Multiple symmetry sustaining phase transitions in spin ice

We present the full phase diagram of the dumbbell model of spin ice as a function of temperature, chemical potential and staggered chemical potential which breaks the point group symmetry of the underlying diamond lattice in favor of a magnetic charge crystal. We observe a double winged structure with five possible phases, monopole fluid (spin ice), fragmented single monopole crystal phases, and double monopole crystal, the zinc-blende structure. Our model provides a skeleton for liquid-liquid phase transitions and for the winged structures observed for itinerant magnets under pressure and external field. We relate our results to recent experiments on ${\mathrm{Ho}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ and propose a wide ranging set of new experiments that exploit the phase diagram, including high-pressure protocols, dynamical scaling of Kibble-Zurek form, and universal violations of the fluctuation-dissipation theorem.

Figure 1

From spins to dumbbells. (Left) The pyrochlore lattice of corner sharing tetrahedra. Tetrahedron centers form a diamond lattice. Blue and red spheres illustrate the reduction of point group symmetry from the diamond to the zinc-blende structure. (Right) The point dipoles are extended to needles touching at the diamond lattice sites. The needles carry magnetic flux and charge $q=\pm m/a$ at each end. In a 2in-2out configuration (top) the vertex is charge neutral. A 3in-1out (3out-1in) configuration carries a monopole charge $Q=2m/a$ ($-Q=-2m/a$) (center). A 4in (4out) configuration carries a double monopole charge $2Q=4m/a$ ($-2Q=-4m/a$) (bottom).

Figure 2

Dumbbell model phase diagram. (0) monopole fluid (${\varphi }_0$), (1) monopole crystal (${\varphi }_1$) and (2) double monopole crystal (${\varphi }_2$). Surfaces show first-order and solid lines second-order transitions and dotted lines show the multicritical region. The long dashed lines show the extension of the second-order lines to infinity. Chemical potentials for single and double monopoles are $\mu =-\nu ,{\mu }_2=-4\nu$ [see Eq. (3)]. For the phase diagram of the $S=2$ Blume-Capel model $\overline{\mathrm{\Delta }}$ replaces $\mathrm{\Delta }$ and $\overline{\nu }$ replaces $\nu$.

Figure 3

Blume-Capel $S=1$. (0) Paramagnet and (1) antiferromagnet. Surfaces show first- and lines second-order transitions and the point shows a tricritical point.

Figure 4

$\mathrm{\Delta }\text{−}T$ plane. (Left) shaded plane through the full phase diagram at fixed $\nu$ - dotted black lines show the intercepts of the phase boundaries with the plane. (Right) The fixed $\nu$ plane rotated to give $\mathrm{\Delta }$ vs $T$. Green lines show the phase boundaries, the dotted black line shows an isothermal trajectory in the plane.

Figure 5

Multiple monopole crystallization. (Top) The order parameter $\varphi$ vs $\mathrm{\Delta }$ simulated from the dumbbell model at fixed $\nu =4.35$ K. Simulations for $N_0=4096$ ($L=8$) and periodic boundaries. All values are in Kelvin. (Bottom) Probability density $P\left(\varphi \right)$ for $\mathrm{\Delta }=2.05$ K, $T=0.3\mathrm{K}$, and $\nu =4.35$ K.

Figure 6

Locating the critical point. (Top) The Binder cumulant $B_4$ for the emergent order parameter $\phi =\varphi -{\varphi }_c$ for fixed $\nu =4.35$ K and $\mathrm{\Delta }={\mathrm{\Delta }}_c=2.03745$ K and ${\varphi }_c=0.42$ at the ${\varphi }_1:{\varphi }_0$ phase boundary (see text). (Bottom) Probability density $P\left(\varphi \right)$ at the critical end point, $\nu =4.35$ K, ${\mathrm{\Delta }}_c=2.03745,\text{and}{T}_c=0.36752$.

Figure 7

Low temperature ($T\ll \nu$). (Left) $\nu ={\nu }^{*}$ and (right) $\nu <{\nu }^{*}$. (0) monopole fluid (${\varphi }_0$), (1) monopole crystal (${\varphi }_1$), and (2) double monopole crystal (${\varphi }_2$).

Figure 8

Symmetry breaking in the $S=2$ Blume-Capel model: the $\overline{\mathrm{\Delta }}=0$ plane of the $S=2$ Blume-Capel model [36]. Phases are defined in Fig. 1.

Figure 9

Critical correlations at equilibrium. Autocorrelation function for the monopole crystal $C_{\varphi }\left(t\right)$ vs $t$ for temperatures close to the critical temperature. Solid black line shows power law decay with exponent $-1/2$.

Figure 10

Hysteresis loops. $\varphi$ vs $\mathrm{\Delta }$ for different sweep rates. System size $L=8$. The $\varphi \left(t\right)$ is a configurational average of 500 samples, each starting at equilibrium at $t=0$.

Figure 11

Kibble-Zurek scaling. $\mathrm{\Delta }\phi$ weighted by $t_{\mathrm{KZ}}^{D_{\lambda }/z}$ as a function of $\left(\frac{t}{t_{\mathrm{KZ}}}\right)$ for ${\nu }_{\lambda }=0.4$ and $z=2$.

Figure 12

FDT violations. Parametric plot of the response function against the correlation function, for various fixed times $t$ and using $t_w$ as a running parameter after a quench at the critical point $T_c,{\mathrm{\Delta }}_c$. The system size is $N_0=13824$ ($L=12$), and data are averaged over 500 independent quenches. The inset show limiting values for long times. The solid lines show the expected universal value $X_{\infty }=0.38$ for the 3D Ising universality class.

Figure 13

Monopole creation in a [111] field. (Top) Connected tetrahedra perpendicular to the [111] axis form kagomé planes of spins which are bases for alternating up and down tetetrahedra. The arrow shows the direction of an applied field. (Bottom) Flipping the spins indicated (left) creates monopole pairs with broken point group symmetry. North pole ($+$): red disk, the south pole ($-$): blue disk.

Figure 14

Monopole phase transition in a [111] field. We reproduce data from Fig. 4 of Ref. [77], which shows the thermal variation of the transition field of ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ for $H$ parallel to [111]. Solid circles (squares) denote the data points obtained for increasing (decreasing) field sweeps. Open triangles are the averaged critical field $H_c$, which show a nearly linear temperature variation with $\frac{d{H}_c}{dT}=0.08$ T ${\mathrm{K}}^{-1}$ (dashed line) at low temperature. The added blue circles are our estimates of the field strength at $T=0$ and at $T_c$ with blue line the phase boundary given by the constant entropy approximation.