The Fractal-Lattice Hubbard Model

TL;DR

This study explores the electronic and magnetic properties of the fractal-lattice Hubbard model on a Sierpinski triangle, revealing localized states, magnetic order, a Mott transition, and effects of spin-orbit coupling across different generations.

Contribution

It provides a comprehensive numerical analysis of the fractal-lattice Hubbard model, highlighting the influence of fractal geometry on electronic states, magnetic phases, and topological features.

Findings

Existence of localized states linked to symmetry and ferrimagnetism.

Persistence of magnetic order across all interaction strengths.

Observation of a Mott transition at U/t approximately 4.5.

Abstract

Here, we investigate the fractal-lattice Hubbard model using various numerical methods: exact diagonalization, the self-consistent diagonalization of a (mean-field) Hartree-Fock Hamiltonian and state-of-the-art Auxiliary-Field Quantum Monte Carlo. We focus on the Sierpinski triangle with Hausdorff dimension $1.58$ and consider several generations. In the tight-binding limit, we find compact localised states, which are also explained in terms of symmetry and linked to the formation of a ferrimagnetic phase at weak interaction. Simulations at half-filling revealed the persistence of this type of magnetic order for every value of interaction strength and a Mott transition for U/t $\sim$ 4.5. In addition, we found a remarkable dependence on the Hausdorff dimension regarding $i)$ the number of compact localised states in different generations, $ii)$ the scaling of the total many-body ground-state energy in the tight-binding limit, and $iii)$ the density of the states at the corners of the lattice for specific values of electronic filling. Moreover, in the presence of an intrinsic spin-orbit coupling, the zero-energy compact localized states become entangled and give rise to inner and outer corner modes.