7 research outputs found
Critical points in the model
The space of solutions of the exact renormalization group fixed point
equations of the two-dimensional model, which we recently obtained
within the scale invariant scattering framework, is explored for continuous
values of . Quasi-long-range order occurs only for , and allows
for several lines of fixed points meeting at the BKT transition point. A rich
pattern of fixed points is present below , while only zero
temperature criticality in the universality class can occur
above this value. The interpretation of an extra solution at requires the
identitication of a path to criticality specific to this value of
On the and universality classes
We recently determined the exact fixed point equations and the spaces of
solutions of the two-dimensional and models using scale
invariant scattering theory. Here we discuss subtleties hidden in some
solutions and related to the difference between ferromagnetic and
antiferromagnetic interaction
Critical points in coupled Potts models and correlated percolation
We use scale invariant scattering theory to exactly determine the renormalization group fixed points of a q-state Potts model coupled to an r-state Potts model in two dimensions. For integer values of q and r the fixed point equations are very constraining and show in particular that scale invariance in coupled Potts ferromagnets is limited to the Ashkin-Teller case (q = r = 2). Since our results extend to continuous values of the number of states, we can access the limit r -> 1 corresponding to correlated percolation, and show that the critical properties of Potts spin clusters cannot in general be obtained from those of Fortuin-Kasteleyn clusters by analytical continuation
Critical points of coupled vector-Ising systems. Exact results
We show that scale-invariant scattering theory allows to exactly determine the critical points of two-dimensional systems with coupled O(N) and Ising order parameters. The results are obtained for N continuous and include criticality of the loop gas type. In particular, for N = 1 we exhibit three critical lines intersecting at the Berezinskii Kosterlitz Thouless transition point of the Gaussian model and related to the Z4 symmetry of the isotropic Ashkin Teller model. For N = 2 we classify the critical points that can arise in the XY-Ising model and provide exact answers about the critical exponents of the fully frustrated XY model
Exact results for the O(N) model with quenched disorder
We use scale invariant scattering theory to exactly determine the lines of renormalization group fixed points for O(N)-symmetric models with quenched disorder in two dimensions. Random fixed points are characterized by two disorder parameters: a modulus that vanishes when approaching the pure case, and a phase angle. The critical lines fall into three classes depending on the values of the disorder modulus. Besides the class corresponding to the pure case, a second class has maximal value of the disorder modulus and includes Nishimori-like multicritical points as well as zero temperature fixed points. The third class contains critical lines that interpolate, as N varies, between the first two classes. For positive N , it contains a single line of infrared fixed points spanning the values of N from 2 121 to 1. The symmetry sector of the energy density operator is superuniversal (i.e. N -independent) along this line. For N = 2 a line of fixed points exists only in the pure case, but accounts also for the Berezinskii-Kosterlitz-Thouless phase observed in presence of disorder
Critical lines in the pure and disordered O(N) model
We consider replicated symmetry in two dimensions within the exact framework of scale invariant scattering theory and determine the lines of renormalization group fixed points in the limit of zero replicas corresponding to quenched disorder. A global pattern emerges in which the different critical lines are located within the same parameter space. Within the subspace corresponding to the pure case (no disorder) we show how the critical lines for non-intersecting loops () are connected to the zero temperature critical line () via the BKT line at . Disorder introduces two more parameters, one of which vanishes in the pure limit and is maximal for the solutions corresponding to Nishimori-like and zero temperature critical lines. Emergent superuniversality (i.e. -independence) of some critical exponents in the disordered case and disorder driven renormalization group flows are also discussed