Scaling analysis and instantons for thermally-assisted tunneling and Quantum Monte Carlo simulations

TL;DR

This paper develops an instantonic calculus to analytically connect thermally-assisted quantum tunneling rates with Quantum Monte Carlo simulation escape rates in a fully connected quantum spin model, revealing their identical exponential scaling.

Contribution

It introduces an instantonic framework for analyzing thermally-assisted tunneling and demonstrates the equivalence of quantum and QMC escape rates in a spin model.

Findings

Quantum tunneling and QMC escape rates share the same exponential scaling.

Derived exact solutions for instanton trajectories using nonlinear dynamical mean-field theory.

Established scaling relations for barrier shapes in different regimes.

Abstract

We develop an instantonic calculus to derive an analytical expression for the thermally-assisted tunneling decay rate of a metastable state in a fully connected quantum spin model. The tunneling decay problem can be mapped onto the Kramers escape problem of a classical random dynamical field. This dynamical field is simulated efficiently by path integral Quantum Monte Carlo (QMC). We show analytically that the exponential scaling with the number of spins of the thermally-assisted quantum tunneling rate and the escape rate of the QMC process are identical. We relate this effect to the existence of a dominant instantonic tunneling path. The instanton trajectory is described by nonlinear dynamical mean-field theory equations for a single site magnetization vector, which we solve exactly. Finally, we derive scaling relations for the "spiky" barrier shape when the spin tunnelling and QMC rates scale polynomially with the number of spins $N$ while a purely classical over-the-barrier activation rate scales exponentially with $N$.