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Physical review. B, Solid state

Application of the renormalization group to phase transitions in disordered systems

G. Grinstein, Alan H. Luther

Published Feb 1, 1976• 529 citations
Publisher

Abstract

The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute $m$-vector model, the $n=0$ 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 $m$-component fixed point, which characterizes the critical behavior of the pure system, is stable. For $m<{m}_{c}$, 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 $m=1$. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.

DOI
10.1103/physrevb.13.1329

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Critical properties of random-spin models from the ε expansion

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Physical review. B, Solid state•Jan 1, 1976•64 citations
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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion to order \ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For m>mc\ensuremath≡4\ensuremath−4\ensuremathϵ+O(\ensuremathϵ2)m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})m>mc​\ensuremath≡4\ensuremath−4\ensuremathϵ+O(\ensuremathϵ2) 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 \ensuremathη=[(5m2\ensuremath−8m)256(m\ensuremath−1)2]\ensuremathϵ2+O(\ensuremathϵ3)\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})\ensuremathη=[256(m\ensuremath−1)2(5m2\ensuremath−8m)​]\ensuremathϵ2+O(\ensuremathϵ3), \ensuremathν=12+[3m32(m\ensuremath−1)]\ensuremathϵ+[m(127m2\ensuremath−572m\ensuremath−32)4096(m\ensuremath−1)3]\ensuremathϵ2+O(\ensuremathϵ3)\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})\ensuremathν=21​+[32(m\ensuremath−1)3m​]\ensuremathϵ+[4096(m\ensuremath−1)3m(127m2\ensuremath−572m\ensuremath−32)​]\ensuremathϵ2+O(\ensuremathϵ3) for m\ensuremath≠1m\ensuremath{\ne}1m\ensuremath=1, and \ensuremathη=\ensuremath−\ensuremathϵ106+O(\ensuremathϵ32)\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})\ensuremathη=\ensuremath−106\ensuremathϵ​+O(\ensuremathϵ23​), \ensuremathν=12+(6\ensuremathϵ53)124+O(\ensuremathϵ)\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})\ensuremathν=21​+4(536\ensuremathϵ​)21​​+O(\ensuremathϵ) for m=1m=1m=1. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.
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.
"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."
We consider the Ginzburg-Landau MN\mathrm{MN}MN model that describes MNMNMN-vector cubic models with O(M)O(M)O(M)-symmetric couplings. We compute the renormalization-group functions to six-loop order in d=3.d=3.d=3. We focus on the limit $\stackrel{\ensuremath{\rightarrow}}{N}0$ which describes the critical behavior of an M-vector model in the presence of weak quenched disorder. We perform a detailed analysis of the perturbative series for the random Ising model (M=1).(M=1).(M=1). We obtain for the critical exponents: $\ensuremath{\gamma}=1.330(17),\ensuremath{\nu}=0.678(10),\ensuremath{\eta}=0.030(3),\ensuremath{\alpha}=\ensuremath{-}0.034(30),\ensuremath{\beta}=0.349(5),\ensuremath{\omega}=0.25(10).$ For M> 2M>~2M> 2 we show that the O(M)O(M)O(M) fixed point is stable, in agreement with general nonperturbative arguments, and that no random fixed point exists.
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.