Geometry of bounded critical phenomena
Year: 2020
Authors: Gori G., Trombettoni A.
Autors Affiliation: Heidelberg Univ, Inst Theoret Phys, D-69120 Heidelberg, Germany; Univ Padua, Dipartimento Fis & Astron Galileo Galilei, Via Marzolo 8, I-35131 Padua, Italy; CNR, IOM DEMOCRITOS Simulat Ctr, Via Bonomea 265, I-34136 Trieste, Italy; SISSA, Via Bonomea 265, I-34136 Trieste, Italy; Ist Nazl Fis Nucl, Sez Trieste, Via Bonomea 265, I-34136 Trieste, Italy; Univ Trieste, Dipartimento Fis, Str Costiera 11, I-34151 Trieste, Italy.
Abstract: 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.
Journal/Review: JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT
Volume: 2020 (6) Pages from: 63210-1 to: 63210-25
More Information: The authors wish to thank Maria del Mar Gonzalez and Robin C Graham for providing essential insights into the mathematics needed for this project. We thank John L Cardy and Slava Rychkov for useful correspondence. Discussions with Jacopo Viti and Nicolo Defenu are also gratefully acknowledged. GG acknowledges important discussions with Antonia Ciani. Moreover GG thanks Universidad Autonoma de Madri d for hospitality partially funded under BBVA Foundation grant for Investigadores y Creadores Culturales (actuales becas Leonardo), 2016, during which crucial stages of the work were performed. Early stages of the project were carried on while GG was visiting ICTP, Trieste. Both of the authors acknowledge hospitality during the ’Disordered systems, random spatial processes and some applications’ program held at IHP, Paris, during ’Conformal Field Theories and Renormalization Group Flows in Dimensions d > 2’ workshop held at GGI, Florence, and a visit to ATOMKI, Debrecen. Computational resources were provided by CNR-IOMand SISSA. GG work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 – 390900948 (the Heidelberg STRUCTURES Excellence Cluster).KeyWords: conformal field theory; correlation functions; critical exponents and amplitudes; surface effectsDOI: 10.1088/1742-5468/ab7f32Citations: 4data from “WEB OF SCIENCE” (of Thomson Reuters) are update at: 2025-09-14References taken from IsiWeb of Knowledge: (subscribers only)
