Polarization plateaus in hexagonal water ice $I_h$

The protons in water ice are subject to so-called ice rules resulting in an extensive ground-state degeneracy. We study how an external electric field reduces this ground-state degeneracy in hexagonal water ice $I_h$ within a minimal model. We observe polarization plateaus when the field is aligned along the [001] and [010] directions. In each case, one plateau occurs at intermediate polarization with reduced but still extensive degeneracy. The remaining ground states can be mapped to dimer models on the honeycomb and the square lattice, respectively. Upon tilting the external field, we observe an order-disorder transition of Kasteleyn type into a plateau at saturated polarization and vanishing entropy. This transition is investigated analytically using the Kasteleyn matrix and numerically using a modified directed-loop Monte Carlo simulation. The protons in both cases exhibit algebraically decaying correlations. Moreover, the features of the static structure factor are discussed.

Figure 1

(a) Illustration of the structure of hexagonal ice $I_h$. An external field along ${\mathit{E}}_0^{\left[001\right]}\parallel \left[001\right]$ or ${\mathit{E}}_0^{\left[010\right]}\parallel \left[010\right]$ is applied with an additional tilt in the directions (b) ${\mathit{E}}_{\perp }^{\left[001\right]}\parallel \left[100\right]$ and (c) ${\mathit{E}}_{\perp }^{\left[010\right]}\parallel \left[101\right]$. Oxygen ions are drawn in red, protons in gray.

Figure 2

Relative polarization (blue dots) and specific heat (green triangles) vs temperature for a field applied along $\mathit{E}=(1,0,100)$. Three regimes exist: (I) a high-$T$ regime with only small polarization only subject to the ice rules (tetrahedron with a two-in–two-out configuration), (II) an intermediate polarization plateau (random dimers), and (III) a low-$T$ regime with fully saturated polarization and long-range order (ordered dimers). The polarization plateau has a relative polarization of $\left|\mathit{P}\right|/P_{\text{sat}}=\frac{2}{\sqrt{10}}\approx 0.63$ (horizontal dashed line). The degrees of freedom within the plateau form a stack of two-dimensional planes, where for each of the planes a microscopic description in terms of dimers on a honeycomb lattice exists. A Kasteleyn transition into a fully ordered state occurs at $T_K=3.96$ (vertical dashed line). The inset shows the relative polarization near the Kasteleyn transition. The Kasteleyn matrix approach enables us to compute the relative polarization (solid black line). Excellent agreement is found with the numerical data (blue dots).

Figure 3

Same plot as in Fig. 2 but for a field applied along $\mathit{E}=\left(\frac{1}{\sqrt{2}},100,\frac{1}{\sqrt{2}}\right)\approx (0.71,100,0.71)$. The Kasteleyn transition occurs at $T_K=3.385$ (vertical dashed line). The polarization in the intermediate plateau points along [010] with magnitude $\left|\mathit{P}\right|/P_{\text{sat}}=\frac{3}{2\sqrt{5}}\approx 0.67$ (horizontal dashed line). Here, the degrees of freedom within the plateau form a stack of two-dimensional planes, where for each of the planes a microscopic description in terms of dimers on a square lattice exists. The inset shows the relative polarization near the Kasteleyn transition of the effective dimers into a fully ordered state. The numerical data agree with the polarization obtained analytically (solid black line) utilizing the Kasteleyn-matrix approach; cf. Sec. 5.

Figure 4

Upon applying a field along [001] and lowering the temperature below the corresponding energy scale, the spins with the largest projection onto the field are pinned. These are highlighted as green spins in (a). The remaining degrees of freedom, blue spins in (a), form a bipartite honeycomb lattice with two-in–one-out or one-in–two-out configurations; see (b). A one-to-one map between the low-energy manifold of spin configurations and dimers on the edges of a honeycomb lattice exists: Edges with a minority spin, i.e., orange spins in (b), map to edges containing a dimer as illustrated in (c).

Figure 5

Upon applying a field along [010] and lowering the temperature below the corresponding energy scale, the spins with the largest projection onto the field get pinned. These are highlighted as green spins in (a). The remaining degrees of freedom, blue spins in (a), form a honeycomb-like lattice with two neighboring vertices being of the same two-in–one-out configuration (or vice versa). Such double vertices have one internal spin, colored dark blue in (b), which is fixed by the configuration of the outer spins, colored light blue or orange. Upon contracting such a double vertex by neglecting the dark blue spin, a one-to-one map to dimers on the edges of a rhombic lattice exists: Edges with a minority spin, i.e., orange spin in (b), map to edges of the rhombic lattice containing a dimer as illustrated in (c).

Figure 6

Unit-cell convention for the computation of the partition sum utilizing a Kasteleyn matrix with bond weights and corresponding bonds highlighted in orange.

Figure 7

Poles of the integral in Eq. (14) with respect to $z$. Three different cases exist depending on the choice of weights $a,b$: (a) $z_1\left(w\right)$ is completely inside, (b) $z_1\left(w\right)$ is partially inside, and (c) $z_1\left(w\right)$ is completely outside of the integration contour $\left|z\right|=1$.

Figure 8

Specific heat at the Kasteleyn transition as obtained using (a) the Kasteleyn-matrix approach (solid black line) of the emergent dimer model on a square lattice and (b) within directed-loop Monte Carlo (black dots). The critical behavior is $C\propto {(T-T_K)}^{-\alpha }$ with $\alpha =1/2$ for $T>T_K$, as illustrated by the dashed blue line in the main plot and the inset. A system of size $160\times 160\times 80$ exhibits a maximum in the specific heat at a temperature slightly lifted by about ${10}^{-2}$ with respect to the critical temperature in the thermodynamic limit. Above this temperature, the numerical and analytical results agree. The error bars display the standard deviation of $C$ as obtained from 16 independent simulations. In addition, the inset shows data for a system of size $40\times 40\times 20$ illustrating possible finite-size effects: smaller systems exhibit a broadening of the peak as well as an increased shift of the maximum to higher temperatures.

Figure 9

Diffuse part of the static structure factor at high temperatures or equivalently without external field. The upper row shows the $\left[hk0\right], \left[h0l\right]$, and $\left[0kl\right]$ planes for $S_I\left(\mathit{k}\right)$ [see Eq. (22)] of the effective spin model with Ising variables $s_i$. The lower row shows the same scattering planes for the proton system, $S_H\left(\mathit{k}\right)$ given in Eq. (21), which can be compared to experimental scattering data of ice $I_h$ [31]. The local ice rule constraint results in pinch points with typical bow-tie features.

Figure 10

$\left[hk0\right]$ plane of the static structure factor within the intermediate plateau ($T=40$) when a field is applied along $\mathit{E}=(1,0,100)$ for (a) effective spin model, (b) protons, (c) asymptotic correlations for dimers with equal weights on the hexagonal lattice [34], and (d) effective spin model at $T=5$ within the plane formed by either the lower (left) or upper (right) spins of a unit cell. The splitting of the high-intensity points in (a) and (b) is caused by the small tilt component of the applied field. Lowering the temperature enhances the splitting as illustrated in (d) until two adjacent peaks merge at the concentric pinch point. The merging occurs at the Kasteleyn transition.

Figure 11

$\left[h0l\right]$ plane of the static structure factor within the intermediate plateau ($T=40$) when a field is applied along $\mathit{E}=(\sqrt{2},100,\sqrt{2})$ for (a) effective spin model, (b) protons, (c) effective spin model without the intermediate spin, and (d) asymptotic correlations for dimers with equal weights on the square lattice [30]. The splitting of the high-intensity points in (a), (b), and (c) is caused by the small tilt component of the applied field. A one-to-one map exists from the double-vertices with three-in–one-out (one-in–three-out) spin configurations to dimers, enabling us to compare (c) and (d).