Loop algorithm for classical Heisenberg models with spin-ice type degeneracy

In many frustrated Ising models, a single-spin flip dynamics is frozen out at low temperatures compared to the dominant interaction energy scale because of the discrete “multiple valley” structure of degenerate ground-state manifold. This makes it difficult to study low-temperature physics of these frustrated systems by using Monte Carlo simulation with the standard single-spin flip algorithm. A typical example is the so-called spin-ice model, frustrated ferromagnets on the pyrochlore lattice. The difficulty can be avoided by a global-flip algorithm, the loop algorithm, that enables to sample over the entire discrete manifold and to investigate low-temperature properties. We extend the loop algorithm to Heisenberg spin systems with strong easy-axis anisotropy in which the ground-state manifold is continuous but still retains the spin-ice type degeneracy. We examine different ways of loop flips and compare their efficiency. The extended loop algorithm is applied to two models, a Heisenberg antiferromagnet with easy-axis anisotropy along the $z$ axis, and a Heisenberg spin-ice model with the local $⟨111⟩$ easy-axis anisotropy. For both models, we demonstrate high efficiency of our loop algorithm by revealing the low-temperature properties which were hard to access by the standard single-spin flip algorithm. For the former model, we examine the possibility of order from disorder and critically check its absence. For the latter model, we elucidate a gas-liquid-solid transition, namely, crossover or phase transition among paramagnet, spin-ice liquid, and ferromagnetically ordered ice-rule state.

Figure 1

(Color online) The pyrochlore lattice composed of a three-dimensional network of corner-sharing tetrahedra. A 16-site cubic unit cell is shown. A and B represent upward and downward tetrahedra, respectively. Ising spins are denoted by arrows. Each spin axis is along the local $⟨111⟩$ axis, which goes from the site to the center of a neighboring tetrahedron. Black (filled) circles represent spins pointing inward while white (open) circles represent spins pointing outward in terms of type-A tetrahedra. The hexagon with a bold dashed line denotes one of the shortest loops on which a flip of all spins (colors) transforms an ice-rule state to another ice-rule state. See the text for details.

Figure 2

(Color online) Schematic picture for a loop construction by tracing a path through ice-rule tetrahedra. The path is made of alternating black and white sites. For simplicity, the figure shows a $⟨111⟩$ kagome layer with connected tetrahedra. The path is denoted by a dashed line, and its lower part represents an example of a closed loop.

Figure 3

(Color online) Two different ways to flip black and white for the spin ${\vec{S}}_i$ at site $i$: (a) flip $xyz$ and (b) flip parallel. See the text for details.

Figure 4

(Color online) (a) Temperature dependence of the acceptance rate of the single-spin flip $\left(P_{\text{single}}\right)$, the probability of formation of closed loops $\left(P_{\text{loop}}\right)$, the acceptance rates of flip of a formed loop by flip $xyz$ $\left(P_{xyz}\right)$ and by flip parallel $\left(P_{\text{parallel}}\right)$. The data are calculated for the model (7) at $D_{\text{I}}=5.0$. The data for the system sizes $L=2$ and $L=4$ are denoted by crosses and filled squares, respectively. (b) $D_{\text{I}}$ dependence of $P_{\text{parallel}}$, $P_{xyz}$, $P_{\text{loop}}$, and $P_{\text{single}}$ at $T=0.1$ for $L=2$.

Figure 5

(Color online) Autocorrelation functions for spin configurations [Eq. (8)] calculated at (a) $T=1.0$, (b) $T=0.5$, and (c) $T=0.05$. The data are at $D_{\text{I}}=5.0$ for $L=2$. The offset $A\left(\infty \right)$ was obtained by averaging $A\left(n\right)$ in the range of $5000\le n\le 10000$. The inset of (c) shows the same data as in (c) in a wider range of $n$.

Figure 6

(Color online) Temperature dependences of (a) the specific heat $C$ and (b) the uniform magnetic susceptibility ${\chi }_0$ for the model (7) at $D_{\text{I}}=5.0$.

Figure 7

(Color online) (a) Schematic picture for a ground state of the Heisenberg spin-ice model given by Eq. (11) on an isolated tetrahedron. ${\theta }_{\text{c}}$ is a canting angle. The big arrow in the center of tetrahedron denotes a net moment by forming the canted two-in two-out spin configuration. (b) A ferromagnetically ordered ice-rule state with aligning net magnetizations along the $z$ axis. (c) Schematic picture for a spin-ice liquid, in which each tetrahedron is in the two-in two-out state but the induced net moments are disordered among tetrahedra.

Figure 8

(Color online) (a) Temperature dependence of the acceptance rates of different updates. The notations are the same as in Fig. 4. (b) MC dynamics of the squared uniform magnetization $m^2$ in the MC runs with and without flip parallel at $T=0.05$. Two data for each case show the results starting from different initial spin configurations. All the data are for $L=2$.

Figure 9

(Color online) Temperature dependences of (a) the squared uniform magnetization $m^2$ and (b) the specific heat $C$ at $D_{\text{I}}=50.0$. A broad peak in the specific heat is denoted by an arrow. The inset shows an enlarged view of the main panel.

Figure 10

(Color online). Temperature dependences of (a) the squared uniform magnetization $m^2$ and (b) the specific heat $C$. (c) $D_{\text{I}}$ dependence of $m^2$ at the lowest temperature in (a); the dotted line denotes the ground-state magnetization calculated from Eq. (13). The system size is $L=4$.

Figure 11

(Color online). $D_{\text{I}}\text{−}T$ phase diagram of the Heisenberg spin-ice model given by Eq. (11). The transition temperature $T_{\text{c}}$ and the crossover temperature $T^{\ast }$ are estimated from a sharp peak and a broad one in the specific heat for $L=4$, respectively.