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Physical review. B, Condensed matter

Critical behavior of the p -component Potts model with correlated impurities

B. Stolan, Erling Pytte, G. Grinstein

Published Aug 1, 1984• 20 citations
Publisher

Abstract

The problem of critical behavior in systems with correlated random impurities has recently attracted a great deal of attention. For the $m$-component Heisenberg model, renormalization-group calculations using a novel double expansion predict second-order transitions described by a new random fixed point, whereas the usual single $\ensuremath{\epsilon}$ expansion leads to a runaway, often interpreted as a smeared transition. In this paper the corresponding results for the $p$-state Potts model with impurities which are perfectly correlated in certain directions are presented. (For $p\ensuremath{\le}2$ the pure Potts model has a continuous transition; $p=1$ corresponds to the percolation problem.) In this case the new random fixed point is, in direct contrast to the $m$-component Heisenberg situation, never both physical and stable. Thus for the Potts model the runaway obtained in the single $\ensuremath{\epsilon}$ expansion persists. For sufficiently strong disorder the runaway is always obtained, even above the upper critical dimension where the Gaussian fixed point is (locally) stable. We speculate that the runaways may be symptomatic of a "smearing" of the phase transition by the strongly correlated impurities.

DOI
10.1103/physrevb.30.1506

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Self-consistent approach to quenched impurity effects on quantum phase transitions

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Critical behavior of random n≥4 vector models studied by the renormalization-group technique in 4−ε dimensions: Crossover from first-order to smeared transitions

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Physical review. B, Solid state•Nov 1, 1976•22 citations
The problem of critical behavior in systems with correlated random impurities has recently attracted a great deal of attention. For the mmm-component Heisenberg model, renormalization-group calculations using a novel double expansion predict second-order transitions described by a new random fixed point, whereas the usual single \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion leads to a runaway, often interpreted as a smeared transition. In this paper the corresponding results for the ppp-state Potts model with impurities which are perfectly correlated in certain directions are presented. (For p\ensuremath≤2p\ensuremath{\le}2p\ensuremath≤2 the pure Potts model has a continuous transition; p=1p=1p=1 corresponds to the percolation problem.) In this case the new random fixed point is, in direct contrast to the mmm-component Heisenberg situation, never both physical and stable. Thus for the Potts model the runaway obtained in the single \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion persists. For sufficiently strong disorder the runaway is always obtained, even above the upper critical dimension where the Gaussian fixed point is (locally) stable. We speculate that the runaways may be symptomatic of a "smearing" of the phase transition by the strongly correlated impurities.
We study the critical behavior of the q-state Potts model with long-range interactions decaying asymptotically as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d+\ensuremath{\sigma})}$ in the presence of random-Tc{T}_{c}Tc​ impurities correlated over large distances such that the correlations fall off as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}a)}$, a<d. We find that the renormalization-group scaling equations have a new fixed point in the appropriate double \ensuremath{\epsilon}, x expansion, where \ensuremath{\epsilon}=3\ensuremath{\sigma}-d and x=a-\ensuremath{\sigma}. This fixed point, however, is never both stable and physical.
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.
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 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.
Double expansion renormalization group treatments are still employed in literature to describe the effects of symmetry-conserving quenched disorder on quantum phase transitions as an alternative to more or less phenomenological models. If one assumes = 4 – d and the dimensionality τ of the imaginary-time space involved in a generalized quantum action as simultaneously small parameters, a stable random fixed point is found which governs a new disorder-induced quantum criticality usually considered meaningful also after extrapolation to the physical time-like dimension τ = 1. In contrast, by considering τ = 1 at the beginning, a runaway takes place in the parameter space for dimensionalities d < 4. To give some insight into these contrasting results, we employ here a simple self-consistent approach for case of short-range correlated quenched disorder which allows us to analyze analytically what happens by continuous variation of the fictitious time-like dimension τ We find that a τ *( n ) < 1 exists, depending on the symmetry index n , above which the disorder inhibits the occurrence of a conventional quantum phase transition. This suggests that the usual procedure to extrapolate the small τ -predictions to the value of interest τ = 1 may have no real physical meaning.
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.