# Geometry of bounded critical phenomena

## TL;DR

This paper introduces a geometric framework for describing bounded critical systems using a fractional Yamabe equation, leading to new predictions for correlation functions that match numerical results in the 3D Ising model.

## Contribution

It develops a novel geometric approach based on fractional Q-curvature to analyze critical phenomena in bounded domains, connecting geometry with conformal field theory.

## Key findings

- Predicted the critical magnetization profile and extracted the order parameter's scaling dimension.
- Derived the structure of two-point correlators depending on fractional Q-hyperbolic distance.
- Numerical simulations in 3D Ising model agree with the geometric predictions.

## Abstract

We devise a geometric description of bounded systems at criticality in any dimension $d$. This is achieved by altering the flat metric with a space dependent scale factor $\gamma(x)$, $x$ belonging to a general bounded domain $\Omega$. $\gamma(x)$ is chosen in order to have a scalar curvature to be constant and negative, the proper notion of curvature being -- as called in the mathematics literature -- the fractional Q-curvature. The equation for $\gamma(x)$ is found to be the Fractional Yamabe Equation (to be solved in $\Omega$) that, in absence of anomalous dimension, reduces to the usual Yamabe Equation in the same domain. From the scale factor $\gamma(x)$ we obtain novel predictions for the scaling form of one-point correlation functions. A (necessary) virtue of the proposed approach is that it encodes and allows to naturally retrieve the purely geometric content of two-dimensional boundary conformal field theory. From the critical magnetization profile in presence of boundaries one can extract the scaling dimension of the order parameter, $\Delta_\phi$. For the 3D Ising model we find $\Delta_\phi=0.518142(8)$ which favorably compares (at the fifth decimal place) with the state-of-the-art estimate. A nontrivial prediction is the structure of two-point correlators at criticality. They should depend on the fractional Q-hyperbolic distance calculated from the metric, in turn depending only on the shape of the bounded domain and on $\Delta_\phi$. Numerical simulations of the 3D Ising model on a slab geometry are found to be in agreement with such predictions.