) e − S eff (ˆ n , t ) n (2 � i λ σ � f ˆ n = i ) dt f (ˆ Z ˆ � � n ′ ← ˆ 1 , Z ˆ n ′ → importance sampling, P acc (ˆ n ) = min . Z ˆ n ◮ Can be generalized to more than one thimble: σ n σ Z σ � Oe i ω � σ � � O � = � σ n σ Z σ � e i ω � σ

On the Lefschetz thimbles structure of the Thirring model The theory ◮ Let’s consider the 0 + 1-dimensional Thirring model � S = β (1 − cos ( z n )) + log detD 1 � detD = 2 L − 1 ( cosh ( L ˆ µ + i z n ) + cosh ( L ˆ m )) , ˆ µ ≡ a µ , ˆ m = asinh ( am ) ◮ It has been shown before that one thimble is not enough to capture the full content of the theory (i.e. JHEP1511(2015)078; but see also JHEP 1605(2016)053) ◮ We wanted to try collecting contributions from different thimbles with our approach

On the Lefschetz thimbles structure of the Thirring model Critical points of the theory ( L = 2, β = 1) ◮ To test if we are able to recover the results of the full theory, we first start from the simple case of L = 2, with a coupling β = 1; the critical points are determined by imposing sinh ( L µ + i � z n ) ∂ S cosh ( L µ + i � z n ) + cosh ( Lm ) = 0 ∂ z n = β sin ( z n ) − i The second term depends on the fields only through the sum s ≡ � z n , then it must be sin ( z n ) = sin ( z ) ∀ n and z n can be either z or π − z . ◮ Critical points in the n − = 0 sector for µ = 0 . . . 2 3 2 Imag(z) 1 0 -1 -4 -2 0 2 4 Real(z)

On the Lefschetz thimbles structure of the Thirring model Stokes phenomenon ( L = 2, β = 1) ◮ We look for changes in the intersection numbers after Stokes phenomena 6 20 4 10 2 Real(S) Imag(S) 0 0 -10 -2 -20 -4 0 0.5 1 1.5 2 0 0.5 1 1.5 2 mu mu (a) SI (b) SR Note: due to a simmetry, the two thimbles that enter the thimbles decomposition at µ � 0 . 30 give conjugate contributions.

On the Lefschetz thimbles structure of the Thirring model Results from MC ( L = 2, β = 1) ◮ How to collect the contributions from more than one thimble? ◮ Last year we proposed to calculate the corrections to the semiclassical weight as � Z G n � − 1 , it turns out this doesn’t work ˆ n Z ˆ very well for this theory. ◮ Nevertheless, we can resort to the method we proposed previously: � O � = n 0 Z 0 e − iS I 0 � Oe i ω 0 � 0 + n 12 Z 12 e − iS I 12 � Oe i ω 12 � 12 n 0 Z 0 e − iS I 0 � e i ω 0 � 0 + n 12 Z 12 e − iS I 12 � e i ω 12 � 12 ≡ � Oe i ω 0 � 0 + α � Oe i ω 12 � 12 . � e i ω 0 � 0 + α � e i ω 12 � 12

On the Lefschetz thimbles structure of the Thirring model Results from MC ( L = 2, β = 1) ◮ Fermion number density 0.8 0.8 � n � � n � 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 µ µ (c) Results from σ 0 (d) Results from σ 0 , i ◮ Condensate 0.8 0.8 � ¯ χχ � � ¯ χχ � 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 µ µ (e) Results from σ 0 (f) Results from σ 0 , i

On the Lefschetz thimbles structure of the Thirring model Numerical integration ◮ At least in this case, where we only have 2 degrees of freedom, one may also compute expectation values by direct numerical integration. 2 11 0 -2 10.5 log(Zn) -4 n -6 10 -8 -10 -12 -1e-09 -5e-10 0 5e-10 1e-09 -1e-09 -5e-10 0 5e-10 1e-09 n0 n0 (g) LogZ n vs n 0 (h) n vs n 0 ◮ Strong dependence of Z n on n 0 , that is the component of the initial displacement along the tangent space at the critical point associated to the larger Takagi value. Sharp peaks in some regions of n 0 . These observations appear to hold also for higher L.

On the Lefschetz thimbles structure of the Thirring model Numerical integration ( L = 2, β = 1) ◮ Fermion number density 0.8 0.8 � n � � n � 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 µ µ (i) Results from σ 0 (j) Results from σ 0 , i ◮ Condensate 0.8 0.8 � ¯ χχ � � ¯ χχ � 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 µ µ (k) Results from σ 0 (l) Results from σ 0 , i

On the Lefschetz thimbles structure of the Thirring model Results from MC ( L = 4, β = 1) ◮ Fermion number density 1 1 � n � � n � 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 µ µ (m) Results from σ 0 (n) Results from σ 0 , i ◮ Condensate 1 1 � ¯ χχ � � ¯ χχ � 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 µ µ (o) Results from σ 0 (p) Results from σ 0 , i

On the Lefschetz thimbles structure of the Thirring model Reweighting and Taylor expansion for thimbles ◮ Reweighting: � dx O [ x ] e − β � (1 − cos ( z n )) − S F = � dx O [ x ] e − ( β − β ′ ) � (1 − cos ( z n )) e − β ′ � (1 − cos ( z n )) − S F = ◮ Taylor expansion: � � O � ( µ ) = � O � ( µ 0 ) + ∂ O � ( µ − µ 0 ) + . . . � ∂µ � µ 0 0.8 � n � 0.6 0.4 0.2 0 0 0.5 1 1.5 2 µ

Conclusions Conclusions ◮ We have studied the (0 + 1)-dimensional Thirring model for L = 2 and L = 4 with a strong coupling β = 1. ◮ Discrepancies between the analytical solution and the results from one-thimble simulations seem to disappear after collecting the contribution from the sub-leading thimble. ◮ Very preliminary, we explored reweighting and Taylor expansion for thimbles.