Bulk Property on Cayley Tree with Smooth Boundary Condition

We study a nearest-neighbor hopping model on the Cayley tree under the smooth boundary condition with the modulation function $f_s={\sin }^2[\pi s\mathrm{/}(2M+1)]$, 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.