Ortoleva P, Iyengar S S
Center for Cell and Virus Theory, Department of Chemistry, Indiana University, Bloomington, Indiana 47405, USA.
J Chem Phys. 2008 Apr 28;128(16):164716. doi: 10.1063/1.2877226.
A quantum nanosystem (such as a quantum dot, nanowire, superconducting nanoparticle, or superfluid nanodroplet) involves widely separated characteristic lengths. These lengths range from the average nearest-neighbor distance between the constituent fermions or bosons, or the lattice spacing for a conducting metal, to the overall size of the quantum nanosystem (QN). This suggests the wave function has related distinct dependencies on the positions of the constituent fermions and bosons. We show how the separation of scales can be used to generate a multiscale perturbation scheme for solving the wave equation. Results for electrons or other fermions show that, to lowest order, the wave function factorizes into an antisymmetric (fermion) part and a symmetric (bosonlike) part. The former manifests the short-range/exclusion-principle behavior, while the latter corresponds to collective behaviors, such as plasmons, which have a boson character. When the constituents are bosons, multiscale analysis shows that, to lowest order, the wave function can also factorize into short- and long-scale parts. However, to ensure that the product wave function has overall symmetric particle label exchange behavior, there could, in principle, be states of the boson nanosystem where both the short- and long-scale factors are either boson- or fermionlike; the latter "dual fermion" states are, due to their exclusion-principle-like character, of high energy (i.e., single particle states cannot be multiply occupied). The multiscale perturbation analysis is used to argue for the existence of a coarse-grained wave equation for bosonlike collective behaviors. Quasiparticles, with effective mass and interactions, emerge naturally as consequences of the long-scale dynamics of the constituent particles. The multiscale framework holds promise for facilitating QN computer simulations and novel approximation schemes.
量子纳米系统（如量子点、纳米线、超导纳米粒子或超流纳米液滴）涉及广泛分离的特征长度。这些长度范围从组成费米子或玻色子之间的平均最近邻距离，或导电金属的晶格间距，到量子纳米系统（QN）的整体尺寸。这表明波函数对组成费米子和玻色子的位置具有相关的不同依赖性。我们展示了如何利用尺度分离来生成用于求解波动方程的多尺度微扰方案。电子或其他费米子的结果表明，在最低阶，波函数分解为反对称（费米子）部分和对称（类玻色子）部分。前者表现出短程/泡利不相容原理行为，而后者对应于集体行为，如具有玻色子特性的等离子体激元。当组成部分是玻色子时，多尺度分析表明，在最低阶，波函数也可以分解为短尺度和长尺度部分。然而，为了确保乘积波函数具有整体对称的粒子标签交换行为，原则上玻色子纳米系统可能存在短尺度和长尺度因子都是类玻色子或类费米子的状态；后者的“对偶费米子”状态由于其类似泡利不相容原理的特性而具有高能量（即单粒子态不能被多重占据）。多尺度微扰分析用于论证类玻色子集体行为的粗粒化波动方程的存在。具有有效质量和相互作用的准粒子自然地作为组成粒子长尺度动力学的结果出现。多尺度框架有望促进量子纳米系统的计算机模拟和新颖的近似方案。
