Quantum aspects of hyperbolic lattices

Hyperbolic lattices constitute a unique platform to study
various novel quantum phenomena on negatively curved space and identify
the imprints of spatial curvatures therein. In this talk, first I will
show a simple classification scheme on the family of two-dimensional
bipartite hyperbolic lattices, resulting from simple but canonical
spin-independent tight-binding model for free electrons, yielding three
classes of ballistic quantum fluids on negatively curved space, namely
Dirac liquids, Fermi liquids, and flat bands, that are, respectively,
characterized by a vanishing, a finite, and a diverging density of states
near the band center or zero-energy [1]. Next, I will show that such
systems can be susceptible to different types of spontaneously ordered
phases (such as the charge and spin density-waves) once Hubbard-like
electronic interactions are considered, giving rise to dynamic mass
generations in half-filled systems [2]. In the second half of the talk, I
will exclusively focus on hyperbolic Dirac systems in which orderings can
only take place beyond critical strengths of interactions via a quantum
phase transition. I will propose two realistic approaches to trigger such
dynamic mass generations on curved space Dirac liquids at sufficiently
weak interactions. I show that (a) application of external magnetic fields
[3, 4] and (b) rotational symmetric strain [5] give rise to a finite
density of states near zero-energy in noninteracting systems, conducing
various mass ordering at weak coupling. Finally, I will discuss the role
of disorder or impurities on hyperbolic Dirac liquids. I show that in
contrary to our traditional wisdom, planar dirty hyperbolic Dirac liquids
show two distinct quantum phase transitions as the disorder strength is
gradually increased in the system. First the ballistic system undergoes a
semimetal-to-metal transition at moderate disorder strength, which is
subsequently followed by the Anderson metal-to-insulator transition, both
being absent in two-dimensional Euclidean Dirac liquids, realizable on
honeycomb lattices,
for example [6]. I will close the talk with some possible future
directions to further unfold unique quantum aspects of hyperbolic
crystals.

References:
1. N. Gluscevich, A. Samanta, S. Manna, and B. Roy, Phys. Rev. B 111, L121108
(2025).
2. C. A. Leong and B. Roy, arXiv:2501.18591
3. B. Roy, Phys. Rev. B 110, 245117 (2024).
4. C. A. Leong and B. Roy, arXiv:2510.02304
5. C. A. Leong and B. Roy, arXiv:2511.16667
6. C. A. Leong, D. J. Salib, and B. Roy, arXiv:2512.05109