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Application of Renormalization-Group Techniques to Random Magnetic Systems.

Ciriyam Jayaprakash

Published Feb 1, 1979• 0 citations
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Abstract

"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."

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Thermodynamic behavior of quenched random magnets from a position-space renormalization group

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Shmuel Fishman, Amnon Aharony A

The phase diagrams and critical properties of a quenched random alloy of a ferromagnet and an antiferromagnet (or of two antiferromagnets with different periodicities) are studied in the mean-field approximation and by renormalization-group techniques in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, using the $n\ensuremath{\rightarrow}0$ replica method. If only nearest-neighbor interactions are assumed, then one finds ferromagnetic ($F$), antiferromagnetic (AF), and spin-glass (SG) phases. As the average strength of the next-nearest-neighbor interactions is increased (e.g., by an additional component in the alloy), the $F$ and the AF phases approach each other, and one may find regions in parameter space where a mixed $F$-AF phase exists, as discussed in Paper II of this series. The phase diagram exhibits lines of multicritical points where the paramagnetic ($P$), $F$ (or AF) and SG phases coexist ($P$-SG-$F$ or $P$-SG-AF), or where the $P$, $F$, and AF phases coexist ($P\ensuremath{-}F$-AF). These lines meet at a new multicritical point, where all four phases coexist ($P$-SG-$F$-AF). Recent renormalization-group analysis of $P$-SG-$F$, by Chen and Lubensky, yields complex exponents for the $\mathrm{XY}$ and Heisenberg cases. We show that the phase diagram predicted by that analysis is difficult to understand even for the Ising case. The same method of analysis yields similar difficulties for $P$-SG-$F$-AF. A modified way to take the limit $n\ensuremath{\rightarrow}0$, which resolves these difficulties, is presented. This modified way still lacks rigorous justification.

Physical review. B, Condensed matter•Jan 1, 1980•56 citations

Real-space renormalization-group investigation of the randomly diluteq-state Potts model

A. Bakchich, A. Benyoussef, T. Biaz +1 more

The randomly bond-diluted two-dimensional nearest-neighbor q-state Potts ferromagnet and antiferromagnet on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. In the bond-diluted ferromagnet Potts model, differential recursion relations yield a phase diagram which is in quantitative agreement with all known results for q\ensuremath{\le}4. In the bond-diluted antiferromagnet Potts model, the phase diagram obtained shows that the threshold ${p}^{\mathrm{*}}$ at which the critical temperature goes to zero depends upon q.

Physical review. B, Condensed matter•Jun 1, 1988•2 citations
"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."
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 randomly bond-dilute two-dimensional nearest-neighbor Ising model on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. 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{p}_{c}pc​. This provides a possible explanation for the apparent specific-heat rounding in certain experiments.
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.
The phase diagrams and critical properties of a quenched random alloy of a ferromagnet and an antiferromagnet (or of two antiferromagnets with different periodicities) are studied in the mean-field approximation and by renormalization-group techniques in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, using the $n\ensuremath{\rightarrow}0$ replica method. If only nearest-neighbor interactions are assumed, then one finds ferromagnetic (FFF), antiferromagnetic (AF), and spin-glass (SG) phases. As the average strength of the next-nearest-neighbor interactions is increased (e.g., by an additional component in the alloy), the FFF and the AF phases approach each other, and one may find regions in parameter space where a mixed FFF-AF phase exists, as discussed in Paper II of this series. The phase diagram exhibits lines of multicritical points where the paramagnetic (PPP), FFF (or AF) and SG phases coexist (PPP-SG-FFF or PPP-SG-AF), or where the PPP, FFF, and AF phases coexist ($P\ensuremath{-}F$-AF). These lines meet at a new multicritical point, where all four phases coexist (PPP-SG-FFF-AF). Recent renormalization-group analysis of PPP-SG-FFF, by Chen and Lubensky, yields complex exponents for the XY\mathrm{XY}XY and Heisenberg cases. We show that the phase diagram predicted by that analysis is difficult to understand even for the Ising case. The same method of analysis yields similar difficulties for PPP-SG-FFF-AF. A modified way to take the limit $n\ensuremath{\rightarrow}0$, which resolves these difficulties, is presented. This modified way still lacks rigorous justification.
The randomly bond-diluted two-dimensional nearest-neighbor q-state Potts ferromagnet and antiferromagnet on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. In the bond-diluted ferromagnet Potts model, differential recursion relations yield a phase diagram which is in quantitative agreement with all known results for q\ensuremath{\le}4. In the bond-diluted antiferromagnet Potts model, the phase diagram obtained shows that the threshold p∗{p}^{\mathrm{*}}p∗ at which the critical temperature goes to zero depends upon q.