Hamilton dynamics for the Lefschetz thimble integration akin to the complex Langevin method

TL;DR

This work addresses the sign problem in complex-action path integrals by recasting Lefschetz thimble integration as a Hamiltonian problem in supersymmetric quantum mechanics, with an epsilon regulator to stabilize computations. By embedding the thimble structure into a complexified representation with a BRST-invariant path integral, the authors derive a SUSY QM framework in which thimbles correspond to ground-state wave-functions and demonstrate this on a zero-dimensional quartic model. Finite-epsilon calculations reveal multiple independent SUSY vacua and localized, thimble-associated wave-functions, while the $ varepsilon o0$ limit recovers conventional thimble behavior; insights into Stokes phenomena and the potential to combine thimbles via initial-condition choices are highlighted. The approach promises a numerically robust route to multi-thimble and real-time problems, offering a complementary perspective to complex Langevin methods while emphasizing the role of BRST structure and initial-state control.

Abstract

The Lefschetz thimble method, i.e., the integration along the steepest descent cycles, is an idea to evade the sign problem by complexifying the theory. We discuss that such steepest descent cycles can be identified as ground-state wave-functions of a supersymmetric Hamilton dynamics, which is described with a framework akin to the complex Langevin method. We numerically construct the wave-functions on a grid using a toy model and confirm their well-localized behavior.

Figures (4)

• Figure 1: Changes of the Lefschetz thimbles for $\omega=1-0.9\mathrm{i}$ (left) and $\omega=1-1.1\mathrm{i}$ (right) with $\lambda=1.5\mathrm{i}$ fixed. (Left) One of three thimbles as shown by the solid line contributes to the integral and two are to be dropped when $\omega=1-0.9\mathrm{i}$. (Right) All three thimbles contribute to the integral when $\omega=1-1.1\mathrm{i}$.
• Figure 2: SUSY ground state $\Psi'^{(z_{\sigma})}$ corresponding to the saddle points $z_0=0$ and $z_+=\sqrt{-\omega/\lambda}$ with $\varepsilon=1$, $\omega=1-\mathrm{i}$, and $\lambda=10^{-3}+1.5\mathrm{i}$. Denoting $\Psi'^{(z_{\sigma})}=\Psi'^{(z_{\sigma})}_1\mathrm{d} x_1 + \Psi'^{(z_{\sigma})}_2\mathrm{d} x_2$, (a) and (b) represent $\Psi'^{(z_0)}$, while (c) and (d) represent $\Psi'^{(z_+)}$.
• Figure 3: SUSY ground state $\tilde{\Psi}'^{(z_{0})}$, which is obtained starting with the saddle point $z_0$ but with the initial relative weight equal to both components.
• Figure 4: Weight functions $P(x_1,x_2)$ corresponding to the saddle point $z_0$ with $\varepsilon=0.2$. (a) and (b) show the result for $\omega=1-\mathrm{i}$, and $\lambda=10^{-3}+1.5\mathrm{i}$. (c) and (d) show the result for $\omega=1+\mathrm{i}$, and $\lambda=10^{-3}+1.5\mathrm{i}$.