Superconductivity in hyperbolic spaces: Cayley trees, hyperbolic continuum, and BCS theory

TL;DR

This paper explores how negative curvature in hyperbolic geometries influences superconductivity, revealing boundary-localized phases and distinct critical temperatures using mean-field theories on Cayley trees and hyperbolic planes.

Contribution

It introduces a unified mean-field framework for superconductivity in hyperbolic spaces, highlighting boundary effects and the stabilization of boundary superconductivity in negatively curved geometries.

Findings

Superconductivity localizes at the boundary in Cayley trees.

Two distinct critical temperatures for edge and bulk superconductivity.

Boundary effects are more pronounced in discrete tree geometries.

Abstract

We investigate $s$-wave superconductivity in negatively curved geometries, focusing on Cayley trees and the hyperbolic plane. Using a self-consistent Bogoliubov-de Gennes approach for trees and a BCS treatment of the hyperbolic continuum, we establish a unified mean-field framework that captures the role of boundaries in hyperbolic spaces. For finite Cayley trees with open boundaries, the superconducting order parameter localizes at the edge while the interior can remain normal, leading to two distinct critical temperatures: $T_\textrm{c}^\textrm{edge} > T_\textrm{c}^\textrm{bulk}$. A corresponding boundary-dominated phase also emerges in hyperbolic annuli and horodisc regions, where radial variations of the local density of states enhance edge pairing. We also demonstrate that the enhancement of the density of states at the boundary is significantly more pronounced for the discrete tree geometry. Our results show that, owing to the macroscopic extent of the boundary, negative curvature can stabilize boundary superconductivity as a phase that persists in the thermodynamic limit on par with the bulk superconductivity. These results highlight fundamental differences between bulk and boundary ordering in hyperbolic matter, and provide a theoretical framework for future studies of correlated phases in negatively curved systems.