Scaling behavior of quantum nanosystems: Emergence of quasi-particles, collective modes, and mixed exchange symmetry states

Examples of quantum nanosystems are graphene nanoribbons, molecular wires, and superconducting nanoparticles. The objective of the multiscale theory presented here is to provide a new perspective on the coupling of processes across scales in space and time underlying the dynamics of these systems. The long range objective for this multiscale approach is to serve as an efficient computational algorithm. Long space-time dynamics is derived using a perturbation expansion in the ratio ɛ 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 Schrödinger-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 nanoassembly. 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 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.