Energy Scale Deformation on Regular Polyhedra

Takuya Eguchi,Satoshi Oga,Andrej Gendiar,Tomotoshi Nishino,Hosho Katsura

doi:10.7566/jpsj.91.034001

Abstract

A variant of energy scale deformation is considered for the S = 1/2 antiferromagnetic Heisenberg model on polyhedra. The deformation is induced by the perturbations to the uniform Hamiltonian, whose coefficients are determined by the bond coordinates. On the tetrahedral, octahedral, and cubic clusters, the perturbative terms do not affect the ground state of the uniform Hamiltonian when they are sufficiently small. On the other hand, for the icosahedral and dodecahedral clusters, it is numerically confirmed that the ground state of the uniform Hamiltonian is almost insensitive to the perturbations unless they lead to a discontinuous change in the ground state. The obtained results suggest the existence of a generalization of sine-square deformation in higher dimensions.

Highlights

A variant of energy scale deformation is considered for the S = 1/2 antiferromagnetic Heisenberg model on polyhedra
The deformation is induced by the perturbations to the uniform Hamiltonian, whose coefficients are determined by the bond coordinates
For the icosahedral and dodecahedral clusters, it is numerically confirmed that the ground state of the uniform Hamiltonian is almost insensitive to the perturbations unless they lead to a discontinuous change in the ground state

Summary

Introduction

Uniformity in quantum states is one of the fundamental properties in condensed matter physics. The ground state of a quantum model is expected to be uniform when the Hamiltonian is translationally invariant unless spatial modulations are spontaneously stabilized. In the case of the free-fermion hopping model on the lattice, it is straightforward to show that the ground state | ψ0 of the uniform part H0 is an eigenstate of the modulated part HM with eigenvalue zero. From the construction of HM in Eq (6), which is related to the N-sided regular polygon, it is possible to state that HM corresponds to the most slowly varying sinusoidally modulated function on the finite lattice. This geometric observation suggests a new type of two-dimensional generalization of the SSD. We discuss possible generalizations of the SSD in higher dimensions

Energy Scale Deformation on the Tetrahedral Cluster

On the Octahedral and Cubic Clusters

On the Icosahedral and Dodecahedral Clusters

Conclusion and Discussion