Conformal Invariance and Critical Percolation
Conformal Invariance and Critical Percolation
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Date
2013-06-30
Auteurs
Sahabi Toufik
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Université oran1 Ahmed Ben Bella
Résumé
Quantum Field Theory (QFT) is the framework for studying systems with an infinite number of degrees of freedom. Initially, it is developed to describe Quantum Electrodynamics (QED), and becomes after a powerful tool for studying elementary particles, condensed matter, string theory and statistical physics.
In what follows, we are particularly interested in the study of the critical behavior of 2D statistical systems as Conformal Field Theories defined either on the full plane or on bounded geometry. Such a quantum field theories have the property to be invariant by a group symmetry which contains in addition to the Euclidean symmetries, conformal transformations, i.e. transformations that preserve angles but not necessary lengths, as the scale transformation. Because of their scale invariance at criticality, the statistical systems will be naturally described in the continuum limit (when the lattice spacing is very small) by conformaly invariant QFT. Ising model, Q-States Potts model, as well as Percolation are among the statistical systems which are well studied as CFT.
The most interesting case is that of two dimensional theories. In that case, the conformal invariance is of special importance since the corresponding algebra is infinite dimensional. Then, as a consequence, we have an infinite number of conserved quantities which permit to derive many quantities, as the correlation functions by symmetry considerations alone. We will see in this thesis that we can derive in this way an important physical quantity in percolation theory: The Crossing probability.
The critical behavior of statistical systems defined on a finite-size region in which the boundary conditions must conserve the conformal symmetry, are studied in the framework of Boundary conformal field theory (BCFT) where we exclude all unphysical conditions (which break the conformal symmetry) and determine the appropriate boundary states. These constraints induce some changes in partition functions and correlation functions. The geometry with boundary is the upper half-plane or any bounded region obtained from it by conformal mapping.
In this thesis, we will provide some techniques and results from the Coulomb-Gas Formalism adapted to boundary conformal field theory to study of critical percolation phenomena and to determine the probability that exist a connect path between two exterior disjoint segments of a medium.
The thesis is organized as follows: Chapter 1 contains basic material of conformal field theory, which serves as the foundation for the remainder of the thesis. We discuss its fundamental techniques such as: Conformal group, primary fields, correlation functions, Virasoro algebra, and on a more abstract level, we study the effects of the insertion of boundaries on the conformal properties of such systems.
Chapter 2 gives the basic steps in the Coulomb-gas Formalism, which permits to obtain the minimal models from some deformation of the free boson theory. It provides also a powerful method for calculating correlation functions without having to solve any differential equation. In the last section we will focus in an approach, developed by S. Kawai to calculate correlation functions in boundary conformal field theories.
In chapter 3, we will present a full description of the general percolation theory as a geometrical phase transition and describe with few details the critical behavior of certain fundamental quantities such as percolation probability, mean size of clusters, and correlation length. We will also discuss some properties and applications of percolation theory in one and two dimension. In advanced level, we will explain the conformal invariance of percolation theory at criticality, and then give its appropriate conformal model. Finally, we will describe the Cardy's method for deriving the crossing probability.
The chapter 4 is devoted to develop our approach to compute the percolation crossing probability based on Kawai's method in the Coulomb-gas formalism. This method was first developed to obtain the BCFTs describing the critical behavior of discrete statistical models (as Ising model) in bounded geometries. The last chapter is a general conclusion.
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Mots-clés
Kawai’s Method,Percolation Theory,Coulomb-Gas Formalism,Critical Statistical Systems,
Crossing Probability,Boundary Conformal Field Theory