Department of Chemistry, Prairie View A & M University, Prairie View, Texas 77446, USA.
J Phys Chem A. 2010 Feb 11;114(5):2213-20. doi: 10.1021/jp908441s.
The behavior of long space-time excitations in many-electron systems with ground state degeneracy is explored via multiscale analysis. The analysis starts with an ansatz for the wave function's dual dependence on the N-electron configuration (i.e., both by direct means and by indirect means via a set of order parameters). It is shown that a Dirac-like equation form of the wave equation emerges in the limit where the ratio epsilon (of the average nearest-neighbor distance to the characteristic length of the long-scale phenomenon of interest) is small. Examples of the long scale are the size of a quantum dot, nanotube, or wavelength of a density disturbance. The velocities in the Dirac-like equation are the transition moments of the single-particle momentum operator connecting degenerate ground states. While detailed band structure and the independent quasi-particle picture could underlie the behavior of some systems (as commonly suggested for graphene), the present scaling law results show it is not necessarily the only explanation. Rather, it can follow from the scaling properties of low-lying, long spatial scale excitations and ground state degeneracy, even in strongly interacting systems. The generality of our findings suggests graphene may be just one of many examples of Dirac-like equation behavior. A preliminary validation of our quantum scaling law for molecular arrays is presented. As our scaling law constitutes a coarse-grained wave equation, path integral or other methods derived from it hold great promise for calibration-free, long-time simulation of many-particle quantum systems.
通过多尺度分析探索了基态简并多电子系统中长时空激发的行为。分析从波函数对 N 电子构型的双重依赖的假设（即直接通过直接方法和通过一组阶参数间接）开始。结果表明，在电子平均最近邻距离与感兴趣的长尺度现象的特征长度的比值 ε 小的极限下，波方程出现类似狄拉克的方程形式。长尺度的例子是量子点、纳米管或密度扰动的波长。类似狄拉克的方程中的速度是连接简并基态的单粒子动量算符的跃迁矩。虽然详细的能带结构和独立的准粒子图像可能是某些系统行为的基础（如通常对石墨烯的建议），但目前的标度律结果表明，这不一定是唯一的解释。相反，它可能来自低能、长空间尺度激发和基态简并的标度性质，即使在强相互作用的系统中也是如此。我们的发现具有普遍性，表明石墨烯可能只是许多类似狄拉克方程行为的例子之一。我们提出了对分子阵列的量子标度律的初步验证。由于我们的标度律构成了一个粗粒化的波动方程，因此从它导出的路径积分或其他方法为无校准、长时间模拟多粒子量子系统提供了很大的前景。
