Experimental measures of topological sector fluctuations in the F-model

The two-dimensional F-model is an ice-rule-obeying model, with a low-temperature antiferroelectric state and high-temperature critical Coulomb phase. Polarization in the system is associated with topological defects in the form of system-spanning windings which makes it an ideal system on which to observe topological sector fluctuations, as have been discussed in the context of spin ice and Berezinskii-Kosterlitz-Thouless (BKT) systems. In particular, the F-model offers a useful counterpoint to the BKT transition in that winding defects are energetically suppressed in the ordered state, rather than dynamically suppressed in the critical phase. In this paper we develop Lieb and Baxter's historic solutions of the F-model to exactly calculate relevant properties, several apparently for the first time. We further calculate properties not amenable to exact solution by an approximate cavity method and by referring to established scaling results. Of particular relevance to topological sector fluctuations are the exact results for the applied field polarization and the “energetic susceptibility.” The latter is a both a measure of topological sector fluctuations and, surprisingly, in this case, a measure of the order parameter correlation exponent. In the high-temperature phase, the temperature tunes the density of topological defects and algebraic correlations, with the energetic susceptibility undergoing a jump to zero at the antiferroelectric ordering temperature, analogous to the “universal jump” in BKT systems. We discuss how these results are relevant to experimental systems, including to spin-ice thin films, and, unexpectedly, to three-dimensional dipolar spin ice and water ice, where we find that an analogous “universal jump” has previously been established in numerical studies. This unexpected result suggests a universal limit on the stability of perturbed Coulomb phases that is independent of dimension and of the order of the transition. Experimental results on water ice Ih are not inconsistent with this proposition. We complete the paper by relating our new results to experimental studies of artificial spin-ice arrays.

Figure 1

(a) The six vertices of the F-model, in which the vertices with no net polarization (5 and 6) are lower in energy than the remaining vertices (1, 2, 3, 4) by an energy gap $\epsilon$. (b) An excited F-model configuration with no net polarization. Excitations may consist of closed loops (green) which carry no net polarization as well as system-spanning loops (red and blue), which “wrap” the periodic boundary conditions and carry a net polarization.

Figure 2

(a) The polarization $m$ of the F-model as a function of the reduced temperature in varying applied fields $H$. (b) The specific heat divided by $T$ (or differential entropy increment) $C/T$ for varying fields. In applied fields $H<\epsilon$ there is a second-order phase transition at a temperature $T_{\text{C}}\left(H\right)$ as well as a maximum in the polarization at $T_{\text{max}}\left(H\right)\ge T_{\text{C}}\left(H\right)$. For fields at $H=\epsilon$ and greater (not shown), the system is aligned with the field at low temperatures and there is no phase transition.

Figure 3

The F-model polarization as a function of applied field at various temperatures between $0.05T_0$ and $0.95T_0$. For small temperatures the critical field $H_{\text{C}}$ is proportional to $|T-T_0|$ (inset).

Figure 4

The temperatures $T_{\text{C}}$ and $T_{\text{max}}$ as functions of the applied field. In the limit of very small fields $T_{\text{C}}\to T_0$ and $T_{\text{max}}\to \sim 1.213T_0$, while in the limit $H\to \epsilon$ the two temperatures converge to zero.

Figure 5

The energetic susceptibility as a function of temperature for different applied fields (with the susceptibility calculated as $\chi =m/H$) and in zero field [using the exact result of Eq. (14)] together with the exact zero-field internal energy. At the transition temperature $T_0$, the internal energy and the zero-field energetic susceptibility intersect at $U=\tilde{\chi }=1/3$ (indicated by a black circle).

Figure 6

The staggered susceptibility ${\chi }^{†}$ of the F-model on a vertex and plaquette tree, together with the asymptotic staggered susceptibility predicted for temperatures close to $T_0$ by scaling theory and the exact result for the staggered polarization. In all three cases, the staggered susceptibility diverges as the transition temperature is approached.

Figure 7

(a) The F-model ground state on a section of the pyrochlore lattice. In pyrochlore systems the staggered polarization in the (100) plane is proportional to the direct polarization along the [100] direction. (b) The same state projected onto the (100) plane. Fields in the (100) plane couple to a vertical and horizontal sublattice as for the square-lattice F-model. Note that on the pyrochlore lat- tice the minimal closed loop contains six spins (an example is highlighted in red).