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arXiv:1709.00364 · Journal of Statistical Mechanics Theory and Experiment

On superuniversality in the q -state Potts model with quenched disorder

Gesualdo Delfino, Elena Tartaglia

Published Dec 1, 2017• 20 citations
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Abstract

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.

DOI
10.1088/1742-5468/aa9bad

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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.
We introduce an exact replica method for the study of critical systems with quenched bond randomness in two dimensions. For the q-state Potts model, we show that a line of renormalization group fixed points interpolates from weak to strong randomness as q-2 grows from small to large values. This theory exhibits a q-independent sector, and allows at the same time for a correlation length exponent which keeps the Ising value and continuously varying magnetization exponent and effective central charge. These findings appear to solve long-standing numerical and theoretical puzzles, and to illustrate the peculiarities which may characterize the conformal field theories of random fixed points.
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.
Abstract 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.
The random qqq-state quantum Potts model is studied on hypercubic lattices in dimensions 2 and 3 using the numerical implementation of the strong disorder renormalization group introduced by Kovacs and Igl\'oi [Phys. Rev. B 82, 054437 (2010)]. Critical exponents $\ensuremath{\nu},$ df,{d}_{f},df​, and $\ensuremath{\psi}$ at the infinite disorder fixed point are estimated by finite-size scaling for several numbers of states qqq between 2 and 50. When scaling corrections are not taken into account, the estimates of both df{d}_{f}df​ and $\ensuremath{\psi}$ systematically increase with qqq. It is shown, however, that qqq-dependent scaling corrections are present and that the exponents are compatible within error bars, or close to each other, when these corrections are taken into account. This provides evidence of the existence of a superuniversality of all two- and three-dimensional random Potts models.
The two-dimensional q-state Potts model is subjected to a Z(q) symmetric disorder that allows for the existence of a Nishimori line. At q=2, this model coincides with the +/- J random-bond Ising model. For q>2, apart from the usual pure- and zero-temperature fixed points, the ferro/paramagnetic phase boundary is controlled by two critical fixed points: a weak disorder point, whose universality class is that of the ferromagnetic bond-disordered Potts model, and a strong disorder point which generalizes the usual Nishimori point. We numerically study the case q=3, tracing out the phase diagram and precisely determining the critical exponents. The universality class of the Nishimori point is inconsistent with percolation on Potts clusters.