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arXiv:1810.10334 · Journal of Statistical Mechanics Theory and Experiment

Critical lines in the pure and disordered O ( N ) model

Gesualdo Delfino, Noel Lamsen

Published Feb 1, 2019• 12 citations
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Abstract

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.

DOI
10.1088/1742-5468/aaf716

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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.
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 numerically study the phase diagram and critical properties of the two-dimensional disordered O(n) loop model by using the transfer matrix and the worm Monte Carlo methods. The renormalization group flow is extracted from the landscape of the effective central charge obtained by the transfer matrix method based on the Zamolodchikov's C-theorem. We find a line of random fixed points (FPs) for nc<n<1n_c < n <1nc​<n<1, with nc∼0.5n_c \sim 0.5nc​∼0.5, for which the central charge and critical exponents agree well with the results of the 1−n1-n1−n perturbative expansion. Furthermore, for n>ncn> n_cn>nc​, we find a line of multicritical FPs at strong randomness. The FP at n=1n=1n=1 has c=0.4612(4)c=0.4612(4)c=0.4612(4), which suggests that it belongs to the universality class of the Nishimori point in the ±J\pm J±J random-bond Ising model. For n>2n>2n>2, we find another critical line that connects the hard-hexagon FP in the pure model to a finite-randomness zero-temperature FP.
Abstract We use scale invariant scattering theory to obtain the exact equations determining the renormalization group fixed points of the two-dimensional CP N −1 model, for N real. Also due to special degeneracies at N = 2 and 3, the space of solutions for N ⩾ 2 reduces to that of the O ( N 2 − 1) model, and accounts for a zero temperature critical point. For N < 2 the space of solutions becomes larger than that of the O ( N 2 − 1) model, with the appearance of new branches of fixed points relevant for criticality in gases of intersecting loops.
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.