Critical exponents of domain walls in the two-dimensional Potts model

Abstract

We address the geometrical critical behavior of the two-dimensional Q-state Potts model in terms of the spin clusters (i.e., connected domains where the spin takes a constant value). These clusters are different from the usual Fortuin-Kasteleyn clusters, and are separated by domain walls that can cross and branch. 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. This leads to an infinite series of fundamental critical exponents h 1 − 2 ,2 1 , valid for 0 ≤ Q ≤ 4, that describe the insertion of 1 thin and 2 thick domain walls.

Bootstrap approach to geometrical four-point functions in the two-dimensional critical Q-state Potts model: a study of the s-channel spectra

“…More interesting and general results can however be obtained by moving to the cluster or loop formulations, in which the correlation functions acquire a geometrical content. In the same vein, the spin correlators can be analytically continued from Q ∈ N to arbitrary real values, in which case they also acquire a geometrical interpretation [12][13][14] to which we shall return in a short while. Such generalisations to Q ∈ R are not only useful, but actually indispensable in our case, since our main objective is to study correlation functions in the generic case where q is not a root of unity.…”

Fields, particles and universality in two dimensions

“…For the crossover line in the dilute Ising model, the mass scale M appearing in (99) is associated to the coupling conjugated to the relevant field ϵ ′ , which is proportional to M 2−2∆ ε ′ . The high energy limit M 2 /s → 0 in which this coupling becomes negligible is that towards the tricritical point, while the critical point of the undilute model is approached as M 2 /s → ∞.…”

“…[96,97] for reviews). When passing from Ising to the three-state Potts model, the paths connecting the boundary condition changing points can branch even on the honeycomb lattice, and this case has been studied at criticality using Monte Carlo [98] and transfer matrix [99] techniques. Looking for a relation between these lattice paths at criticality and the off-critical interfaces characterized earlier in this section directly in the continuum appears as an interesting subject for future research.…”

Connectivities of Potts Fortuin–Kasteleyn clusters and time-like Liouville correlator

“…Exceptions are the results derived from the complementary SLE [4,13,14] or boundary CFTs approaches [15][16][17] which are mainly based on the use of Fuchsian-type partial differential equations satisfied by probability functions. On the other hand, the effects of discrete symmetries which arise in pure [18][19][20] and disordered [21,22] models is not understood: for example, the behavior of spin domain walls is basically unknown. A better knowledge of the CFTs describing extended objects will pave the way to the computation of important unknown observables.…”