Geometry of bounded critical phenomena

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 $γ(x)$, $x$ belonging to a general bounded domain $Ω$. $γ(x)$ 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 $γ(x)$ is found to be the Fractional Yamabe Equation (to be solved in $Ω$) that, in absence of anomalous dimension, reduces to the usual Yamabe Equation in the same domain. From the scale factor $γ(x)$ 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, $Δ_ϕ$. For the 3D Ising model we find $Δ_ϕ=0.518142(8)$ 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 $Δ_ϕ$. Numerical simulations of the 3D Ising model on a slab geometry are found to be in agreement with such predictions.

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