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キーワード:
High Energy Physics - Theory, hep-th, Condensed Matter, Statistical Mechanics, cond-mat.stat-mech,General Relativity and Quantum Cosmology, gr-qc,Quantum Physics, quant-ph
要旨:
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.
