Bulk Property on Cayley Tree with Smooth Boundary Condition

TL;DR

This paper investigates a hopping model on a Cayley tree with a smooth boundary condition, demonstrating that the particle density becomes nearly uniform in the bulk and aligns well with Bethe lattice results.

Contribution

It introduces a smooth boundary condition with a specific modulation function that improves bulk property estimation on Cayley trees.

Findings

Particle density in the bulk becomes nearly uniform due to smoothing.

Calculated properties agree well with exact Bethe lattice results.

The method effectively approximates bulk properties on Cayley trees.

Abstract

We study a nearest-neighbor hopping model on the Cayley tree under the smooth boundary condition with the modulation function $f_s=\sin^2[\pi s/(2M+1)]$, where $s$ is a distance from the central site, and $M$ is the number of shells on the tree. As a result of this smoothing, the particle density in the ground state becomes nearly uniform in the bulk region even when $M$ is relatively small. We compare the calculated particle density at the center with exact result on the Bethe lattice, and they show a good agreement. The calculated bond energy at the center also agrees with that on the Bethe lattice.