Thimble regularization at work: from toy models to chiral random matrix theories

TL;DR

The paper investigates Lefschetz thimble regularization as a strategy to tame the sign problem in complex actions, applying it to a simple $0$-dimensional $\phi^4$ toy model and to a chiral random matrix theory (CRM). By decomposing the path integral into thimbles tied to complex critical points and using steepest ascent/descent flows, the authors explore how many thimbles contribute and how to sample configurations on them, validating against exact results. They implement and compare several algorithms (Aurora Langevin, Gaussian thimble, Metropolis) and address the role of the residual phase, finding that in the CRM setup a single thimble often dominates in the studied region, while the sign problem persists in others; a crude Monte Carlo over ascents demonstrates the viability of the approach. The work highlights both the viability of thimble regularization for non-perturbative problems and the key algorithmic challenges ahead, notably efficient importance sampling across multiple thimbles and accurate density-of-states calculations for larger systems.

Abstract

We apply the Lefschetz thimble formulation of field theories to a couple of different problems. We first address the solution of a complex 0-dimensional phi^4 theory. Although very simple, this toy-model makes us appreciate a few key issues of the method. In particular, we will solve the model by a correct accounting of all the thimbles giving a contribution to the partition function and we will discuss a number of algorithmic solutions to simulate this (simple) model. We will then move to a chiral random matrix (CRM) theory. This is a somehow more realistic setting, giving us once again the chance to tackle the same couple of fundamental questions: how many thimbles contribute to the solution? how can we make sure that we correctly sample configurations on the thimble? Since the exact result is known for the observable we study (a condensate), we can verify that, in the region of parameters we studied, only one thimble contributes and that the algorithmic solution that we set up works well, despite its very crude nature. The deviation of results from phase quenched results highlights that in a certain region of parameter space there is a quite important sign problem. In view of this, the success of our thimble approach is quite a significant one.

Figures (8)

• Figure 1: Thimbles structure for $\sigma = 0.5 + i 0.75$, $\lambda = 2$ (left panel). In this case only the unstable thimble attached to $z = 0$ intersects the real axis and thus only one critical point contributes. On the right we can see how the Langevin simulation correctly covers the relevant thimble.
• Figure 2: Thimbles structure for $\sigma = -0.5 + i 0.75$, (left panel) and $\sigma = i 0.75$ (right); in both cases $\lambda = 2$. For $\sigma_R < 0$ (left) all the three critical points contribute. $\sigma_R = 0$ (right) is an example of a Stokes phenomenon.
• Figure 3: Left panel: the three thimbles associated to $\sigma = -0.5 + i 0.75$ correctly sampled by a Metropolis simulation. Right panel: for the same choice of parameters, the computed values of $\langle \phi^8 \rangle$ over a range of both $\sigma_R>0$ and $\sigma_R<0$.
• Figure 4: Exact (solid red line) and phase quenched (dashed blue line) results for the condensate, at fixed $N_f=2$, $\tilde{\mu}=2$.
• Figure 5: Exact (solid red line), phase quenched (dashed blue line) and gaussian approximation results for the condensate, at fixed $N_f=2$, $\tilde{\mu}=2$.