Speaker
Description
Hyperbolic lattices constitute synthetic matter in which sites are more densely interconnected than in ordinary lattices, leading to an effective negative curvature at large length scales. Such systems have in recent years been successfully realized across a range of experimental platforms, including coupled microwave resonators, electric-circuit networks, and silicon photonics. This experimental progress has also fueled theoretical efforts to characterize models on hyperbolic lattices, which notably include a non-Abelian generalization of the Bloch theorem with infinite-dimensional reciprocal space [1,2,3].
In Ref. [4], we introduced an approximate numerical scheme rooted in computational group theory, the hyperbolic supercell method, which enables a controlled and convergent treatment of these generalized Bloch states. We have since implemented this scheme in a pair of tandem software packages: HyperCells and HyperBloch. In this contribution, I will present excerpts from an upcoming work [5] that demonstrates the efficiency and the broad applicability of the supercell method across a diverse set of hyperbolic lattice models, including systems with topological and flat energy bands, mean-field-treated density waves in hyperbolic Hubbard models, and quantum spin liquids in hyperbolic Kitaev models.
References:
[1] J. Maciejko and S. Rayan, Automorphic Bloch theorems for hyperbolic lattices, Proc. Natl. Acad. Sci. U.S.A. 119, e2116869119 (2022).
[2] N. Cheng, F. Serafin, J. McInerney, Z. Rocklin, K. Sun, and X. Mao, Band Theory and Boundary Modes of High-Dimensional Representations of Infinite Hyperbolic Lattices, Phys. Rev. Lett. 129, 088002 (2022).
[3] G. Shankar and J. Maciejko, Hyperbolic Lattices and Two-Dimensional Yang-Mills Theory, Phys. Rev. Lett. 133, 146601 (2024).
[4] P. M. Lenggenhager, J. Maciejko, and T. Bzdušek, Non-Abelian Hyperbolic Band Theory from Supercells, Phys. Rev. Lett. 131, 226401 (2023).
[5] M. Looser, M. Pavliuk, J. Maciejko, T. Bzdušek, and P. M. Lenggenhager, Band structures of hyperbolic tight-binding models: Leveraging the HyperCells and HyperBloch software packages (in preparation, 2026).