Emergent classical geometries on boundaries of randomly connected tensor networks

TL;DR

This paper demonstrates how classical geometries, including curved spaces and flat tori, emerge on the boundaries of randomly connected tensor networks, with dynamics governed by an effective diffeomorphism-invariant action.

Contribution

It introduces a method to generate and control emergent geometries from tensor networks, including explicit examples and analysis of phase transitions between different space configurations.

Findings

Emergent flat tori in arbitrary dimensions demonstrated.

Effective action invariant under spatial diffeomorphisms derived.

Various phase transitions among different space geometries identified.

Abstract

It is shown that classical spaces with geometries emerge on boundaries of randomly connected tensor networks with appropriately chosen tensors in the thermodynamic limit. With variation of the tensors, the dimensions of the spaces can be freely chosen, and the geometries, which are curved in general, can be varied. We give the explicit solvable examples of emergent flat tori in arbitrary dimensions, and the correspondence from the tensors to the geometries for general curved cases. The perturbative dynamics in the emergent space is shown to be described by an effective action which is invariant under the spatial diffeomorphism due to the underlying orthogonal group symmetry of the randomly connected tensor network. It is also shown that there are various phase transitions among spaces, including extended and point-like ones, under continuous change of the tensors.