Scaling ansatz for the ac magnetic response in two-dimensional spin ice

A theory for frequency-dependent magnetic susceptibility $\chi \left(\omega \right)$ is developed for thermally activated magnetic monopoles in a two-dimensional (2D) spin ice. By modeling the system in the vicinity of the ground-state manifold as a 2D Coulomb gas with an entropic interaction, and then as a 2D sine-Gordon model, we have shown that the susceptibility has a scaling form $\chi \left(\omega \right)/\chi \left(0\right)=\mathcal{F}(\omega /{\omega }_1)$, where the characteristic frequency ${\omega }_1$ is related to a charge correlation length between diffusively moving monopoles, and to the principal-breather excitation. The dynamical scaling is universal and applicable not only for kagome ice, but also for superfluid and superconducting films and generic 2D ices possibly including the artificial spin ice.

Figure 1

Spin configurations on ${\Lambda }^{*}$ are given in (a) and (c); corresponding monopole distributions on $\Lambda$ are in (b) and (d). (a) The loop ($\mathcal{C}$) and strings (${\mathcal{S}}_{1,2}$) are given by thick gray lines along which spins line up tail to nose. (b) The red (blue) circle represents a positive (negative) charge monopole; $a$ indicates the site spacing. Thick gray arrows show the displacements.

Figure 2

ac magnetic susceptibility: (a) Eq. (12) and (b) Eq. (14). The frequency is given in units of ${\omega }_1$, and is plotted on a logarithmic scale ($b=\frac{1}{4}$). The cutoff dependence of ${\chi }^{\prime }$ obtained by MC simulations are given in the inset of (a); $1/\chi \left(0\right)$ is plotted in the inset of (b), where a fitting line is also given by using data at the lowest two temperatures. The scaled MC data using monopole density and $\chi \left(0\right)$ are given in (a).

Figure 3

Honeycomb lattice $\Lambda$ is given by dotted lines. Red (blue) triangles represent upward (downward) pointing tetrahedra, and include sites in ${\Lambda }_{\mathrm{a}}$ (${\Lambda }_{\mathrm{b}}$). The two axes ${\mathit{e}}_v$ ($\parallel \left[1\overline{1}0\right]$) and ${\mathit{e}}_h$ ($\parallel \left[11\overline{2}\right]$) as well as the lattice spacings are given by arrows. Open circles show a triangular lattice ${\Lambda }_{\mathrm{t}}$ for the cell charges $q_{\lambda }$ (see Fig. 4). A rhombic plaquette exhibits one unit cell of $\Lambda$.

Figure 4

Coarse-graining procedure. In a left rhombus, two vertexes of dotted lines show the two sites $l\left(\lambda \right)$ in a $\lambda \mathrm{th}$ unit cell of $\Lambda$; in a right one, an open circle gives a $\lambda \mathrm{th}$ site in ${\Lambda }_{\mathrm{t}}$. In both rhombi, monopoles (antimonopoles) are denoted by red (blue) circles. (a) and (b) exhibit one-monopole states with $q_l=\pm 1$ and the corresponding states with $q_{\lambda }=\pm 1$. (c) A pair of monopoles with $q_l=\pm 1$ are mapped to $q_{\lambda }=0$ as in (d).