Geometry of bounded critical phenomena - Gori, Giacomo; Trombettoni, Andrea (Journal of statistical mechanics: theory and experiment, 2020)
The quest for a satisfactory understanding of systems at criticality in dimensionsd> 2 is a major field of research. We devise here a geometric description of bounded systems at criticality in any dimensiond. This is achieved by altering the flat metric with a space dependent scale factor gamma(x),xbelonging to a bounded domain omega.gamma(x) is chosen in order to have a scalar curvature to be constant and matching the one of the hyperbolic space, the proper notion of curvature being-as called in the mathematics literature-the fractionalQ-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 order parameter 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)= 0.518 142(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 spin-spin correlators at criticality. They should depend on the fractionalQ-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.
Gesehen am 01.06.2021