The Fractal-Lattice Hubbard Model

Monica Conte; Vinicius Zampronio; Malte Röntgen; Cristiane Morais Smith

doi:10.22331/q-2024-09-11-1469

The Fractal-Lattice Hubbard Model

Monica Conte¹, Vinicius Zampronio^2,1, Malte Röntgen³, and Cristiane Morais Smith¹

¹Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584CC Utrecht, The Netherlands,
²Departamento de Física Teórica e Experimental - UFRN, Av. Sen. Salgado Filho 3000, 59078-970, Natal - RN, Brazil,
³Laboratoire d’Acoustique de l’Université du Mans, Unite Mixte de Recherche 6613, Centre National de la Recherche Scientifique, Avenue O. Messiaen, F-72085 Le Mans Cedex 9, France

Published:	2024-09-11, volume 8, page 1469
Eprint:	arXiv:2310.07813v2
Doi:	https://doi.org/10.22331/q-2024-09-11-1469
Citation:	Quantum 8, 1469 (2024).

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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.

Featured image: Local magnetization per site, it shows a ferrimagnetic state; red (blue) arrows pointing up (down) indicate positive (negative) local magnetization. The length of the arrows represents the intensity of the local magnetization. The magnetic order is preserved across all interaction values, but the system undergoes a Mott transition at U~4.5.

Popular summary

The Hubbard model goes beyond conventional band theory by introducing short-range interactions between particles in periodic potentials. Its first success was in detecting the Mott transition, but its importance was later reinforced by its ability to describe many phases of matter, such as metals, insulators, and even superconductors.

Despite its simplicity, an analytic solution has been found only for the one-dimensional case. For higher dimensions, numerical approaches are required and have been used to study a wide variety of two-dimensional lattice configurations. However, the Hubbard model has never been studied at a non-integer dimension. Fractal geometries can serve as a playground to investigate physics in non-integer spaces, and many interesting findings have recently been reported, ranging from anomalous behaviour in quantum transport to higher-order topological insulators. Moreover, fractal geometries lack periodicity, which creates an environment where the Bloch theorem, a cornerstone of solid-state physics, is not valid.

To address the fundamental question of how electrons behave in a non-integer dimension and a non-periodic space, we numerically solve the Hubbard model on a fractal lattice built from the Sierpinski triangle, which has a Hausdorff dimension d = 1.58. We start by examining the tight-binding limit, where we find remarkable dependencies on the Hausdorff dimension of the fractal. Then, we introduce interactions and use a mean-field Hartree-Fock approach together with a Quantum Monte Carlo algorithm to solve the system. We find that the system undergoes a Mott transition from metal to insulator while maintaining ferrimagnetic order for every value of interaction. The emergence of a phase in which magnetism coexists with metallicity at half filling is a novel feature, never found in any regular two-dimensional lattice in the context of the Hubbard model, despite a constant search. It requires the richness of the Sierpinski gasket fractal lattice, in which the number of sites in the two sublattices is not equal, and the connectivity of the sites is non-uniform.

► BibTeX data

@article{Conte2024fractallattice,
  doi = {10.22331/q-2024-09-11-1469},
  url = {https://doi.org/10.22331/q-2024-09-11-1469},
  title = {The {F}ractal-{L}attice {H}ubbard {M}odel},
  author = {Conte, Monica and Zampronio, Vinicius and R{\"{o}}ntgen, Malte and Smith, Cristiane Morais},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1469},
  month = sep,
  year = {2024}
}

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