Avalanches and hysteresis in frustrated superconductors and $XY$ spin glasses

We study avalanches along the hysteresis loop of long-range interacting spin glasses with continuous $XY$ symmetry, which serves as a toy model of granular superconductors with long-range and frustrated Josephson couplings. We identify sudden jumps in the $T=0$ configurations of the $XY$ phases as an external field is increased. They are initiated by the softest mode of the inverse susceptibility matrix becoming unstable, which induces an avalanche of phase updates (or spin alignments). We analyze the statistics of these events and study the correlation between the nonlinear avalanches and the soft mode that initiates them. We find that the avalanches follow the directions of a small fraction of the softest modes of the inverse susceptibility matrix, similarly as was found in avalanches in jammed systems. In contrast to the similar Ising spin glass (Sherrington-Kirkpatrick) studied previously, we find that avalanches are not distributed with a scale-free power law but rather have a typical size which scales with the system size. We also observe that the Hessians of the spin-glass minima are not part of standard random matrix ensembles as the lowest eigenvector has a fractal support.

Figure 1

Distribution $P\left(h\right)$ of the modulus of the local fields $h\equiv |{\vec{h}}_i|$, obtained for a system size $N=1024$ and averaged over 1000 disorder realizations. The external magnetic field was set to $H=0$. A hard gap is clearly visible. The straight line is a fit $\left[f\left(h\right)=0.96(h-0.61)\right]$ to the roughly linear increase of $P\left(h\right)$, as described in Ref. [38].

Figure 2

The forward branch of the hysteresis loop in the average magnetization per spin, $m_x$, for a representative small sample of size $N=16$ (red thick curve). At the points $H=\pm H_c$ the hysteresis curve starts deviating from full polarization. A big jump at $H=0$ arises since the magnetization spontaneously swivels by $180{}^{\circ }$ to realign with the external field which changes sign. Apart from that trivial jump, two avalanche events are seen. Their magnitude ${\left|\right|\Delta \varphi \left|\right|}_1$ is indicated by two peaks (dashed blue).

Figure 3

Schematic plot of the functional $\mathcal{H}(\Psi =({\Psi }_1,0,\cdots ,0))$, Eq. (19), vs ${\Psi }_1$, for various values of the external field $H$. The field increases from the top to the bottom curve. Before the instability ($H<H_{\alpha }$), $\mathcal{H}\left({\Psi }_1\right)$ has a locally stable minimum at ${\Psi }_1=0$. However, in general, there already exist lower-lying minima at nonzero values of ${\Psi }_1$. As the field $H$ approaches the critical value $H_{\alpha }$ (the dashed curve), the minimum ${\Psi }_1=0$ becomes locally unstable and a spontaneous rearrangement (avalanche) is triggered.

Figure 4

The upward hysteresis curve, averaged over disorder for different system sizes $N=32,64,128,256,512,1024,2048,4096$ (top to bottom curves for $H>0$ and bottom to top curves for $H<0$, respectively) with the number of disorder samples averaged over respectively being $1000,500,200,100,100,100,100,100$. We show the vicinity of $H=0$. The vertical span of the curve decreases with increasing $N$. Inset: The average magnitude of the magnetization at $H=0^{-}$, $s\equiv |\overline{m_x}\left(0^{-}\right)|$ is a measure for the vertical span of the hysteresis curve. The decrease of $s$ (red points) with $N$ is consistent with a power law $f\left(N\right)=b{N}^{-c}$. A fit yields $b=0.56\pm 0.05$ and $c=0.37\pm 0.02$ (blue dashed curve in the inset).

Figure 5

The distribution of the participation ratio $Y_2$, as defined by Eq. (23) for several system sizes $N$. The peak of the distribution increases with $N$ and shifts towards $Y_2=0$. Even though the thermodynamic limit is not yet clearly reached at $N=4096$ [curve with the highest peak (orange)], the data suggest that, as $N\to \infty$, $P\left(Y_2\right)$ remains peaked at a finite $Y_2\approx 0.05$. This would imply that there is a typical avalanche size of order $N$.

Figure 6

Fraction of the total magnetization reversal, which occurs in the form of discontinuous avalanches. In the thermodynamic limit, the curve saturates, conceivably to a fraction less than 1. However, larger system sizes would be needed to yield a reliable estimate of $f_{\mathrm{disc}}(N\to \infty )$.

Figure 7

Average spectral density of the inverse susceptibility matrix $T$, averaged over all instability points occurring for $\left|H\right|<1$ (except the trivial jump at $H=0$) in a single disorder sample of size $N=1024$ (red dots). The blue dashed curve is a fit to the function $\rho \left(e\right)=a\sqrt{e}$ in the region $[0,1.5]$, with $a=0.335\pm 0.003$. This confirms the gapless “semicircle law” (24) for small eigenvalues.

Figure 8

The log-log plot of the average interavalanche spacing $\delta H$ within the range $\left|H\right|\le 1$ of external fields. The (red) dots are the numerical data. For comparison we show the two power laws $N^{-1/2}$ [top line (black)] and $N^{-2/3}$ [lower line (brown)] that are suggested by scaling arguments. The best fit to the numerical data is $\delta H\sim N^{-a}$ with $a=0.57\pm 0.01$ [middle line (blue)].

Figure 9

Participation ratio of the soft mode divided by the system size, $n_1/N$, plotted versus $N$. The (red) dots are the numerical values, the solid (blue) curve is the fit to the data: $ln(n_1/N)=1.26-0.65ln\left(N\right)$, which is compatible with the theoretically anticipated scaling $n_1\sim N^{1/3}$.

Figure 10

Data collapse of scaled densities of two different measures of avalanches: (a) $s_1=N^{-a}\Delta M_x$, $r_1\left(s_1\right)=N^{2a-1}\rho \left(\Delta M_x\right)$ and (b) $s_2=N^{-a}\left|\Delta \vec{M}\right|$, $r_2\left(s_2\right)=N^{2a-1}\rho \left(\right|\Delta \vec{M}\left|\right)$. The body of the data collapses best with an exponent $a=1-\alpha =0.43$, which is in agreement with $\alpha =0.57$ obtained in Fig. 8. The solid (black) line corresponds to a power law $\sim s^{1/2}$.

Figure 11

Average contribution $W_L$ [see Eq. (37)] of longitudinal modes to the avalanches, plotted as a function of system size. Note the large standard deviations, indicating that jumps come in all sizes.

Figure 12

Top: Distribution of the participation ratio $Y_{\omega }$ of transverse modes, as defined in Eq. (39). Bottom: Cumulative distribution function of $Y_{\omega }$, rescaled by its average, for different system sizes. The absence of a clear collapse onto a single curve indicates the presence of many scales in the distribution of $Y_{\omega }$.