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Physical review. B, Solid state

Critical properties of random-spin models from the ε expansion

T. C. Lubensky

Published May 1, 1975• 398 citations
Publisher

Abstract

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 $n$-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 $n>1$, 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 $n=1$, there is no stable fixed point, which is again interpreted as a smeared transition.

DOI
10.1103/physrevb.11.3573

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Physical review. B, Solid state•Jan 1, 1976•64 citations
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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion to second order in \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ. 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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ 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 \ensuremathαH{\ensuremath{\alpha}}^{H}\ensuremathαH of the pure system is negative. If \ensuremathαH>0{\ensuremath{\alpha}}^{H}>0\ensuremathαH>0 and n>1n>1n>1, there is a sharp transition with new exponents \ensuremathη=[(5n2\ensuremath−8n)256(n\ensuremath−1)2]\ensuremathϵ2\ensuremath{\eta}=[\frac{(5{n}^{2}\ensuremath{-}8n)}{256{(n\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}\ensuremathη=[256(n\ensuremath−1)2(5n2\ensuremath−8n)​]\ensuremathϵ2 and \ensuremathν=12+[3n32(n\ensuremath−1)]\ensuremathϵ+[n(127n2\ensuremath−572n\ensuremath−32)4096(n\ensuremath−1)3]\ensuremathϵ2\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}\ensuremathν=21​+[32(n\ensuremath−1)3n​]\ensuremathϵ+[4096(n\ensuremath−1)3n(127n2\ensuremath−572n\ensuremath−32)​]\ensuremathϵ2. 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.
"Renorma1ization-group methods have been applied in the study of quenched random magnetic systems in recent years. We begin with a brief review of second-order phase transitions in pure, homogeneous systems and also of the .. renorma1ization group framework. Then we provide an introduction to quenched random magnetic systems. Next, momentum-space methods and position-space techniques as applied to quenched random magnets are outlined and compared. Grinstein and Luther applied the Wilson-Fisher E-expansion to random n-vector models; Khme1'nitsky discovered that the random Ising model (n = 1) possessed a ""random"" fixed point of 0(squareroot(e)). This fixed point was found to have one marginal and one irrelevant operator. We have investigated the stability of this fixed point using Ca11an-Symanzik equations and renorma1ized perturbation theory. We find the fixed point stable in the next order; we have also obtained critical exponents to one higher order. Next, position-space techniques are used to study some simple model systems. In addition to critical exponents, global thermodynamic properties are determined. These calculations are based on the Migda1-Kadanoff approximate recursion relations suitably generalized to the inhomogeneous case. Firstly we study the randomly bond-dilute two-dimensional nearest- neighbor Ising model on a square lattice. Calculations give both thermal and magnetic exponents associated with the percolative fixed point. Differential recursion relations yield a phase diagram which is in quantitative agreement with all known results. Curves for the specific heat, percolation probability, and magnetization are displayed. The critical region of the specific heat becomes unobservably narrow well above the percolation threshold Pc. This provides a possible explanation for the apparent specific-heat rounding in certain experiments. We then study the Edwards-Anderson model of a spin glass. The current theoretical situation, which is far from satisfactory at present, is briefly reviewed. We treat the spin-l/2 Ising model with independently random nearestneighbor interactions in dimensionalities d = 2, 3, and 4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spinglass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific heat anomaly. Global specific-heat curves are also displayed for d = 2."
The phase transitions in a large class of physical systems are described by $n\ensuremath{\ge}4$ component order parameters. Here, the critical behavior of quenched random $n\ensuremath{\ge}4$ vector models is studied by means of renormalization-group theory in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. Recursion relations for average potentials are constructed following the methods derived by Lubensky. For several Hamiltonians describing homogeneous $n\ensuremath{\ge}4$ systems there exist no stable fixed points in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions which explains the first-order transitions actually observed in these systems. It is shown that the recursion relations for the corresponding quenched random systems are also unstable. However, the runaway in this case is of a fundamentally different nature. The fluctuations of the local mean-field transition temperature diverge, and this behavior is interpreted as a "smeared" transition. This interpretation is consistent with existing experiments. On the other hand, in the cases where the homogeneous Hamiltonian possesses a stable fixed point in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions, this fixed point remains stable against random perturbations, so no change in the critical behavior is expected. For most of the models studied there is at least one fixed point of order ${\ensuremath{\epsilon}}^{\frac{1}{2}}$. These fixed points are all unstable. It is suggested that experiments should be performed to determine the critical behavior of random $n\ensuremath{\ge}4$ systems. Of particular interest are the systems with no stable fixed points, such as Cr, Eu, MnO, and UO2{\mathrm{UO}}_{2}UO2​, where crossover from first-order to a smeared transition is predicted.
Renormalization-group techniques are used to study a model of nnn-coupled mmm-component spin systems, in the limit of large dipole-dipole interactions. As recently shown by Emery, this model describes the critical behavior of a constrained dipolar system in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ and that of a dipolar system with a quenched random perturbation in the limit $n\ensuremath{\rightarrow}0$. In both cases, the unperturbed dipolar fixed point is unstable, and there is a crossover to a new behavior. For the constrained system, this leads to another dipolar fixed point (if $\ensuremath{\alpha}<0$, $\ensuremath{\alpha}$ being the dipolar specific-heat exponent) with the same thermodynamic critical exponents, or to one with renormalized dipolar exponents (if $\ensuremath{\alpha}>0$). These results are different from those of previous "spherical" dipolar models. For the random case, the crossover is either to a new fixed point, with very different exponents, e.g., $2\ensuremath{\nu}\ensuremath{\simeq}1+1.183\ensuremath{\epsilon}$ for $m=d=4\ensuremath{-}\ensuremath{\epsilon}$, or away from all the fixed points found to order $\ensuremath{\epsilon}$. One of the new fixed points in this case has complex eigenvalues of the linearized recursion relations. This is related to the fact that the recursion-relation flow is not of a gradient type for random systems.
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}=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 ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ 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 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.
The renormalization-group treatment of the critical behavior of random systems is augmented by including all the cumulants of the disorder distribution. The results substaniate Lubensky's argument that the $n\ensuremath{\rightarrow}0$ isotropic fixed point is "unphysical," and that the phase transition of random mmm-component systems which have a positive specific-heat exponent will be sharp, with exponents determined by the "random" fixed point. The renormalization-group transformations of a random Gaussian model lead to a "runaway," which is shown to be unrelated to a first-order transition. This is related to the problem of a particle in a random potential with and without an absorptive part.