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

Abstract

The geometrical critical behaviour of the two-dimensionalQ-state Potts model is usually studied in terms of the Fortuin–Kasteleyn (FK) clusters, ortheir surrounding loops. In this paper we study a quite different geometrical object: thespin clusters, defined as connected domains where the spin takes a constant value. Unlikethe 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 separatespin clusters of different or identical colours. For these reasons their critical behaviour isdifferent from, and richer than, those of FK clusters. We develop a transfer matrixtechnique enabling the formulation and numerical study of spin clusters even whenQ is not an integer. We further identify geometrically the crossing events which give rise toconformal correlation functions. We study the critical behaviour both in the bulk,and at a boundary with free, fixed, or mixed boundary conditions. This leads toinfinite series of fundamental critical exponents, in the bulk and at the boundary, valid for 0 ≤ Q ≤ 4, that describe the insertion of thin and 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 derivedanalytically from a massless scattering approach.