Quantum-tunneling transitions and exact statistical mechanics of bistable systems with parametrized Dikand\'e-Kofan\'e double-well potentials

TL;DR

This paper investigates how shape deformability in parametrized double-well potentials influences quantum tunneling transitions, revealing a first-order transition and providing exact solutions for the associated statistical mechanics problem.

Contribution

It introduces a class of parametrized double-well potentials and demonstrates their impact on tunneling transition order, along with deriving exact solutions for the transfer-integral operator.

Findings

Shape deformability favors a first-order quantum tunneling transition.

Exact eigenvalues and eigenfunctions of the transfer-integral operator are obtained.

The critical deformability parameter triggers the transition, consistent across potential families.

Abstract

We consider a one-dimensional system of interacting particles, in which particles are subjected to a bistable potential the double-well shape of which is tunable via a shape deformability parameter. Our objective is to examine the impact of shape deformability on the order of transition in quantum tunneling in the bistable system, and on the possible existence of exact solutions to the transfer-integral operator associated with the partition function of the system. The bistable potential is represented by a class composed of three families of parametrized double-well potentials, whose minima and barrier height can be tuned distinctly. It is found that the extra degree of freedom, introduced by the shape deformability parameter, favors a first-order transition in quantum tunneling, in addition to the second-order transition predicted with the $\phi^4$ model. This first-order transition in quantum tunneling, which is consistent with Chudnovsky's conjecture of the influence of the shape of the potential barrier on the order of thermally-assisted transitions in bistable systems, is shown to occur at a critical value of the shape-deformability parameter which is the same for the three families of parametrized double-well potentials. Concerning the statistical mechanics of the system, the associate partition function is mapped onto a spectral problem by means of the transfer-integral formalism. The condition that the partition function can be exactly integrable, is determined by a criterion enabling exact eigenvalues and eigenfunctions for the transfer-integral operator. Analytical expressions of some of these exact eigenvalues and eigenfunctions are given, and the corresponding ground-state wavefunctions are used to compute the probability density which is relevant for calculations of thermodynamic quantities such as the correlation functions and the correlation lengths.