Bose-Einstein condensation in exotic lattice geometries

TL;DR

This paper explores how exotic lattice geometries like fractals and hyperbolic graphs influence Bose-Einstein condensation, revealing unique effects on critical temperature, fluctuations, and phase transitions, with implications for quantum simulation.

Contribution

It demonstrates the profound impact of lattice geometry on Bose-Einstein condensation and quantum phase transitions, including effects in fractal and hyperbolic lattices, and introduces methods to analyze these phenomena.

Findings

Fractal lattices lower condensation temperature and increase fluctuations.

Hyperbolic lattices mimic 3D condensation behavior despite 2D embedding.

Strong interactions lead to a Mott insulator with geometry-dependent features.

Abstract

Modern quantum engineering techniques allow for synthesizing quantum systems in exotic lattice geometries, from self-similar fractal networks to negatively curved hyperbolic graphs. We demonstrate that these structures profoundly reshape Bose-Einstein condensation. Fractal lattices dramatically lower the condensation temperature and enhance condensation fluctuations. In a Sierpi\'nski carpet, quasi-degeneracies in the tight-binding spectrum fragment the condensate. Hyperbolic lattices, on the other hand, exhibit condensation features similar to regular three-dimensional lattices, despite their embedding in only two dimensions: The critical temperature increases as the system grows, and the temperature-dependence of the condensate fraction follows the same power-law as for cubic lattices. We explain these similarities through the similarity of the densities of state at low energies. When strong repulsive interactions are included, the gas enters a Mott insulating state. Using a multi-site Gutzwiller approach as well as a simple strong-coupling expansion, for the Sierpi\'nski triangle we find a smooth interpolation between the characteristic insulating lobes of one-dimensional and two-dimensional systems. Our findings establish lattice geometry as a powerful tuning knob for quantum phase phenomena and pave the way for experimental exploration in photonic waveguide arrays and Rydberg-atom tweezer arrays.