# Statistical Properties of Correlated Semiclassical Bands in Tight-Binding Small-World Networks

## TL;DR

This paper explores how correlations in energy bands of small-world networks affect electron behavior, with implications for superconductivity in 2D materials.

## Contribution

The study reveals how band correlations influence the local density of states in tight-binding small-world networks with nonlinearity.

## Key findings

- Correlations shift the nonlinear density of states toward the band edge in the nearest neighbor limit.
- Band collapse and increased nonlinear states occur near the fully connected limit.
- Correlations flatten the band in both limits but with distinct localized features in the fully connected case.

## Abstract

Linear tight-binding models with long-range interactions and small-world geometry have a broad energy spectrum in the nearest neighbor coupling limit, while the spectrum becomes narrow in the fully connected limit due to the emergence of flat bands. A transition to a Wigner-like density of states appears at a low fraction of long-range bonds. Adding nonlinearity to the model introduces correlations among the stationary states, while multiple new states are generated as a result of the nonlinearity. In this work, we study the effect of band correlations on the local density of states for small-world networks as a function of the number of long-range bonds. We find that close to the nearest neighbor limit, the onset of correlations shifts the nonlinear density of states towards the band edge of the spectrum. Close to the opposite limit of the fully connected model, the band collapses in the band center, accompanied by a large increase in the new states induced by the nonlinearity. While in both limits the effect of correlations is to flatten the band, close to the mean field fully connected limit, the states are correlated and generally have distinct localized features. These effects may have implications for the dynamics of electrons in two-dimensional moiré structures and the onset of superconductivity in these systems.