A novel method to evaluate real-time path integral for scalar $\phi^4$ theory

TL;DR

This paper introduces a new numerical scheme for evaluating real-time path integrals in scalar  theory, combining complex contour deformation, quadrature, and tensor network techniques for improved stability and efficiency.

Contribution

The authors develop a novel method that reformulates the real-time path integral into a convergent form using complex contour deformation and tensor networks, enabling practical computations.

Findings

Successfully evaluated time-correlator in 1+1 dimensions.

Results agree with exact solutions at small volumes.

Stable results in larger volumes with coarse-graining.

Abstract

We present a new scheme which numerically evaluates the real-time path integral for $\phi^4$ real scalar field theory in a lattice version of the closed-time formalism. First step of the scheme is to rewrite the path integral in an explicitly convergent form by applying Cauchy's integral theorem to each scalar field. In the step an integration path for the scalar field is deformed on a complex plane such that the $\phi^4$ term becomes a damping factor in the path integral. Secondly the integrations of the complexified scalar fields are discretized by the Gauss-Hermite quadrature and then the path integral turns out to be a multiple sum. Finally in order to efficiently evaluate the summation we apply information compression technique using the singular value decomposition to the discretized path integral, then a tensor network representation for the path integral is obtained after integrating the discretized fields. As a demonstration, by using the resulting tensor network we numerically evaluate the time-correlator in 1+1 dimensional system. For confirmation, we compare our result with the exact one at small spatial volume. Furthermore, we show the correlator in relatively large volume using a coarse-graining scheme and verify that the result is stable against changes of a truncation order for the coarse-graining scheme.