Unsolved problems in the theory of short-range spin glasses: the freezing region, low-energy excitations, and low-temperature thermodynamics of dilute magnets with short-range exchange

A. A. Abrikosov's review 'Spin glasses with short range interaction' (Advances in Physics 29, 869, 1980) constructed a percolation-cluster theory for dilute magnets whose exchange is effectively short-ranged (interaction range much smaller than the mean spin spacing, $n_{\mathrm{m}}^{1/3}l\ll1$ ; prototype Eu $_x$ Sr $_{1-x}$ S), and explicitly listed the problems this theory left open. The framework-internal questions are: the nature of the freezing at the percolation temperature $\Theta$ in the critical window where clusters exceed the frustration length $r_0$ and lose collinearity; the microscopic value of the surface exponent $q$ in the kinetic equation for the cluster-size distribution; and the long-range limit of the one-dimensional problem. The underlying physical questions are whether short-range Heisenberg spin glasses have a genuine thermodynamic transition in three dimensions, what the low-energy excitation spectrum of the frozen state is (propagating spin waves versus damped, diffusive, or localized modes), and what the quantitative low-temperature thermodynamics (linear specific heat, logarithmic susceptibility, commensurate case, dipolar regime) is. The core of the problem is the completion of a controlled microscopic theory of the freezing region and frozen state of site-disordered short-range spin glasses, together with its experimental test.

Background

Spin glasses are dilute magnetic alloys or doped insulators in which localized magnetic moments sit at random positions in a non-magnetic host; their hallmark is that at low temperature the moments freeze into a static but spatially random, non-collinear configuration with no net magnetization. In metallic spin glasses the moments interact via the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, an indirect exchange mediated by conduction electrons that oscillates in sign with distance and decays as $r^{-3}$ . In a sufficiently dirty metal (short electron mean free path $l$ ) or in a semiconducting host, the RKKY-type interaction acquires an exponential cutoff $e^{-r/l}$ (or $e^{-\alpha r}$ ), and when the interaction range is much smaller than the mean distance between magnetic atoms, $n_{\mathrm{m}}^{1/3}l\ll1$ with $n_{\mathrm{m}}$ the volume concentration of spins, each spin couples appreciably only to a few nearest neighbours. This is the short-range spin glass regime; its canonical experimental realization is the insulating mixed crystal Eu $_x$ Sr $_{1-x}$ S, where Eu $^{2+}$ moments occupy a fraction $x$ of the cation sites and the superexchange reaches only the first two neighbour shells.

A. A. Abrikosov's review 'Spin glasses with short range interaction' (Advances in Physics 29, 869 (1980)), developed in companion papers (Journal of Low Temperature Physics 33, 505 (1978); 37, 111 (1979); 39, 217 (1980)), built a systematic theory of this regime. Its central device is the thermal radius $r(T)$ , the interspin distance at which the interaction energy equals $T$ : spins closer than $r(T)$ bind into clusters, so the thermodynamics maps onto continuum percolation (spheres of radius $r(T)/2$ , percolation-threshold filling fraction $\approx0.347$ ), and the percolation temperature defines the freezing temperature $\Theta$ . Well below $\Theta$ a one-neighbour approximation (each spin feels the effective field of its nearest neighbour, with Poisson-distributed distances) gives closed forms for the specific heat and susceptibility; below a lower crossover temperature $T_0$ the dominant saddle point sits at the frustration length $r_0=(4\pi n_{\mathrm{m}}l)^{-1/2}$ , beyond which many neighbours compete, and the entropy is carried instead by rare small values of the oscillatory sign factor of the exchange. This yields a linear specific heat $C\propto T$ and a logarithmic susceptibility $\chi\propto\ln(T_0/T)$ for incommensurate spin glasses (where the sign factor is continuously distributed), but qualitatively different, random-ferromagnet-like behaviour for commensurate ones (where the sign factor takes discrete values fixed by the host band structure, e.g. $\pm1$ , $\pm3$ ). The review also derived the exchange contribution to the electrical resistivity, which rises linearly in $T$ at the lowest temperatures; an exact cluster-size distribution $a(n)=\lambda^2(1-\lambda)^{n-1}$ for the one-dimensional case, with $z(T)$ the one-dimensional analogue of the thermal radius (the interspin distance at which the pair interaction energy equals $T$ ) and $\lambda=e^{-n_{\mathrm{m}}z(T)}$ (here $n_{\mathrm{m}}$ is the linear spin density, the inverse mean spin spacing - the one-dimensional counterpart of the volume concentration introduced above; the same symbol carries this linear-density meaning throughout the one-dimensional problem below); and, in an appendix, a kinetic equation for the cluster-size distribution near the percolation threshold, containing a phenomenological surface exponent $q$ (the effective surface dimension governing the merging probability of two clusters).

The review closed by stating what remained unsolved: the nature of the freezing at $\Theta$ , since the percolation picture itself fails in the critical window where clusters larger than $r_0$ develop effective long-range frustrated interactions and lose collinearity; the existence of spin waves in the frozen state, where the hydrodynamic continuous-medium theory of Halperin and Saslow (Physical Review B 16, 2154 (1977)) predicts gapless linearly dispersing spin waves while microscopic treatments indicated strong damping and a diffusive spectrum; the quantitative role of magnetic dipole forces at the lowest temperatures; the thermodynamics of commensurate spin glasses below $T_0$ where the one-neighbour assumption fails; a microscopic determination of the exponent $q$ ; the long-range limit of the one-dimensional problem; and experimental verification of the predicted laws in materials satisfying $n_{\mathrm{m}}^{1/3}l\ll1$ , beyond the encouraging but incomplete comparison with Eu $_x$ Sr $_{1-x}$ S data.

The field later reorganized around the Edwards-Anderson lattice model with random short-range bonds, so much of the relevant subsequent literature is phrased in that language. For isotropic Heisenberg spin glasses it remains contested whether freezing in three dimensions is a genuine thermodynamic transition, whether its order parameter involves spin chirality (the chiral-glass scenario: chirality is the handedness of the triple product of three neighbouring spins, which can order even when the spins themselves do not), and what the low-energy excitations of the frozen state are. These modern questions are the reformulated descendants of the review's open problems, and they are not settled.

Problem Statement

The problem, as posed by the source review and left unresolved, is to complete the microscopic theory of site-disordered (short-range) spin glasses - dilute magnets with exponentially cut-off, sign-oscillating exchange satisfying $n_{\mathrm{m}}^{1/3}l\ll1$ , of which Eu $_x$ Sr $_{1-x}$ S is the prototype - from the freezing transition down to $T\to0$ , and to test it experimentally. The following components are the open problems individually enumerated in the review; each is an open target in its own right and may be attacked on its own. They share the common theme of a controlled, first-principles description of the freezing region and of the frozen state, together with its experimental validation:

(1) Freezing region: determine the true nature of the freezing at the percolation temperature $\Theta$ - in particular the critical window near $\Theta$ , where the percolation approach breaks down because clusters larger than the frustration length $r_0=(4\pi n_{\mathrm{m}}l)^{-1/2}$ (equivalently, clusters of more than $n_0$ spins, $n_0$ being the cluster size at which the cluster radius reaches $r_0$ ) develop effective long-range frustrated interactions and lose collinearity - and replace it, if possible, by a controlled microscopic treatment (percolation-based or otherwise, including a proper treatment of the fully frustrated long-range case). In modern language: does such a system undergo a genuine thermodynamic phase transition in three dimensions, and with what order parameter?

(2) Excitations: determine the existence and properties of the low-energy excitations of the frozen state - whether it supports the gapless linear spin waves suggested by continuous-medium (hydrodynamic) arguments, or the strongly damped, diffusive, or localized modes indicated by microscopic calculations - including the character of the spectrum (gapless and linear, diffusive, gapped, or localized) and its dispersion and damping.

(3) Lowest-temperature thermodynamics: compute the thermodynamic functions quantitatively in the regimes where the one-neighbour assumption fails - below the crossover $T_0$ for incommensurate spin glasses (origin of the linear specific heat and the logarithmic susceptibility) and at all low temperatures for commensurate spin glasses - and incorporate magnetic dipole forces, which become important at the very lowest temperatures, quantitatively rather than only qualitatively.

(4) Cluster kinetics: determine the surface exponent $q$ for $d=2$ and $d=3$ in the kinetic equation for the cluster-size distribution near the percolation threshold, or solve that kinetic equation directly.

(5) One-dimensional long-range limit: extend the exactly solved one-dimensional short-range case (interaction decay length $\alpha^{-1}$ much smaller than the mean spin spacing, $n_{\mathrm{m}}\alpha^{-1}\ll1$ ) to the opposite, long-range-interaction limit ( $n_{\mathrm{m}}\alpha^{-1}\gtrsim1$ ), where qualitatively different behaviour is expected and which the cluster approach does not cover.

(6) Experiment: verify the predicted laws - the linear specific heat, the logarithmic susceptibility, and the linear-in- $T$ resistivity - in materials satisfying the short-range condition $n_{\mathrm{m}}^{1/3}l\ll1$ , beyond the encouraging but incomplete comparison with Eu $_x$ Sr $_{1-x}$ S data.

The problem concerns this physical system and these questions rather than a proxy for them.

References

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