Instanton Solution of a Nonlinear Potential in Finite Size

TL;DR

This paper derives an explicit instanton solution for a double well potential considering finite size effects, extending classical results to finite systems and providing detailed quantum fluctuation analysis.

Contribution

It introduces a finite size instanton solution using Jacobi elliptic functions and derives the corresponding functional determinant for quantum fluctuations.

Findings

Explicit finite size instanton solution in terms of elliptic functions

Generalization of the (anti)kink solution to finite systems

Detailed derivation of quantum fluctuation contributions

Abstract

The Euclidean path integral method is applied to a quantum tunneling model which accounts for finite size ($L$) effects. The general solution of the Euler Lagrange equation for the double well potential is found in terms of Jacobi elliptic functions. The antiperiodic instanton interpolates between the potential minima at any finite $L$ inside the quantum regime and generalizes the well known (anti)kink solution of the infinite size case. The derivation of the functional determinant, stemming from the quantum fluctuation contribution, is given in detail. The explicit formula for the finite size semiclassical path integral is presented.