Taylor expansions on Lefschetz thimbles

Thimble regularization is a possible solution to the sign problem, which is evaded by formulating quantum field theories on manifolds where the imaginary part of the action stays constant (Lefschetz thimbles). A major obstacle is due to the fact that one in general needs to collect contributions coming from more than one thimble. Here we explore the idea of performing Taylor expansions on Lefschetz thimbles. We show that in some cases we can compute expansions in regions where only the dominant thimble contributes to the result in such a way that these (different, disjoint) regions can be bridged. This can most effectively be done via Padé approximants. In this way multi-thimble simulations can be circumvented. The approach can be trusted provided we can show that the analytic continuation we are performing is a legitimate one, which thing we can indeed show. We briefly discuss two prototypal computations, for which we obtained a very good control on the analytical structure (and singularities) of the results. All in all, the main strategy that we adopt is supposed to be valuable not only in the thimble approach, which thing we finally discuss.

Figure 1

Thimbles structure for a 0-dim ${\varphi }^4$ toy model: continuous (blue) lines are stable thimbles; dashed (red) lines are unstable thimbles. The three panels refer to different points in the parameter space of the theory. In the middle, an example of a Stokes phenomenon. For further details, see [20].

Figure 2

(Left panel) The flow lines highlighting the thimbles structure of the 1-dim Thirring model at $\frac{\mu }{m}=1.4$: stable thimbles are depicted in blue, unstable thimbles in magenta. The dominant thimble is associated to the critical point sitting at $\Re (z)=0$. The critical point ${\sigma }_1$ is the closest to the latter to the right (there is a mirror image to the left as well): notice that the unstable thimble associated to it does not intersect the original domain of integration (which is on the real axis). (Center panel) The chiral condensate as obtained from the analytic solution (continuous black line) and from our Padé approximant (we plot points instead of a continuum line so that the size of errors are easier to spot.). The points providing input to the evaluation of Padé are marked as triangles. (Right panel) Singularity of the solution in the complex plane: red point computed from the analytic solution, green point is the only pole of our Padé approximant. We plot the radii of convergence which are relevant for the expansions at hand: our analytic continuation indeed stands on firm ground.

Figure 3

Left panel: semiclassical evaluation of the relative weight of different thimbles at $\frac{\mu }{m}=0.9995$: weights are plotted vs the value of the (real) action. The dominant thimble has de facto weight 1 (it is very hard to detect the weight of the thimble with action near the value $-60$). Center panel: the known analytic value for the quark number density is correctly reproduced over the entire relevant range of $\frac{\mu }{m}$ via the Padé approximant; the points entering the computation are marked as triangles (see main text for details). Right panel: the known singularity of the solution as analytically known (red point) and as reconstructed by the Padé approximant (green point).

Figure 4

Left panel: we plot for the 0-dim ${\varphi }^4$ toy model the semiclassical evaluation of the relative weight of thimbles attached to ${\varphi }_{\pm }$ critical points. Weight is plotted vs ${\sigma }_R$. The weight of ${\varphi }_0$ is $1-W$. Right to the left of the Stokes point (${\sigma }_R=0$) the three thimbles provide comparable contributions, while as ${\sigma }_R$ gets more and more negative the weight of ${\varphi }_{\pm }$ is more and more enhanced. Right panel: the value of the moment $⟨{\varphi }^8⟩$ is correctly reproduced over the entire negative semiaxis ${\sigma }_R<0$ via the Padé approximant; the points entering the computation are marked as triangles, where at the point to the left the expansion is computed on the thimble attached to ${\varphi }_{+}$ (or ${\varphi }_{-}$; see main text for details).