Self-Induced Glassiness and Pattern Formation in Spin Systems Subject to Long-Range Interactions

We study the glass formation in two- and three-dimensional Ising and Heisenberg spin systems subject to competing interactions and uniaxial anisotropy with a mean-field approach. In three dimensions, for sufficiently strong anisotropy the systems always modulate in a striped phase. Below a critical strength of the anisotropy, a glassy phase exists in a finite range of temperature, and it becomes more stable as the system becomes more isotropic. In two dimensions the criticality is always avoided and the glassy phase always exists.

Figure 1

Panel (a) the function $\mathcal{I}({\mathrm{\Sigma }}_G)$ for the two-dimensional Ising model ($D=2$ and $N_s=1$), plotted as a function of ${\mathrm{\Sigma }}_G$ and for three values of the anisotropy parameter ${ϵ}_0$. The function always diverges in the limit ${\mathrm{\Sigma }}_G\to 0$. Inset: the function $\mathcal{J}({\mathrm{\Sigma }}_F)$ for ${ϵ}_0=0.01$ and $T=0.25$. It clearly shows two zeros. Panel (b) same as panel (a) but for the three-dimensional Ising model ($D=3$, $N_s=1$). Note that, unless ${ϵ}_0=0$, the function always converges to a finite value, which defines the minimum temperature $T_{\text{crit}}$ below which the system undergoes a phase transition to the ordered phase. Inset: the function $\mathcal{J}({\mathrm{\Sigma }}_F)$ for ${ϵ}_0=0.01$ and $T=2.5$.

Figure 2

Panel (a) the configurational entropy of the mean-field problem for the two-dimensional Ising model ($D=2$ and $N_s=1$). Note that this curve has been multiplied by a factor 0.1. Inset: the transition temperature $T_A$ as a function of the anisotropy parameter ${ϵ}_0$. Panel (b) same as panel (a) but for the two-dimensional Heisenberg model ($D=2$, $N_s=3$). Inset: the temperature $T_A$ as a function of ${ϵ}_0$.