Modeling Quantum Dot Systems as Random Geometric Graphs with Probability Amplitude-Based Weighted Links

Abstract

This paper focuses on modeling a disorder ensemble of quantum dots (QDs) as a special kind of Random Geometric Graphs (RGG) with weighted links. We compute any link weight as the overlap integral (or electron probability amplitude) between the QDs (=nodes) involved. This naturally leads to a weighted adjacency matrix, a Laplacian matrix, and a time evolution operator that have meaning in Quantum Mechanics. The model prohibits the existence of long-range links (shortcuts) between distant nodes because the electron cannot tunnel between two QDs that are too far away in the array. The spatial network generated by the proposed model captures inner properties of the QD system, which cannot be deduced from the simple interactions of their isolated components. It predicts the system quantum state, its time evolution, and the emergence of quantum transport when the network becomes connected.

Persistence in complex systems

“…The existence of a threshold that separates two different macroscopic phases (absence/ persistence of a disease) is an instance of percolation transitions on complex networks [214]. An illustrative example of percolation during the growth of networks is the emergence of electron transport in some networks that model systems of disordered quantum dots (QDs) [348,349]. In this approach, a QD −which confines most of the wavefunction inside it− is encoded by a node, while electron hopping between two QDs (nodes) is represented by a link.…”

Approaching Disordered Quantum Dot Systems by Complex Networks with Spatial and Physical-Based Constraints

“…A conceptually different approach has recently been proposed in [ 46 ], as it focuses on modeling a disordered ensemble of QDs as Random Geometric Graphs (RGG) with weighted links, with these being the overlap integral (or electron probability amplitude) between the QDs (=nodes) involved. These are networks in which the nodes are spatially embedded [ 74 ] or constrained to sites in a metric space, usually, the Euclidean distance .…”

“…The novelty of our proposal, especially in relation to that in [ 46 ], is threefold. First, although we consider that the QDs are randomly distributed in a metric space (spatial network (SN) [ 34 , 47 ]), they have to fulfill the condition that there is a minimum inter-dot distance that cannot be violated (to avoid localization effects, as described below).…”

Adaptation of Random Binomial Graphs for Testing Network Flow Problems Algorithms

“…The input parameters were as follows: p = 0.2, npath = 4, ncycle = 2, minu = 1, maxu = 7, minc = 1, and maxc = 7. The algorithm generated the following directed paths: P 1 = (1, 2, 3, 6), P 2 = (1, 6), P 3 = (1,4,5,6), and P 4 = (1, 2, 4, 6), as well as the directed cycles C 1 = (4, 5, 2, 3, 4), and C 2 = (2, 4, 5, 1, 2). In the end, from the remaining 19 pairs of nodes were not connected with arcs, based on the considered probability of p = 0.2, and AGRFN generated three more arcs: (1, 5), (3,4), and (3, 5).…”

“…These random graphs are generated based on the values of two parameters: n (the number of nodes) and p ∈ [0, 1] (the probability of introducing any edge in the graph). These kinds of random networks have been applied for Zagreb indices, general sum-connectivity index, general inverse sum indeg index, and general first geometric-arithmetic index [4]. In a network generated in this manner, there is the possibility that the source will poorly communicate to the sink or even not communicate at all.…”