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arXiv:1707.00998 · Physical review. E

Classifying Potts critical lines

Gesualdo Delfino A, Elena Tartaglia

Published Oct 16, 2017• 25 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.96.042137

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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.
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.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.
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 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 2−1 \sqrt{2}-1 2​−1 2 − 1 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.
Abstract The space of solutions of the exact renormalization group fixed point equations of the two-dimensional RP N −1 model, which we recently obtained within the scale invariant scattering framework, is explored for continuous values of N ⩾ 0. Quasi-long-range order occurs only for N = 2, and allows for several lines of fixed points meeting at the Berezinskii–Kosterlitz–Thouless transition point. A rich pattern of fixed points is present below N * = 2.244 21‥, while only zero temperature criticality in the O ( N ( N + 1)/2 − 1) universality class can occur above this value. The interpretation of an extra solution at N = 3 requires the identification of a path to criticality specific to this value of N .
We study a square-lattice three-state Potts antiferromagnet with a staggered polarization field at finite temperature. Numerically treating the transfer matrices, we determine two phase boundaries separating the model-parameter space into three parts. We confirm that one of them belongs to the ferromagnetic three-state Potts criticality, which is in accord with a recent prediction, and another to the Ising-type; these are both corresponding to the massless renormalization-group flows stemming from the Gaussian fixed points. We also discuss a field theory to describe the latter Ising transition.