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Physical review. B, Condensed matter

Phase transitions in systems with correlated disorder

J.C. Lee, Richard L. Gibbs

Published Feb 1, 1992• 17 citations
Publisher

Abstract

For random systems with quenched disorders distributed randomly in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions and perfectly correlated in the remaining ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ dimensions, the renormalization-group theory has been performed in the past in two different ways. In the single-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as finite and expands the perturbation terms only for \ensuremath{\epsilon}=${\mathit{d}}_{\mathit{c}}$-d, where ${\mathit{d}}_{\mathit{c}}$ is the upper critical dimension. In the double-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as infinitesimal and performs an additional expansion for ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$. The former predicts smeared phase transitions, while the latter predicts sharp second-order phase transitions. We wish to find out which of the two predictions holds for a three-dimensional Ising model for which ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$=2 and ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$=1. We argue, using the replica method, that the spin correlation is isotropic for large distances as long as the fluctuation among the random bonds (in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions) is sufficiently smaller than the thermal energy ${\mathit{k}}_{\mathit{B}}$T. With parameters chosen to satisfy this condition in the model, a Monte Carlo simulation has been performed. The results favor a sharp second-order phase transition.

DOI
10.1103/physrevb.45.2217

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For random systems with quenched disorders distributed randomly in \ensuremathϵR{\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}\ensuremathϵR​ dimensions and perfectly correlated in the remaining \ensuremathϵC{\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}\ensuremathϵC​ dimensions, the renormalization-group theory has been performed in the past in two different ways. In the single-expansion method, one treats \ensuremathϵC{\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}\ensuremathϵC​ as finite and expands the perturbation terms only for \ensuremath{\epsilon}=dc{\mathit{d}}_{\mathit{c}}dc​-d, where dc{\mathit{d}}_{\mathit{c}}dc​ is the upper critical dimension. In the double-expansion method, one treats \ensuremathϵC{\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}\ensuremathϵC​ as infinitesimal and performs an additional expansion for \ensuremathϵC{\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}\ensuremathϵC​. The former predicts smeared phase transitions, while the latter predicts sharp second-order phase transitions. We wish to find out which of the two predictions holds for a three-dimensional Ising model for which \ensuremathϵC{\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}\ensuremathϵC​=2 and \ensuremathϵR{\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}\ensuremathϵR​=1. We argue, using the replica method, that the spin correlation is isotropic for large distances as long as the fluctuation among the random bonds (in \ensuremathϵR{\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}\ensuremathϵR​ dimensions) is sufficiently smaller than the thermal energy kB{\mathit{k}}_{\mathit{B}}kB​T. With parameters chosen to satisfy this condition in the model, a Monte Carlo simulation has been performed. The results favor a sharp second-order phase transition.
In quenched random-spin systems, the renormalization group can be used to develop recursion relations for the probability distribution for random potentials. Alternatively, recursion relations for the average values of potentials and their higher cumulants can be obtained. In this paper, the above technique is used to study phase transitions in quenched random nnn-component classical spin systems using the $\ensuremath{\epsilon}$ expansion to second order in $\ensuremath{\epsilon}$. It there are long-range correlations in the random potentials (e.g., all potentials along a line are equal), there are no stable physical fixed points within the $\ensuremath{\epsilon}$ expansion. This is interpreted as a smeared transition. If there are no long-range correlations in the random potentials, there is a sharp transition with pure system exponents if the specific-heat exponent ${\ensuremath{\alpha}}^{H}$ of the pure system is negative. If ${\ensuremath{\alpha}}^{H}>0$ and n>1n>1n>1, there is a sharp transition with new exponents $\ensuremath{\eta}=[\frac{(5{n}^{2}\ensuremath{-}8n)}{256{(n\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}$ and $\ensuremath{\nu}=\frac{1}{2}+[\frac{3n}{32(n\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{n(127{n}^{2}\ensuremath{-}572n\ensuremath{-}32)}{4096{(n\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}$. For n=1n=1n=1, there is no stable fixed point, which is again interpreted as a smeared transition.
The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute mmm-vector model, the n=0n=0n=0 limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic mmm-component fixed point, which characterizes the critical behavior of the pure system, is stable. For m<mcm<{m}_{c}m<mc​, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for m=1m=1m=1. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.
Disorder is an inevitable part of any condensed matter system and therefore its study has always been of great importance. The effect of quenched randomness on a system that exhibits a continuous phase transition in the absence of any impurity has been studied in the past and the results are relatively well understood. However, the effect of quenched randomness on \\emph{first-order} phase transitions is still not well understood. In this dissertation, we study the effect of quenched bond-randomness on the classical and quantum first-order phase transitions.In Chapter 2, we study the effect of the disordered three-color Ashkin-Teller model, whose pure version undergoes a first-order phase transition. We show that the rounding of the first-order transition of the pure model due to the impurities is manifested as a critical point. We conclusively rule out that the model belongs to the universality class of the two-dimensional Ising model. Furthermore, we find that the exponents beta\\betabeta and nu\\nunu vary with disorder and the four-spin coupling strength.In Chapter 3, we extend our study of the disordered three-color Ashkin-Teller model. Utilizing extensive cluster Monte Carlo simulations on large lattice sizes of up to 128times128128 \\times 128128times128 spins, each of which is represented by three colors taking values pm1\\pm 1pm1, we show that the rounding of the first-order phase transition is an emergent criticality.We find that the critical exponents, nu\\nunu and beta\\betabeta, change as the strength of disorder or the four-spin coupling varies, and we show that the correlation length critical exponent violates the lower bound 2/Dlenu2/D \\le \\nu2/Dlenu, where DDD is the dimension of the system.In Chapter 4, we study the \\emph{quantum} three-color Ashkin--Teller model and show that the quantum critical point in (1+1)(1+1)(1+1) dimension is an unusual one, with activated scaling at the critical point and Griffiths-McCoy phase away from it. We find that the behavior is similar to the transverse random field Ising model, even though the pure system has a first-order transition in this case.
The influence of uncorrelated, quenched disorder on the phase transition of two dimensional Potts models will be reviewed. After an introduction where the conditions of relevance of quenched randomness on phase transitions are exemplified by experimental measurements, the results of perturbative and numerical investigations in the case of the Potts model will be presented. The Potts model is of particular interest, since it can have in the pure case a second-order or a first-order transition, depending on the number of states per spin. In 2D, transfer matrix calculations and Monte Carlo simulations were used in order to check the validity of conformal invariance methods in disordered systems. These techniques were then used to investigate the universality class of the disordered Potts model, in both regimes of the pure model phase transitions. A test of replica symmetry became possible through a study of multiscaling properties and a detailed analysis of the probability distribution of the correlation functions was also made possible.
In order to study the influence of quenched disorder on second-order phase transitions, high-temperature series expansions of the susceptibility and the free energy are obtained for the quenched bond-diluted Ising model in d=3−−5d=3--5d=3−−5 dimensions. They are analyzed using different extrapolation methods tailored to the expected singularity behaviors. In d=4d=4d=4 and 5 dimensions we confirm that the critical behavior is governed by the pure fixed point up to dilutions near the geometric bond percolation threshold. The existence and form of logarithmic corrections for the pure Ising model in d=4d=4d=4 are confirmed, and our results for the critical behavior of the diluted system are in agreement with the type of singularity predicted by renormalization group considerations. In three dimensions we find large crossover effects between the pure Ising, percolation, and random fixed points. We estimate the critical exponent of the susceptibility to be $\ensuremath{\gamma}=1.305(5)$ at the random fixed point.
Abstract We consider the Ising model between 2 and 4 dimensions perturbed by quenched disorder in the strength of the interaction between nearby spins. In the interval 2 < d < 4 this disorder is a relevant perturbation that drives the system to a new fixed point of the renormalization group. At d = 2 such disorder is marginally irrelevant and can be studied using conformal perturbation theory. Combining conformal perturbation theory with recent results from the conformal bootstrap we compute some scaling exponents in an expansion around d = 2. If one trusts these computations also in d = 3, one finds results consistent with experimental data and Monte Carlo simulations. In addition, we perform a direct uncontrolled computation in d = 3 using new results for low-lying operator dimensions and OPE coefficients in the 3d Ising model. We compare these new methods with previous studies. Finally, we comment about the O (2) model in d = 3, where we predict a large logarithmic correction to the infrared scaling of disorder.