Trapped surfaces and emergent curved space in the Bose-Hubbard model

TL;DR

This paper demonstrates that in a Bose-Hubbard model on a dynamical lattice, highly connected regions can trap matter and mimic curved spacetime, providing insights into emergent geometry and gravity analogues.

Contribution

It explicitly shows how trapped surfaces emerge in a Bose-Hubbard model with a dynamical lattice, linking connectivity to matter trapping and curved spacetime analogues.

Findings

Highly connected subgraphs trap matter effectively.

The model reduces to a one-dimensional Hubbard model with variable vertex degree.

A wave equation describes the evolution of probability density, indicating curved spacetime effects.

Abstract

A Bose-Hubbard model on a dynamical lattice was introduced in previous work as a spin system analogue of emergent geometry and gravity. Graphs with regions of high connectivity in the lattice were identified as candidate analogues of spacetime geometries that contain trapped surfaces. We carry out a detailed study of these systems and show explicitly that the highly connected subgraphs trap matter. We do this by solving the model in the limit of no back-reaction of the matter on the lattice, and for states with certain symmetries that are natural for our problem. We find that in this case the problem reduces to a one-dimensional Hubbard model on a lattice with variable vertex degree and multiple edges between the same two vertices. In addition, we obtain a (discrete) differential equation for the evolution of the probability density of particles which is closed in the classical regime. This is a wave equation in which the vertex degree is related to the local speed of propagation of probability. This allows an interpretation of the probability density of particles similar to that in analogue gravity systems: matter inside this analogue system sees a curved spacetime. We verify our analytic results by numerical simulations. Finally, we analyze the dependence of localization on a gradual, rather than abrupt, fall-off of the vertex degree on the boundary of the highly connected region and find that matter is localized in and around that region.