Bulk and boundary critical behaviour of thin and thick domain walls in the two-dimensional Potts model

Abstract

The geometrical critical behaviour of the two-dimensional Q-state Potts model is usually studied in terms of the Fortuin-Kasteleyn (FK) clusters, or their surrounding loops. In this paper we study a quite different geometrical object: the spin clusters, defined as connected domains where the spin takes a constant value. Unlike the usual loops, the domain walls separating different spin clusters can cross and branch. Moreover, they come in two versions, "thin" or "thick", depending on whether they separate spin clusters of different or identical colours. For these reasons their critical behaviour is different from, and richer than, those of FK clusters. We develop a transfer matrix technique enabling the formulation and numerical study of spin clusters even when Q is not an integer. We further identify geometrically the crossing events which give rise to conformal correlation functions. We study the critical behaviour both in the bulk, and at a boundary with free, fixed, or mixed boundary conditions. This leads to infinite series of fundamental critical exponents, h_{l_1-l_2,2 l_1} in the bulk and h_{1+2(l_1-l_2),1+4 l_1} at the boundary, valid for 0 <= Q <= 4, that describe the insertion of l_1 thin and l_2 thick domain walls. We argue that these exponents imply that the domain walls are `thin' and `thick' also in the continuum limit. A special case of the bulk exponents is derived analytically from a massless scattering approach.