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.

18 pages, 5 figures, 2 tables. Work based on the invited talk given by Jesper L. Jacobsen at STATPHYS-24 in Cairns, Australia (July 2010). Extended version of arXiv:1008.1216

References in corpus (10)

• 2D growth processes: SLE and Loewner chains
• Conformal boundary loop models
• Topological order from quantum loops and nets
• Conformal boundary conditions in the critical O(n) model and dilute loop models
• SLE in the three-state Potts model - a numerical study
• Critical exponents of domain walls in the two-dimensional Potts model
• Loop models and their critical points
• Some geometric critical exponents for percolation and the random-cluster model
• Critical interfaces of the Ashkin-Teller model at the parafermionic point
• Single-cluster dynamics for the random-cluster model

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• Critical properties of joint spin and Fortuin-Kasteleyn observables in the two-dimensional Potts model
• Critical interfaces and duality in the Ashkin Teller model
• Recursive Percolation
• Critical properties of quantum three- and four-state Potts models with boundaries polarized along the transverse field
• $U_q({\mathfrak{sl}}_n)$ web models and ${\mathbb{Z}}_n$ spin interfaces
• Critical points in coupled Potts models and correlated percolation
• On the CFT describing the spin clusters in 2d Potts model
• Bootstrap approach to geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: A study of the $s$-channel spectra
• $U_{\mathfrak{q}}({\mathfrak{sl}}_3)$ web models: Locality, phase diagram and geometrical defects
• The bulk one-arm exponent for the CLE${}_{{\kappa }^{\prime }}$ percolations