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arXiv:2104.02490 · Physical review. B./Physical review. B

Numerical evidence of superuniversality of the two-dimensional and three-dimensional random quantum Potts models

Valentin Anfray, Christophe Chatelain

Published May 21, 2021• 3 citations
arXivHTMLPDF

Abstract

The random $q$-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 $\ensuremath{\nu},$ ${d}_{f},$ and $\ensuremath{\psi}$ at the infinite disorder fixed point are estimated by finite-size scaling for several numbers of states $q$ between 2 and 50. When scaling corrections are not taken into account, the estimates of both ${d}_{f}$ and $\ensuremath{\psi}$ systematically increase with $q$. It is shown, however, that $q$-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.

DOI
10.1103/physrevb.103.174207

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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 \ensuremathν,\ensuremath{\nu},\ensuremathν, df,{d}_{f},df​, and \ensuremathψ\ensuremath{\psi}\ensuremathψ 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 \ensuremathψ\ensuremath{\psi}\ensuremathψ 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 random quantum q-state clock and Potts models are studied in two and three dimensions. The existence of Griffiths phases is tested in the two-dimensional case with q=6 by sampling the integrated probability distribution of local susceptibilities of the equivalent McCoy-Wu three-dimensional classical models with Monte Carlo simulations. For the random Potts model, numerical evidence of the existence of Griffiths phases is given and the finite-size effects are analyzed. For the clock model, the data also suggest the existence of a Griffiths phase but with much larger finite-size effects. The critical point of the random quantum clock model is then studied with the Strong-Disorder Renormalization Group. Evidence is given that, at strong enough disorder, this critical behavior is governed by the same infinite-disorder fixed point as the Potts model, for all the number of states q considered. At weak disorder, our renormalization group method becomes unstable and does not allow us to make conclusions.
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.
The random quantum qqq-state clock and Potts models are studied in 2 and 3 dimensions. The existence of Griffiths phases is tested in the 2D case with q=6q=6q=6 by sampling the integrated probability distribution of local susceptibilities of the equivalent McCoy-Wu 3D classical modelswith Monte Carlo simulations. No Griffiths phase is found for the clock model. In contrast, numerical evidences of the existence of Griffiths phases in the random Potts model are given and the Finite Size effects are analyzed. The critical point of the random quantum clock model is then studied by Strong-Disorder Renormalization Group. Despite a chaotic behavior of the Renormalization-Group flow at weak disorder, evidences are given that this critical behavior is governed by the same Infinite-Disorder Fixed Point as the Potts model, independently from the number of states qqq.
Using a very efficient numerical algorithm of the strong disorder renormalization group method, we have extended the investigations about the critical behavior of the random transverse-field Ising model in three and four dimensions, as well as for Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi random graphs, which represent infinite dimensional lattices. In all studied cases, an infinite disorder quantum critical point is identified, which ensures that the applied method is asymptotically correct and the calculated critical exponents tend to the exact values for large scales. We have found that the critical exponents are independent of the form of (ferromagnetic) disorder and they vary smoothly with the dimensionality.
Three-state Potts ferromagnets on the simple cubic lattice in random fields are studied by extensive Monte Carlo simulation. The implications of the possible hyperscaling violation to finite-size scaling are discussed. Evidence for a second-order transition is obtained, as well as rough estimates for critical exponents. A qualitative phase diagram is proposed compatible with the predictions of Blankschtein et al. on the random field-induced shift of the tricritical curve q = q c ( d ) in the plane of variables dimensionality d and number of states q .
We study the critical behavior of the random q-state Potts quantum chain by density-matrix renormalization techniques. Critical exponents are calculated by scaling analysis of finite lattice data of short chains (L< 16)(L<~16)(L< 16) averaging over all possible realizations of disorder configurations chosen according to a binary distribution. Our numerical results show that the critical properties of the model are independent of q in agreement with a renormalization group analysis of Senthil and Majumdar [Phys. Rev. Lett. 76, 3001 (1996)]. We show how an accurate analysis of moments of the distribution of magnetizations allows a precise determination of critical exponents, circumventing some problems related to binary disorder. Multiscaling properties of the model and dynamical correlation functions are also investigated.