Subharmonics and Aperiodicity in Hysteresis Loops

We show that it is possible to have hysteretic behavior for magnets that does not form simple closed loops in steady state, but cycles multiple times before returning to its initial state. We show this by studying the low temperature dynamics of the 3D Edwards-Anderson spin glass. The specific multiple varies from system to system and is often quite large and increases with system size. The last result suggests that the magnetization could be aperiodic in the large system limit for some realizations of randomness. It should be possible to observe this phenomenon experimentally.

Figure 1

Steady-state magnetization as a function of magnetic field for a spin glass with $8^3$ spins. It can be seen that two cycles of the magnetic field are needed for the magnetization (and in fact the full spin configuration of the system) to return to its initial value.

Figure 2

Average power spectrum for systems of $8^3$ spins. The peaks at $f=1$ and $f=0$, whose heights are roughly $3\times {10}^4$, are suppressed. The power is concentrated at $f=1/2$, with smaller peaks at other fractions. The data are at temperatures $T=0$ (+) and $T=0.2$ (curve). The inset is a blown up portion of the graph with peaks at different subharmonics for both temperatures.

Figure 3

Low frequency average power spectrum for systems of $4^3$ and ${16}^3$ spins. (To avoid confusion with the horizontal axis, the vertical axis has an origin at $-1$.) There is substantially more power at low frequencies for ${16}^3$ spins, and the spectrum extends down to $1/f\approx 40$, i.e., a 40-cycle that is indivisible. The number of systems is the same as in Table .