Topological Invariants in Microscopic Transport on Rough Landscapes: Morphology, Hierarchical Structure, and Horton Analysis of Riverlike Networks of Vortices

TL;DR

This study investigates whether topological invariants known from large-scale river networks, such as Horton's laws, also apply to microscopic vortex networks in superconductors, revealing similar hierarchical structures at vastly smaller scales.

Contribution

The paper demonstrates that Horton's laws, traditionally observed in large river basins, also hold for microscopic vortex networks in superconductors, supported by realistic simulations.

Findings

Horton's laws are obeyed in certain vortex network morphologies.

A phase diagram of network morphologies is derived.

Microscopic vortex networks exhibit hierarchical structures similar to river basins.

Abstract

River basins as diverse as the Nile, the Amazon, and the Mississippi satisfy certain topological invariants known as Horton's laws. Do these macroscopic (up to 10^3 km) laws extend to the micron scale? Through realistic simulations, we analyze the morphology and statistical properties of networks of vortex flow in flux-gradient-driven superconductors. We derive a phase diagram of the different network morphologies, including one in which Horton's laws of length and stream number are obeyed -- even though these networks are about 10^9 to 10^15 times smaller than geophysical river basins.