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

TL;DR

This work analyzes $s$-wave superconductivity in negatively curved geometries using a BdG framework on Cayley trees and a BCS approach on the hyperbolic plane. It uncovers a boundary-dominated superconducting phase on finite Cayley trees, featuring two distinct critical temperatures $T_c^{\text{edge}}$ and $T_c^{\text{bulk}}$, driven by an edge-localized order parameter and a boundary-enhanced LDOS with discrete spectral contributions. In the continuous hyperbolic space, boundary effects are captured by exact horodisk LDOS calculations and by numerical BdG studies on hyperbolic annuli, showing curvature-controlled boundary amplification that can raise $T_c$ relative to the bulk, albeit with different spectral signatures than the discrete tree. The results reveal fundamental differences between bulk and boundary ordering in hyperbolic matter and establish a theoretical framework for exploring correlated phases in negatively curved systems, with potential connections to boundary phenomena and flat-band physics in hyperbolic lattices.

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.

Figures (13)

• Figure 1: Possible hyperbolic spaces. In this paper, we focus on trees and continuous hyperbolic spaces.
• Figure 2: Density of states (left panel) and phase diagram (right panel) for Bethe lattice with degree of vertices $q=K+1=3$ and for Hubbard potential $U=1$.
• Figure 3: (a) Definition of shells of a Cayley tree. These shells are also used directly in the construction of the symmetric basis states in Eq. \ref{['eq: symm basis']}. The number of shells in the example is set to $M=4$. (b) Definition of shells used for constructing nonsymmetric basis states emanating from the seed node $|{j}\rangle$ as in Eq. \ref{['eq: nonsymm basis']}.
• Figure 4: Superconducting order parameter of finite and infinite Cayley trees with vertex degree $K+1=3$ for parameters $\mu=0$ and $U=1$. We display the computed order parameter at the central (root) site and at the edge sites for (a) small system with $10$ shells, and for (b) larger system with $100$ shells. The right panel compares the value of the order parameter at the center of trees with various radial sizes against the thermodynamic limit of the Bethe lattice.
• Figure 5: The spatial profile of the order parameter for the Caylee tree with connectivity $q=2$, with the total number of layers $M=8$ (left panel) and $M=100$ (right panel). The parameters are $U=1$, $\mu=0$ and $T=0.1$. The profile of the order parameter on the right panel is shown on a logarithmic scale.