Topological Spin Glass in Diluted Spin Ice

It is a salient experimental fact that a large fraction of candidate spin liquid materials freeze as the temperature is lowered. The question naturally arises whether such freezing is intrinsic to the spin liquid (“disorder-free glassiness”) or extrinsic, in the sense that a topological phase simply coexists with standard freezing of impurities. Here, we demonstrate a surprising third alternative, namely, that freezing and topological liquidity are inseparably linked. The topological phase reacts to the introduction of disorder by generating degrees of freedom of a new type (along with interactions between them), which in turn undergo a freezing transition while the topological phase supporting them remains intact.

Figure 1

Phase diagram of spin ice with a small density $x$ of spins removed. The glass transition takes place at $T_c$, while the crossovers from the high-temperature paramagnet to the topological Coulomb phase, and to impurity dominated monopole excitations take place at ${\mathrm{\Theta }}_W$ and $T_{\delta }$, respectively.

Figure 2

Complexity reduction via genesis of ghost spins. The cartoons are for a projection of spin ice onto the two-dimensional plane (top left panel) for clarity; the diamond lattice defined by the centers of the tetrahedra thus turns into a square lattice. At low temperature $T\ll \delta ,\mathrm{\Delta }$, spin ice with a small density $x$ of missing spins (crossed out in top right) is equivalent to a small density of ghost spins (bottom left). This is straightforwardly established by writing each spin as a dumbbell of equal and opposite magnetic charges. At vertices either end of a missing spin, the net charge is nonzero (red and blue circles), so that they form the ends of the dipole of the ghost spin. At all other vertices, the net charge vanishes (green circles) and can thus be omitted, so that this pristine bulk of spin ice only provides an effective medium (shaded green) carrying an interaction between the ghost spins, Eq. (3) and Fig. 3.

Figure 3

Correlations between a pair of ghost spins ${\overrightarrow{\mu }}_0$ and ${\overrightarrow{\mu }}_r$ in an otherwise fully populated sample of spin ice (of linear dimension $L=12$, with $D=1.41$ and $T=0.5\mathrm{K}$), separated by $r$ units in the $[101]$ direction. This is compared to two isolated magnetic dipoles in an otherwise empty unit cell; agreement with numerics is only achieved upon including the entropic interactions mediated by the spin ice bulk, for which the dashed line denotes the asymptotic form, Eq. (3). The inset shows the freezing transition of the ghost spins. Number of spins equals $16L^{3}x$ in a system with $L^3$ unit cells, with $x=1/16$. The data collapse of $\xi /L$ for different system sizes, using $\nu =1.16(8)$ and critical temperature $T_x=0.082(3)$, shows a continuous spin glass transition (see Supplemental Material [18]).