Analytical theory of pyrochlore cooperative paramagnets

The pyrochlore lattice is associated with several potential and actual spin liquid phases as a result of its strong geometric frustration. At finite temperature, these can exhibit an unusually broad crossover regime to a conventional paramagnet. Here, we study this regime analytically by showing how a single-tetrahedron Hamiltonian can extrapolate beyond the first term of a high-temperature expansion and yield insights into the buildup of correlations. We discuss how this unusual behavior is brought about by the structure of the eigenspaces of the coupling matrix. Further interesting behavior can appear for parameter values located near phase transitions: we find coexistence of (111) rods and (220) peaks in the structure factor, as observed in neutron-scattering experiments on ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$.

Figure 1

The single-tetrahedron structure factor derived from $\langle \mathbf{S}{\mathbf{S}}^{⊺}\rangle \sim -\mathcal{J}$ and $\langle \mathbf{S}{\mathbf{S}}^{⊺}\rangle \sim P_0$ (left and right halves, 1 and 2 respectively, within each panel). (a) $(J_{zz},J_{\pm },J_{\pm \pm },J_{z\pm })=(1,0,0,0)$: the $P_0\stackrel{\text{off}}{\propto }-\mathcal{J}$ correspondence holds, and the two structure factors are exactly proportional to one another. (b) $(J_{zz},J_{\pm },J_{\pm \pm },J_{z\pm })=(0,0,0,1)$: the $P_0\stackrel{\text{off}}{\propto }-\mathcal{J}$ correspondence does not hold, and the two structure factors are obviously different. However, a certain degree of qualitative similarity is still apparent.

Figure 2

Eigenvalues of $\mathcal{J}$ and deviation $\mathcal{Q}$ from $\mathcal{J}\stackrel{\text{off}}{\propto }-P_0$ (top and bottom panel, respectively), for $(J_{zz},J_{\pm },J_{\pm \pm },J_{z\pm })=(cos\theta ,sin\theta ,0,0)$ against $\theta \in [0,2\pi ].$ Vertical gridlines are shown at points where $\mathcal{J}\stackrel{\text{off}}{\propto }-P_0$ holds exactly, i.e. where $\mathcal{Q}=0$. One of the eigenvalues, shown as a thicker line, contains two eigenspaces, $T_2$ and $T_{1A^{\prime }}$.

Figure 3

(a) Fig. 4(a) of Ref. [17], showing the experimental structure factor of ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ at $T\approx 50$ mK; single-tetrahedron structure factors from (b) $\langle \mathbf{S}{\mathbf{S}}^{⊺}\rangle \sim P_{T_{1A^{\prime }}}$, (c) $\langle \mathbf{S}{\mathbf{S}}^{⊺}\rangle \sim P_E$, and (d) $\langle \mathbf{S}{\mathbf{S}}^{⊺}\rangle \sim P_{T_{1A^{\prime }}}+(3/2)P_E$. All plots use the $g$ factor $g_{xy}/g_z=2$.

Figure 4

Structure factor from Monte Carlo simulations using the ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ parameters of Ref. [18] and temperature $T=450$ mK: (a) on a system of $20\times 20\times 20$ fcc 16-spin cubic unit cells and periodic boundary conditions; (b) on the same system, but truncating the correlators to nearest-neighbor distance before computing the structure factor (equivalently, calculating single-tetrahedron structure factors, averaged over all tetrahedra); and (c) on a single isolated tetrahedron.