Quantitative cluster statistics and field-dependent thermodynamics in the percolation model of the dilute RKKY spin glass: the cluster-size distribution $A(n,\varepsilon)$, the multidomain net-spin distribution $B(m,n)$, and the predicted cusps of $\partial^2 M/\partial h^2$

A. A. Abrikosov (J. Low Temp. Phys. 33, 505 (1978)) modeled the dilute metallic spin glass with short-range (RKKY) interactions in a magnetic field by cutting the oscillatory interaction at a thermal radius $r(T)$ and mapping the statistics of rigidly bound spin clusters onto three-dimensional percolation of overlapping spheres, predicting weak cusps and power-law field dependences in the magnetic specific heat $\Delta C(h)$ and in $\partial^2 M/\partial h^2$ near the percolation-like transition. The theory was explicitly asymptotic and model-based: the true cluster-size distribution $A(n,\varepsilon)$ of the sphere-percolation problem was replaced by a Bethe-lattice proxy and a two-branch scaling ansatz, the multidomain net-spin distribution $B(m,n)$ rested on unverified decorrelation assumptions, the exponents were imported from separate mid-1970s numerics, and only the asymptotic limits of the crossover behavior were computed. The objective is to complete that program: determine $A(n,\varepsilon)$, $B(m,n)$, the scaling function $f(x)$ and exponents quantitatively, derive the full crossover of the field-dependent observables, and test the predictions against microscopic calculation, simulation, or experiment. The spin-glass-specific core — the realistic multidomain $B(m,n)$, the crossover regime $|\varepsilon|\sim\alpha^{2a}$, and the experimental feasibility of the predicted cusps — remains open in the literature.

Background

A spin glass is a dilute magnetic alloy, such as CuMn or AuFe, in which magnetic impurities sit at random positions and interact through the conduction electrons via the RKKY (Ruderman–Kittel–Kasuya–Yosida) indirect exchange, which in a 'dirty' host with electron mean free path $l$ takes the oscillating, distance-decaying form $V(r)\propto\cos(2p_0 r)\,r^{-3}e^{-r/l}$, with $p_0$ the electron Fermi momentum. Because the sign of the interaction oscillates with distance, randomly placed spins are coupled by a mixture of ferromagnetic and antiferromagnetic bonds (frustration), and at low temperature the system freezes into a disordered state rather than an ordered magnet.

Abrikosov's 1978 approach to this problem was a cluster-percolation picture. He defined a thermal interaction radius $r(T)$ by the condition that the interaction strength at that distance equals $T$, and treated spins separated by less than $r(T)$ as rigidly bound into clusters, ignoring interactions beyond it. On this approximation the cluster statistics of the dilute alloy become those of continuum percolation of overlapping spheres: spheres of radius $r(T)/2$ centered at the random impurity positions (density $n_m$), two spins being connected when their spheres overlap, i.e. when their separation is less than $r(T)$. The control parameter is the sphere concentration $\beta=(4\pi/3)n_m r^3(T)$, which grows as $T$ falls, with a percolation threshold $\beta_c\approx 2.7$ at which an infinite connected cluster first appears (in this convention $\beta$ is eight times the usual overlapping-sphere filling factor $\eta=(4\pi/3)n_m (r/2)^3$, whose three-dimensional critical value is $\eta_c\approx0.34$); the distance from the threshold is $\varepsilon=(\beta_c-\beta)/\beta_c$. Percolation theory is the study of connectivity in such random media; the key object is the cluster-size distribution $A(n,\varepsilon)$, the number of clusters containing $n$ spins per sample. Each finite cluster is then treated as a free paramagnetic moment whose net spin is $Sm$ ($S$ the atomic spin, $m$ an integer), rotating as a whole in the external field $h$, with reduced field $\alpha=\mu h S/T$ ($\mu$ the atomic magnetic moment). Averaging the resulting Schottky-type single-cluster specific heat (a broad anomaly produced when the Zeeman levels of a moment are spread over an energy of order $k_B T$) over the cluster statistics yields predictions for the field-induced magnetic specific heat $\Delta C(h)=C(h)-C(0)$ and for the field derivatives of the magnetization $\partial^2 M/\partial h^2$ near the threshold.

Three ingredients were, however, only modeled, not known. First, no exact form of $A(n,\varepsilon)$ is available for the three-dimensional sphere problem: Abrikosov adopted the large-$n$ asymptotic form exactly solvable on a Bethe lattice (an infinite branching tree with no loops), $A(n,\varepsilon)=\frac{N}{(2\pi)^{1/2}}n^{-5/2}\exp(-n\varepsilon^{2}/2)$ with $N$ the total number of magnetic atoms in the sample, together with a general scaling ansatz

$$A(n,\varepsilon)=N\,n^{-b}f(\varepsilon n^{a}),$$

with $2<b<3$, $a>0$, and an ad hoc two-branch parametrization of the scaling function $f(x)$; in modern percolation notation $b$ is the Fisher exponent $\tau$ and $a$ is the cluster-cutoff exponent $\sigma$, since the distribution is cut off at $n\sim|\varepsilon|^{-1/\sigma}$. Second, large clusters are not single domains of collinear spins: Abrikosov modeled a multidomain cluster as a set of independently oriented domains and derived closed forms for the net-spin distribution $B(m,n)$ — the distribution of the integer net spin $m$ of an $n$-spin cluster — only in the isotropic and planar orientation limits, on untested assumptions that domain orientations decorrelate and can rotate freely at domain boundaries. Third, the exponents were imported from then-available numerical studies of three-dimensional percolation (Stauffer's dilute-Ising scaling work and the Shklovskii–Efros school), giving $a=0.49\pm0.08$, $b=2.17\pm0.04$ and derived index combinations such as $\xi=(b-2)/a$, and only the asymptotic regimes $|\varepsilon|\gg\alpha^{2a}$ and $|\varepsilon|\ll\alpha^{2a}$ of the observables were computed — not the crossover between them.

Problem Statement

Within Abrikosov's mapping of the dilute short-range (RKKY) spin glass in a magnetic field onto three-dimensional overlapping-sphere percolation — spheres of radius $r(T)/2$ centered at the impurity positions, connected on overlap, equivalently spins rigidly bound when their separation is less than the thermal interaction radius $r(T)$, concentration parameter $\beta=(4\pi/3)n_m r^3(T)$ with $n_m$ the magnetic-impurity density, threshold $\beta_c\approx 2.7$, distance to threshold $\varepsilon=(\beta_c-\beta)/\beta_c$, and reduced field $\alpha=\mu h S/T$ with $\mu$ the atomic magnetic moment, $S$ the atomic spin, $h$ the external field, and $T$ the temperature — make the theory quantitative and test its observable predictions. This comprises four linked components serving one objective, converting the asymptotic cluster-percolation theory into a quantitatively verified description of the field-dependent thermodynamics of the dilute spin glass near its percolation-like transition: (1) Determine the true finite-cluster size distribution $A(n,\varepsilon)$ of the three-dimensional overlapping-sphere percolation problem — in closed analytic form or as a quantitatively validated numerical/scaling form — including the critical exponents ($a\equiv\sigma$, $b\equiv\tau$ in modern notation), the scaling function $f(x)$ in $A=N n^{-b}f(\varepsilon n^{a})$ and its amplitude $f(0)$, thereby replacing the Bethe-lattice proxy and the two-branch ansatz. (2) Determine the net-spin distribution $B(m,n)$ — the distribution of the integer net spin $m$ of an $n$-spin cluster — of multidomain clusters for realistic spin-glass microphysics, including the role of the oscillatory interaction sign structure and finite numbers of inter-domain coupling pairs, beyond the isotropic and planar model limits. (3) Derive or compute the full crossover behavior of $\partial^2 M/\partial h^2$ and $\Delta C(h)$ in the intermediate regime $|\varepsilon|\sim\alpha^{2a}$, not only the asymptotic limits. (4) Test the predictions against microscopic calculation, numerical simulation, or measurement. The four components are jointly posed, and a numerically or scaling-derived $A(n,\varepsilon)$ is quantitatively validated when it agrees with an independent microscopic calculation, simulation, or measurement.

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