Cluster algorithm for Monte Carlo simulations of spin ice

We present an algorithm for Monte Carlo simulations of a nearest-neighbor spin-ice model based on its cluster representation. To assess its performance, we estimate a relaxation time, and find that, in contrast to the Metropolis algorithm, our algorithm does not develop spin freezing. Also, to demonstrate the efficiency, we calculate the spin and charge structure factors, and observe pinch points in a high-resolution color map. We then find that Debye screening works among defects and brings about short-range correlations, and that the deconfinement transition triggered by a fugacity of defects $z$ is dictated by a singular part of the free-energy density $f_{\mathrm{s}}\propto z$.

Figure 1

Left: The pyrochlore lattice ${\Lambda }_{\mathrm{p}}$. Blue dots represent magnetic ions. The frame in magenta (cyan) marks the primitive (cubic) cell with 4 (16) spins. The edge length and fundamental vectors of the cubic cell are, respectively, $a$ and $({\mathbf{e}}_x,{\mathbf{e}}_y,{\mathbf{e}}_z)$. The length of primitive vectors, e.g., ${\mathbf{t}}_1=\frac{a}{2}({\mathbf{e}}_x+{\mathbf{e}}_y)$, is $a/\sqrt{2}$. Right: Top three lines give the 16 states per one tetrahedron $S_u^{\mu }$ classified by the number of inward spins $\mu$ (blue arrows). Bottom three lines give the 10 graphs $G_v^{\nu }$, where $\nu$ denotes the number of bonds (bold lines).

Figure 2

(a) ${\overline{A}}_M\left(k\right)$ at $T=0.3$, 0.5, and 1.0. Blue (red) marks plot data by the Metropolis (cluster) algorithm. The forms of exponential relaxation are drawn as fitted curves in blue. (b) and (c) The $n$ and $L$ dependencies of ${\tau }_{\mathrm{dep}}$ in two algorithms. The relaxation times estimated via ${\overline{A}}_M\left(k\right)$ are given as blue dotted lines; the red indicates ${\tau }_{\mathrm{dep}}=\frac{1}{2}$. (d) Temperature dependence of the relaxation times; the fitted line shows its exponential dependence in the Metropolis dynamics.

Figure 3

Color maps of spin and charge structure factors, ${\mathcal{S}}_{\perp }\left(\mathbf{Q}\right)$ and $\mathcal{C}\left(\mathbf{Q}\right)$, at $T=0.3$ (bottom), 1.0 (middle), and 2.0 (top). The data in the left (right) panels are normalized by an amplitude of spin (charge) structure factor at each $T$. The wave vector is on the $(h,h,l)$ plane, and is measured in units of $2\pi /a$.

Figure 4

(a) and (b) $r\mathcal{S}$ and $r\mathcal{C}$ versus $r$ for various $T$. Slopes of fitted lines determine the inverse correlation lengths for the $L=128$ system. (c) The $z$ dependence of ${\lambda }_{\mathrm{s}}$ (diamonds), ${\lambda }_{\mathrm{c}}$ (squares), and $n_{\mathrm{m}}$ (circles). The slope of the line for ${\lambda }_{\mathrm{s},\mathrm{c}}$ is $-\frac{1}{2}$, and for $n_{\mathrm{m}}$ is 1. (d) gives the finite-size-scaling plot of $n_{\mathrm{m}}$ with the critical exponent $\nu =\frac{1}{3}$.