# Directed-Loop Quantum Monte Carlo Method for Retarded Interactions

## TL;DR

This paper extends the directed-loop quantum Monte Carlo method to efficiently handle retarded interactions in fermion-boson and spin-boson models, enabling faster simulations of complex quantum phase transitions.

## Contribution

The authors develop an exact quantum Monte Carlo algorithm that incorporates retarded interactions using path integrals, improving efficiency over previous methods.

## Key findings

- Overcomes long autocorrelation times in simulations.
- Achieves exponential speedup in sampling.
- Enables study of large chains up to 1282 sites.

## Abstract

The directed-loop quantum Monte Carlo method is generalized to the case of retarded interactions. Using the path integral, fermion-boson or spin-boson models are mapped to actions with retarded interactions by analytically integrating out the bosons. This yields an exact algorithm that combines the highly-efficient loop updates available in the stochastic series expansion representation with the advantages of avoiding a direct sampling of the bosons. The application to electron-phonon models reveals that the method overcomes the previously detrimental issues of long autocorrelation times and exponentially decreasing acceptance rates. For example, the resulting dramatic speedup allows us to investigate the Peierls quantum phase transition on chains of up to $1282$ sites.