Emergent gravity in two dimensions

TL;DR

This paper demonstrates emergent metric properties in a two-dimensional lattice sigma-model through numerical simulations, revealing universal critical behavior and long-range correlations indicative of self-tuned criticality.

Contribution

It introduces a discretized two-dimensional sigma-model with lattice diffeomorphism invariance that exhibits emergent metric and self-tuned criticality without parameter tuning.

Findings

Emergent metric with Minkowski or Euclidean signature observed

Universal power-law decay in metric correlation functions

Non-zero zweibein expectation value with long-range correlations

Abstract

We explore models with emergent gravity and metric by means of numerical simulations. A particular type of two-dimensional non-linear sigma-model is regularized and discretized on a quadratic lattice. It is characterized by lattice diffeomorphism invariance which ensures in the continuum limit the symmetry of general coordinate transformations. We observe a collective order parameter with properties of a metric, showing Minkowski or euclidean signature. The correlation functions of the metric reveal an interesting long-distance behavior with power-like decay. This universal critical behavior occurs without tuning of parameters and thus constitutes an example of "self-tuned criticality" for this type of sigma-models. We also find a non-vanishing expectation value of a "zweibein" related to the "internal" degrees of freedom of the scalar field, again with long-range correlations. The metric is well described as a composite of the zweibein. A scalar condensate breaks euclidean rotation symmetry.