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arXiv:1602.07229 · Physical review. B./Physical review. B

Critical behavior of the two-dimensional Ising model with long-range correlated disorder

M. Dudka, Andrei A. Fedorenko, Viktoria Blavatska, Yurij V. Holovatch

Published Jun 22, 2016• 25 citations
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Abstract

We study critical behavior of the diluted two-dimensional Ising model in the presence of disorder correlations which decay algebraically with distance as $\ensuremath{\sim}{r}^{\ensuremath{-}a}$. Mapping the problem onto two-dimensional Dirac fermions with correlated disorder we calculate the critical properties using renormalization group up to two-loop order. We show that beside the Gaussian fixed point the flow equations have a nontrivial fixed point which is stable for $0.995<a<2$ and is characterized by the correlation length exponent $\ensuremath{\nu}=2/a+O({(2\ensuremath{-}a)}^{3})$. Using bosonization, we also calculate the averaged square of the spin-spin correlation function and find the corresponding critical exponent ${\ensuremath{\eta}}_{2}=1/2\ensuremath{-}(2\ensuremath{-}a)/4+O({(2\ensuremath{-}a)}^{2})$.

DOI
10.1103/physrevb.93.224422

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We study critical behavior of the diluted two-dimensional Ising model in the presence of disorder correlations which decay algebraically with distance as \ensuremath∼r\ensuremath−a\ensuremath{\sim}{r}^{\ensuremath{-}a}\ensuremath∼r\ensuremath−a. Mapping the problem onto two-dimensional Dirac fermions with correlated disorder we calculate the critical properties using renormalization group up to two-loop order. We show that beside the Gaussian fixed point the flow equations have a nontrivial fixed point which is stable for 0.995<a<20.995<a<20.995<a<2 and is characterized by the correlation length exponent \ensuremathν=2/a+O((2\ensuremath−a)3)\ensuremath{\nu}=2/a+O({(2\ensuremath{-}a)}^{3})\ensuremathν=2/a+O((2\ensuremath−a)3). Using bosonization, we also calculate the averaged square of the spin-spin correlation function and find the corresponding critical exponent \ensuremathη2=1/2\ensuremath−(2\ensuremath−a)/4+O((2\ensuremath−a)2){\ensuremath{\eta}}_{2}=1/2\ensuremath{-}(2\ensuremath{-}a)/4+O({(2\ensuremath{-}a)}^{2})\ensuremathη2​=1/2\ensuremath−(2\ensuremath−a)/4+O((2\ensuremath−a)2).
In this paper we provide new analytic results on two-dimensional q q -Potts models ( q ≥ 2 q ≥ 2 ) in the presence of bond disorder correlations which decay algebraically with distance with exponent a a . In particular, our results are valid for the long-range bond disordered Ising model ( q=2 q = 2 ). We implement a renormalization group perturbative approach based on conformal perturbation theory. We extend to the long-range case the RG scheme used in [V. Dotsenko et al., Nucl. Phys. B 455 701-23] for the short-range disorder. Our approach is based on a 2 2 -loop order double expansion in the positive parameters (2-a) ( 2 − a ) and (q-2) ( q − 2 ) . We will show that the Weinrib-Halperin conjecture for the long-range thermal exponent can be violated for a non-Gaussian disorder. We compute the central charges of the long-range fixed points finding a very good agreement with numerical measurements.
The critical behavior of the random transverse-field Ising model in finite-dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization-group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in d=1,2d=1,2d=1,2, and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In d=1d=1d=1 the results are related to dimer-dimer correlations in the random antiferromagnetic XXXXXX chain and have been tested by numerical calculations using free-fermionic techniques.
We analyse the critical properties of a weakly diluted (random) Ising model with the long-range interaction decaying with distance x as ∼ x - d - σ in a d-dimensional space. It is known to belong to a new long-range random universality class for certain values of the decay parameter σ. Exploiting the field-theoretic renormalization group approach within the minimal subtraction scheme, we compute the three-loop renormalization group functions. On their basis, with the help of asymptotic series resummation methods, we estimate the correlation length critical exponent ν characterising the new universality class for d = 3 and for those values of σ for which long-range interactions are relevant for the critical behaviour.
We study the critical behavior of the 2D N-color Ashkin-Teller model in the presence of random bond disorder whose correlations decays with the distance r as a power-law r -a .We consider the case when the spins of different colors sitting at the same site are coupled by the same bond and map this problem onto the 2D system of N/2 flavors of interacting Dirac fermions in the presence of correlated disorder.Using renormalization group we show that for N = 2, a "weakly universal" scaling behavior at the continuous transition becomes universal with new critical exponents.For N > 2, the first-order phase transition is rounded by the correlated disorder and turns into a continuous one.
An important yet largely unsolved problem in the statistical mechanics of disordered quantum systems is to understand how quenched disorder affects quantum phase transitions in systems of itinerant fermions. In the clean limit, continuous quantum phase transitions of the symmetry-breaking type in Dirac materials such as graphene and the surfaces of topological insulators are described by relativistic (2+1)-dimensional quantum field theories of the Gross-Neveu-Yukawa (GNY) type. We study the universal critical properties of the chiral Ising, XY, and Heisenberg GNY models perturbed by quenched random-mass disorder, both uncorrelated or with long-range power-law correlations. Using the replica method combined with a controlled triple epsilon expansion below four dimensions, we find a variety of new finite-randomness critical and multicritical points with nonzero Yukawa coupling between low-energy Dirac fields and bosonic order parameter fluctuations, and compute their universal critical exponents. Analyzing bifurcations of the renormalization-group flow, we find instances of the fixed-point annihilation scenario—continuously tuned by the power-law exponent of long-range disorder correlations and associated with an exponentially large crossover length—as well as the transcritical bifurcation and the supercritical Hopf bifurcation. The latter is accompanied by the birth of a stable limit cycle on the critical hypersurface, which represents the first instance of fermionic quantum criticality with emergent discrete scale invariance.