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arXiv:2208.14844 · Journal of Statistical Mechanics Theory and Experiment

Critical points in coupled Potts models and correlated percolation

Noel Lamsen, Youness Diouane, Gesualdo Delfino A

Published Jan 1, 2023• 5 citations
arXivHTMLPDF

Abstract

Abstract 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.

DOI
10.1088/1742-5468/aca901

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Abstract 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.
We use scale invariant scattering theory to exactly determine the lines of renormalization group fixed points invariant under the permutational symmetry S_{q} in two dimensions, and we show how one of these scattering solutions describes the ferromagnetic and square lattice antiferromagnetic critical lines of the q-state Potts model. Other solutions we determine should correspond to new critical lines. In particular, we obtain that a S_{q}-invariant fixed point can be found up to the maximal value q=(7+sqrt[17])/2. This is larger than the usually assumed maximal value 4 and leaves room for a second-order antiferromagnetic transition at q=5.
Determining whether an additional local symmetry affects the universality class of a sta-tistical model is an important issue in the theory of critical phenomena. A basic example is provided by the RP N −1 model, in which N -component spin variables at each lattice site interact through an Hamiltonian invariant under global O(N ) rotations and local spin re-versals. The local symmetry makes the difference with the usual O(N ) model and amounts to the head-tail symmetry characteristic of liquid crystals. In three dimensions, the weak first order transition observed in numerical simulations of the ferromagnetic model is con-sistent with the mean field scenario. On the other hand, in the two-dimensional case –the one we focus on in this thesis – fluctuations are stronger and minimize the reliability of mean field predictions, as illustrated by the phase transition of the three-state Potts model, which becomes continuous on planar lattices. For the RP N −1 model, the absence of spontaneous breaking of continuous symmetry in two dimensions generically suggests that criticality is limited to zero temperature, and Monte Carlo studies for T → 0 showed a fast growth of the correlation length which made particularly hard to reach the asymp-totic limit and draw conclusions. On the other hand, the possibility of finite temperature topological transitions similar to the Berezinskii-Kosterlitz-Thouless (BKT) one – which should definitely occur for RP 1 ∼ O(2) – and mediated by ”disclination” defects has also been debated in numerical studies. While two-dimensional criticality has allowed for an impressive amount of exact solutions thanks to lattice integrability and conformal field theory, models with local symmetries traditionally remained outside the range of application of these methods. In this thesis, however, we show how the renormalization group fixed points of the RP N −1 model and of its complex generalization, the CP N −1 model, can be accessed in an exact way. This is achieved in the scale invariant scattering framework which, as we review in the introductory part of the thesis, implements in the basis of particle excitations the infinite-dimensional conformal symmetry characteristic of critical points in two dimensions and has provided in the last few years new results for pure and disordered systems. In the last part of the thesis, we will exploit the generality of the scale invariant scattering method to progress with another long standing problem of two-dimensional criticality, namely that of spin clusters in Potts correlated percolation. It has been known for long time that the problem can be addressed considering two coupled Potts models, but the need to consider the number of states of one of these models as a continuous variable has severely limited the analytical or numerical study of the critical points. Also here, the ability of the scattering method to enforce conformal invariance for the internal symmetry characteristic of the universality class will allow us to obtain exact equations for the critical points and to determine their solutions in the relevant limits.
Abstract We obtain the exact scale invariant scattering solutions for two-dimensional field theories with replicated permutational symmetry S q . After sending to zero the number of replicas they correspond to the renormalization group fixed points of the q -state Potts model with quenched disorder. We find that all solutions with non-zero disorder possess q -independent sectors, pointing to superuniversality (i.e. symmetry independence) of some critical exponents. The solution corresponding to the random bond ferromagnet, for which disorder vanishes as q → 2 , allows for superuniversality of the correlation length exponent ν (Delfino 2017 Phys. Rev. Lett . 118 250601). Of the two solutions which are strongly disordered for all values of q , one is completely q -independent and accounts for the zero-temperature percolation fixed point of the randomly bond diluted ferromagnet. The other is the main candidate to describe the Nishimori-like fixed point of the Potts model with ± J disorder, and leaves room for superuniversality of the magnetic exponent η , a possibility not yet excluded by available numerical data.
Abstract The N -color Ashkin–Teller model corresponds to N Ising models coupled by four-spin interactions. We consider the two-dimensional case in presence of quenched disorder and use scale invariant scattering theory to determine all the solutions of the exact renormalization group fixed points equations. The weak disorder sector is characterized by a solution that, for any fixed N larger than 1, is a line of fixed points with Ising thermal exponents and continuously varying magnetic exponents. The number of fixed point solutions allowed by the symmetries of the model increases at strong disorder illustrating the growing dependence on the distributions of the two random couplings. The presence of some critical exponents which do not depend on the symmetry parameter N confirms this type of superuniversality as a peculiar feature of random criticality.
We consider the q-state Potts model with correlated random interactions such that their correlations decay at large distances as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}a)}$. Applying the renormalization group and a double (\ensuremath{\epsilon},${\ensuremath{\epsilon}}_{a}$) expansion, we derive the scaling form of the equation of state to linear order in \ensuremath{\epsilon}=6-d and ${\ensuremath{\epsilon}}_{a}$=a-2. In the limit q\ensuremath{\rightarrow}1, which describes correlated percolation, the critical amplitudes ${A}^{\ifmmode\pm\else\textpm\fi{}}$ associated with the mean number of clusters F\ensuremath{\sim}${A}^{\ifmmode\pm\else\textpm\fi{}}$\ensuremath{\Vert}t${\ensuremath{\Vert}}^{2\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$ are shown to diverge when \ensuremath{\alpha} approaches a negative integer as a function of ${\ensuremath{\epsilon}}_{a}$.
Abstract We show that scale-invariant scattering theory allows to exactly determine the critical points of two-dimensional systems with coupled 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 Z 4 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.