Flux $1/f^{\alpha }$ noise in two-dimensional Heisenberg spin glasses: Effects of weak anisotropic interactions

We study the dynamics of a two-dimensional ensemble of randomly distributed classical Heisenberg spins with isotropic RKKY and weaker anisotropic dipole-dipole couplings. Such ensembles may give rise to the flux noise observed in SQUIDs with a $1/f^{\alpha }$ power spectrum with $\alpha \lesssim 1$. We solve numerically the Landau-Lifshitz-Gilbert equations of motion in the dissipationless limit. We find that Ising-type fluctuators, which arise from spin clustering close to a spin-glass critical behavior with $T_c=0$, give rise to $1/f^{\alpha }$ noise. Even weak anisotropic interactions lead to a crossover from the Heisenberg-type criticality to the much stronger Ising-type criticality. The temperature-dependent exponent $\alpha \left(T\right)\lesssim 1$ increases and approaches unity when the temperature is lowered. This mechanism acts in parallel to the spin diffusion mechanism. Whereas the latter is sensitive to the device geometry, the spin-clustering mechanism is largely geometry independent.

Figure 1

Spin fluctuation power spectra ${\mathcal{S}}_{\nu }(k,\omega )$ for the case of purely isotropic interactions. The spectra are independent of the spin direction $\nu =x,y,z$. Temperatures are $T=0.2J_0$ (circles) and $3.2J_0$ (triangles). The system size is $N^2$ spins with $N=80$. Dashed lines are Lorentzian fits. Inset: $T$ dependence of the spin diffusion coefficient $\mathcal{D}\left(T\right)$. The dashed line is a logarithmic fit.

Figure 2

The spin-glass correlation function, $C\left(r\right)$, for temperatures $T=0.1J_0,0.2J_0,0.4J_0$, and $0.8J_0$. The dashed lines are fits to $C\left(r\right)\propto e^{-r/\xi \left(T\right)}/r^b$. The inset depicts the temperature dependence of the spin-glass correlation length $\xi \left(T\right)$. The system size is $N^2$ spins with $N=80$. The number of different realizations of the random coupling is four.

Figure 3

Distribution of clusters defined as groups of spins for which the magnitude of the mutual correlator, $c_{\nu ij}\equiv \langle s_{\nu i}{s}_{\nu j}\rangle -\langle s_{\nu i}\rangle \langle s_{\nu j}\rangle$, exceeds 0.95. (a) Large clusters $\left(\nu =x\right)$ of sizes between 10 $(▵)$ and 23 $(\circ )$ are depicted for a given realization of the random couplings at temperature $T=0.5J_0$. The labels $({\mathcal{N}}_c,n)$ indicate the number of clusters, $n$, with size ${\mathcal{N}}_c$. The system size is $60\times 60$ spins. (b) Probability distribution, $P\left({\mathcal{N}}_c\right)$, for the sizes of clusters (with magnetization parallel and perpendicular to the plane) at different temperatures.

Figure 4

Properties of an Ising-like cluster. (a) Spatial distribution of the correlator $c_{xij}$ between one spin $\left(i\right)$ belonging to a cluster of size ${\mathcal{N}}_c=23$ (white circles) and the other spins $\left(j\right)$. Configuration and parameters as in Fig. 3. Red and blue colors denote ferromagnetic and antiferromagnetic correlations, respectively. (b) Time dependence of the magnetization, $\mu \left(t\right)$, of the cluster shown in (a).

Figure 5

Power spectra ${\mathcal{S}}_{\nu }(k,\omega )$ of the in-plane $\left(\nu =x,y\right)$ magnetization fluctuations, for different wave numbers and temperatures for a system with $N^2$ spins $\left(N=60\right)$.