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Physical review. B, Condensed matter

Phase diagrams and multicritical points in randomly mixed magnets. III. Competing spin-glass and magnetic ordering

Shmuel Fishman, Amnon Aharony A

Published Jan 1, 1980• 56 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.21.280

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The Ising model with a quenched random distribution of ferromagnetic and antiferromagnetic interactions has been investigated by tracking the probability distribution of interactions under rescaling. As the initial probability of a ferromagnetic interaction is increased, the phase diagram in three dimensions shows three ordered phases: antiferromagnetic, spin glass and ferromagnetic. We find that, within the spin-glass region, effective interactions between pairs of spins at successive length scales occur in a chaotic sequence, while the average magnitudes of the ferromagnetic and the antiferromagnetic interactions both flow to infinity under iteration. Our calculated spin-glass multicritical points are on the Nishimori line in cases of both bimodal (±J) and diluted (trimodal) distributions, and these points flow under rescaling to a common fixed distribution with a specific heat exponent α of approximately −3.7.
"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 three-dimensional site-random Ising model consisting of two types of magnetic ions, A and B, is investigated by means of the exchange Monte Carlo method. The nearest-neighbor interactions are antiferromagnetic (AF) for the B-B pair and ferromagnetic (FM) otherwise. A uniform FM or AF ordering is found to exist in a wide range of concentration c of A ions, in contrast with the bond-random Ising model such as the ± J model. The phase diagram is found to be symmetric on the c – T plane, though the ratio of the FM bonds to AF ones is asymmetric with respect to c =1/2. Spin configurations of degenerate ground states and their cluster structure are also studied. Even in the system with c =0.55, there exists only a unique rigid cluster which includes about 90% of spins and that cluster shows nonvanishing uniform magnetization. These results strongly suggest that the present model has neither a spin glass phase nor an FM-AF mixed phase, though their existence in a quite narrow region around c =1/2 has not been completely excluded yet.
We study the quenched disordered magnetic system, which is obtained from the 2D SO(3) quantum Heisenberg model, on a square lattice, with nearest neighbors interaction, by taking a Gaussian random distribution of couplings centered in an antiferromagnetic coupling, J¯>0 and with a width ΔJ. Using coherent spin states we can integrate over the random variables and map the system onto a field theory, which is a generalization of the SO(3) nonlinear sigma model with different flavors corresponding to the replicas, coupling parameter proportional to J¯ and having a quartic spin interaction proportional to the disorder (ΔJ). After deriving the CP1 version of the system, we perform a calculation of the free energy density in the limit of zero replicas, which fully includes the quantum fluctuations of the CP1 fields zi. We, thereby obtain the phase diagram of the system in terms of (T,J¯,ΔJ). This presents an ordered antiferromagnetic (AF) phase, a paramagnetic (PM) phase and a spin-glass (SG) phase. A critical curve separating the PM and SG phases ends at a quantum critical point located between the AF and SG phases, at T=0. The Edwards–Anderson order parameter, as well as the magnetic susceptibilities are explicitly obtained in each of the three phases as a function of the three control parameters. The magnetic susceptibilities show a Curie-type behavior at high temperatures and exhibit a clear cusp, characteristic of the SG transition, at the transition line. The thermodynamic stability of the phases is investigated by a careful analysis of the Hessian matrix of the free energy. We show that all principal minors of the Hessian are positive in the limit of zero replicas, implying in particular that the SG phase is stable.
We study the criticality at the spin-glass-ferromagnetic-paramagnetic multicritical point in the d=3, \ifmmode\pm\else\textpm\fi{}J distribution, random-bond Ising model. Using high-temperature expansions to order ${\mathit{T}}^{\mathrm{\ensuremath{-}}34}$, we estimate that the multicritical point N lies on the Nishimori line at Tc{\mathit{T}}_{\mathit{c}}Tc​/J=1.690\ifmmode\pm\else\textpm\fi{}0.016. Along this line the critical exponents are found to be \ensuremath{\gamma}=1.80\ifmmode\pm\else\textpm\fi{}0.15 and \ensuremath{\nu}=0.85\ifmmode\pm\else\textpm\fi{}0.08. The latter is clearly consistent with the rigorous exponent inequality \ensuremath{\nu}\ensuremath{\ge}2/d. We also calculate the crossover exponent \ensuremath{\varphi} and show that the scaling axes at N are in agreement with the recent predictions of Le Doussal and Harris.
The Edwards-Anderson model of a spin glass is studied by position-space renormalization-group techniques, using an inhomogeneous generalization of Migdal's approximate recursion relation. We treat the spin-1/2 Ising model with independently random nearest-neighbor interactions in dimensionalities d=2,3,and4d=2, 3, \mathrm{and} 4d=2,3,and4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spin-glass 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.