Monopole ordered phases in dipolar and nearest-neighbors Ising pyrochlore: From spin ice to the all-in–all-out antiferromagnet

We study Ising pyrochlores by means of Monte Carlo simulations. We cover a set of exchange constants ranging from the frustrated ferromagnetic case (spin-ice) to the fully-ordered “all-in–all-out” antiferromagnet in the dipolar model, reinterpreting the results—as in an ionic system—in terms of a temperature vs magnetic charge density phase diagram. In spite of its spin nature and the presence of both double and single nonconserved magnetic charges, the dipolar model gives place to a phase diagram which is quite comparable with those previously obtained for on-lattice systems of electric charges, and on spin ice models with a conserved number of single magnetic charges. The contrast between these systems, to which we add results from the nearest-neighbors model, put forward other features of our phase diagram—notably, a monopole fluid with charge order at high monopole densities that persists up to arbitrarily high temperatures—that can only be explained taking into account construction constraints forced by the underlying spin degrees of freedom.

Figure 1

Conventional unit cell of the pyrochlore lattice. In spin ice compounds, Ising-like spins occupy the vertices of corner-sharing “up” (pink) and “down” (lilac) tetrahedra. Black/blue spins point inwards/outwards (along the local $\langle 111\rangle$ directions) of an up tetrahedron. Via the dumbbell model [4], the diverse spin configurations are mapped to different types of magnetic charges: single/double (small/big) and positive/negative (green/red) monopoles (note that since this unit cell is only a part of a bigger system, the constraint of local magnetic neutrality is not satisfied). This configuration snapshot was obtained for a material with $J_{nn}/D_{nn}=-0.947$ at $T/D_{nn}\approx 0.57$, resulting in a monopole density (defined as the average number of charges per tetrahedron) of $\rho \approx 1.1$.

Figure 2

Number of single charges per tetrahedron vs temperature for $L=4$ and different values of $J_{nn}/D_{nn}$. It is possible to distinguish two very different regimes: Some of the curves (those with $|J_{nn}/D_{nn}|\le 0.911)$ reach $\rho =0$ when $T\to 0$, while the rest approach $\rho =2$. Among the latter it is easy to notice that the jump between low and high density becomes smoother as $|J_{nn}/D_{nn}|$ increases.

Figure 3

Top: molar specific heat for $L=4$ as a function of temperature; the sharp, deltalike peak broadens as $|J_{nn}/D_{nn}|$ increases—a sign of a first order phase transition that becomes second order. This idea is reinforced by the double monopole staggered density (bottom), which displays a steplike jump that becomes smooth and continuous.

Figure 4

For $J_{nn}/D_{nn}=-0.919$ the specific heat data displays finite size effects consistent with a first order phase transition. Moreover, the value of the specific heat and the double monopole susceptibility at their maximum grow linearly with the volume of the system, as shown in the inset.

Figure 5

For $J_{nn}/D_{nn}=-1.064$ the double monopole susceptibility displays finite size effects consistent with a second order phase transition. The values of ${\rho }_S^d(T_c\left(L\right)),C(T_c\left(L\right))/R$, and ${\chi }_S^d(T_c\left(L\right))$ evolve as power laws with the size of the system and present good correspondence with the behavior expected for the three dimensional Ising model universality class [27] (gray lines).

Figure 6

Temperature vs monopole density phase diagram with dipolar (long-range) interactions. At low temperatures the system undergoes a first order phase transition (white dome), where a low density gas and a crystal coexist. The transition becomes second order (dot-dashed line) at a tricritical point near ${\rho }_t\approx 1.17,T_t/D_{nn}\approx 0.34$. The diagram is overlaid on top of an interpolated contour plot of ${\rho }_S^d$ for $L=4$. The vertical dotted line at ${\rho }_c\approx 1.4$ indicates the critical density of the geometric transition in the NNSIM (see Sec. 4a).

Figure 7

In the case of the nearest neighbor model, the specific heat displays the finite size effects characteristic of a second order phase transition for $J_{nn}/D_{nn}=-1.064$. Inset: again, the values of ${\rho }_S^d(T_c\left(L\right)),C(T_c\left(L\right))/R$, and ${\chi }_S^d(T_c\left(L\right))$ evolve as power laws with the size of the system and present good correspondence with the behavior expected for the three dimensional Ising model [27] (gray lines).

Figure 8

Temperature vs monopole density phase diagram with nearest-neighbors interactions. The second order phase transition (dot-dashed line) between the gas and the staggered charge fluid occurs at ${\rho }_c\approx 1.4$ at all temperatures, hence we call it a geometric phase transition. The diagram is overlaid on top of an interpolated contour plot of ${\rho }_S^d$ for $L=4$.

Figure 9

$T/D_{nn}$ vs $J_{nn}/D_{nn}$ phase diagram for both dipolar and nearest-neighbors spin ice models. Dotted lines represent crossovers, while solid and dot-dashed lines stand for first and second order phase transitions, respectively. The effect of the dipolar interactions is clearly seen: The system orders at the alternating charge configuration even for values of $J_{\mathrm{eff}}>0$, because the Coulomb attraction between monopoles of opposite charge makes it energetically favorable to create and order these particles. Melko-Gingras-den Hertog first order transition [25] is not visible because we used a single spin-flip algorithm that mimics the real material and thus freezes at low temperatures.

Figure 10

Staggered charge density (main figure) and its fluctuations (inset) vs particle density for a system of “plus” and “minus” noninteracting hard spheres within the Grand Canonical Ensemble. Finite size scaling of these quantities is consistent with the three dimensional Ising universality class [27] (not shown).