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Mesoscopic quantum systems exhibit complex many-body quantum phenomena, where interactions between spins and charges give rise to collective modes and topological states. Even simple, non-interacting theories display a rich landscape of energy states—distinct many-particle configurations connected by spin- and energy-dependent transition rates. The ways in which these energy states interact is difficult to characterize or predict, especially in regimes of frustration where many-body effects create a multiply degenerate landscape. Here, we use network science to characterize the complex interconnection patterns of these energy-state transitions. Using an experimentally verified computational model of electronic transport through quantum antidots, we construct networks where nodes represent accessible energy states and edges represent allowed transitions. We find that these networks exhibit Rentian scaling, which is characteristic of efficient transportation systems in computer circuitry, neural circuitry, and human mobility, and can be used to measure the interconnection complexity of a network. We find that the topological complexity of the state transition networks—as measured by Rent’s exponent— correlates with the amount of current flowing through the antidot system. Furthermore, networks corresponding to points of frustration (due, for example, to spin-blockade effects) exhibit an enhanced topological complexity relative to non-frustrated networks. Our results demonstrate that network characterizations of the abstract topological structure of energy landscapes capture salient properties of quantum transport. More broadly, our approach motivates future efforts to use network science to understand the dynamics and control of complex quantum systems.

 

Abstract 

Executive function is a quintessential human capacity that emerges late in development and displays different developmental trends in males and females. Sex differences in executive function in youth have been linked to vulnerability to psychopathology as well as to behaviors that impinge on health, wellbeing, and longevity. Yet, the neurobiological basis of these differences is not well understood, in part due to the spatiotemporal complexity inherent in patterns of brain network maturation supporting executive function. Here we test the hypothesis that sex differences in impulsivity in youth stem from sex differences in the controllability of structural brain networks as they rewire over development. Combining methods from network neuroscience and network control theory, we characterize the network control properties of structural brain networks estimated from diffusion imaging data acquired in males and females in a sample of 879 youth aged 8–22 years. We summarize the control properties of these networks by estimating average and modal controllability, two statistics that probe the ease with which brain areas can drive the network towards easy versus difficult-to-reach states. We find that females have higher modal controllability in frontal, parietal, and subcortical regions while males have higher average controllability in frontal and subcortical regions. Furthermore, controllability profiles in males are negatively related to the false positive rate on a continuous performance task, a common measure of impulsivity. Finally, we find associations between average controllability and individual differences in activation during an n-back working memory task. Taken together, our findings support the notion that sex differences in the controllability of structural brain networks can partially explain sex differences in executive function. Controllability of structural brain networks also predicts features of task-relevant activation, suggesting the potential for controllability to represent context- specific constraints on network state more generally.