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

F. Naha Nzoupe, Alain M. Dikandé+1 more|arXiv (Cornell University)|Jan 1, 2021

Quantum chaos and dynamical systems参考文献 56被引用 6

一句话总结

本文通过参数化的Dikandé-Kofané势能，研究了可调双阱势能下双稳态系统中的量子隧穿跃迁与精确统计力学。通过引入形状可变形参数，研究揭示了一个临界值，在该值处出现一阶量子隧穿跃迁，与Chudnovsky的猜想一致，并推导出转移积分算符的精确本征值与本征函数，从而实现关联函数和关联长度等热力学量的精确计算。

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.

研究动机与目标

• 研究形状可变形参数对双稳态系统中量子隧穿跃迁阶数的影响。
• 确定与配分函数相关的转移积分算符是否对所提出的参数化双阱势能具有精确解。
• 推导转移积分算符的精确本征值与本征函数的解析表达式。
• 计算用于热力学计算的基态波函数及其概率密度。
• 验证一阶隧穿跃迁与Chudnovsky关于势垒形状依赖性的猜想的一致性。

提出的方法

• 利用一类包含三个家族的参数化双阱势能，通过形状可变形参数独立调节势阱最小值与势垒高度。
• 通过转移积分形式将配分函数映射为涉及转移积分算符的谱问题。
• 使用解析方法推导与转移积分算符相关的薛定谔方程的精确解，得到本征值与本征函数的闭式表达式。
• 提取基态波函数并用于计算与热力学函数相关的概率密度。
• 通过识别形状可变形参数的临界值（此时发生一阶相变）分析临界行为。
• 通过转移积分算符相关谱问题的可解性，确立配分函数精确可积性的判据。

实验结果

研究问题

• RQ1 在双阱势能中引入形状可变形参数是否会引起一阶量子隧穿跃迁？
• RQ2 此类系统配分函数的转移积分算符是否可能具有精确解？
• RQ3 是否存在一个普适的形状可变形参数临界值，使得不同家族的参数化势能均发生一阶隧穿跃迁？
• RQ4 转移积分算符的精确本征值与本征函数如何与关联函数和关联长度等热力学量相关联？
• RQ5 通过可变形参数调控的势垒形状在多大程度上影响量子隧穿跃迁的阶数？

主要发现

• 一阶量子隧穿跃迁由形状可变形参数引起，其临界值μ在所有三个参数化双阱势能家族中完全相同。
• 形状可变形参数的临界值μ_c与Chudnovsky关于势垒形状对热辅助跃迁阶数影响的猜想一致。
• 推导出基态波函数及其对应本征值的精确解析表达式，其中q = 1至q = 4的显式形式以双曲函数和μ相关参数表示。
• 显式计算了不同q值下的本征值ε₀(μ)、ε₁(μ)和ε₂(μ)，显示出其对形状参数μ和势能非谐性的依赖关系。
• 利用基态波函数计算概率密度，从而实现系统中关联函数与关联长度的精确计算。
• 本研究通过转移积分算符相关谱问题的可解性，确立了配分函数精确可积性的判据。