Scaling Behavior of Quantum Nanosystems: Emergence of Quasi-particles, Collective Modes, and Mixed Exchange Symmetry States

TL;DR

This paper develops a multiscale approach to analyze quantum nanosystems, deriving a coarse-grained wave equation that captures long-scale dynamics and introduces a novel mean-field approximation for efficient simulation.

Contribution

It introduces a new multiscale framework and a coarse-grained wave equation for quantum nanosystems, along with a distinct mean-field approximation for long-scale dynamics.

Findings

Derived a Schrödinger-like coarse-grained wave equation for nanosystems.

Established a balance criterion for space-time scaling to ensure physical relevance.

Proposed an efficient algorithm for simulating quantum nanosystems.

Abstract

Quantum nanosystems such as graphene nanoribbons or superconducting nanoparticles are studied via a multiscale approach. Long space-time dynamics is derived using a perturbation expansion in the ratio of the nearest-neighbor distance to a nanometer-scale characteristic length, and a theorem on the equivalence of long-time averages and expectation values. This dynamics is shown to satisfy a coarse-grained wave equation (CGWE) which takes a Schr\"odinger-like form with modified masses and interactions. The scaling of space and time is determined by the orders of magnitude of various contributions to the N-body potential. If the spatial scale of the coarse-graining is too large, the CGWE would imply an unbounded growth of gradients; if it is too short, the system's size would display uncontrolled growth inappropriate for the bound states of interest, i.e., collective motion or migration within a stable nano-assembly. The balance of these two extremes removes arbitrariness in the choice of the scaling of space-time. Since the long-scale dynamics of each fermion involves its interaction with many others, we hypothesize that the solutions of the CGWE have mean-field character to good approximation, i.e., can be factorized into single-particle functions. This leads to a Coarse-grained Mean-field (CGMF) approximation that is distinct in character from traditional Hartree-Fock theory. A variational principle is used to derive equations for the single-particle functions. This theme is developed and used to derive an equation for low-lying disturbances from the ground state corresponding to long wavelength density disturbances or long-scale migration. An algorithm for the efficient simulation of quantum nanosystems is suggested.