Chiral RKKY interaction in Pr${}_2$Ir${}_2$O${}_7$

Motivated by the potential chiral spin liquid in the metallic spin ice Pr${}_2$Ir${}_2$O${}_7$, we consider how such a chiral state might be selected from the spin-ice manifold. We propose that chiral fluctuations of the conducting Ir moments promote ferrochiral couplings between the local Pr moments, as a chiral analog of the magnetic RKKY effect. Pr${}_2$Ir${}_2$O${}_7$ provides an ideal setting to explore such a chiral RKKY effect, given the inherent chirality of the spin-ice manifold. We use a slave-rotor calculation on the pyrochlore lattice to estimate the sign and magnitude of the chiral coupling, and find it can easily explain the 1.5 K transition to a ferrochiral state.

Figure 1

(a) For any given tetrahedra, the net chirality (a vector sum over the faces) will be proportional to the net magnetization. (b) Thus for the system to exhibit net magnetization without net chirality, the chirality must originate from independent triangles, like the isosceles triangles that make up the hexagons of the kagome planes, whose chirality will have an associated direction given by the normal to the triangular plaquette.

Figure 2

Hexagons which contain alternating “in” and “out” spins contain local modes in which every spin around the hexagon is flipped, resulting in another state in the spin-ice manifold. These “flippable” hexagons have zero chirality. Here, we indicate “in” spins with blue (darker) sphere and “out” with white (lighter) spheres, and have defined “in” and “out” self-consistently around the hexagon by picking the axes pointing into the alternating tetrahedra pointing out of the plane.

Figure 3

The chiral coupling $J_{\kappa }\left(r\right)$ calculated in the classical, $J_K\to \infty$ approximation for a parabolic dispersion. The $y$ axis is in kelvins, for a possible set of parameters for Pr${}_2$Ir${}_2$O${}_7$. For $r=a=10.4$ Å, the chiral coupling is $-200$ K. Just like the magnetic RKKY interaction, the chiral RKKY interaction is oscillatory and decays rapidly.

Figure 4

Bands on the pyrochlore lattice. (a) The bosonic bands plotted through the high-symmetry points of the pyrochlore Brillouin zone, for $t_2=-0.03t$, $Q_X^{\left(1\right)}=0.14$, $Q_X^{\left(1\right)}=0.20$, where $t$ is the nearest-neighbor oxygen-mediated hopping magnitude. (b) The fermionic bands with $t_{\sigma }=-t$, $t_{\pi }=2/3t$, and $t_2=-0.03t$. These are plotted for $U=0$, and where $\mu =-0.31t$ has been included so that $E_F$ is at zero when the Ir sites are half filled. (c) The fermionic density of states. Inset: The pyrochlore Brillouin zone.

Figure 5

(a) Fermionic transverse current-current correlator, ${\Pi }_F\left(\mathbf{q}\right)$, on the pyrochlore lattice for varying proximity to the Mott transition: $X_0^2=0$ (red, $U=1.4t$); $X_0^2=0.05$ (orange, $U=1.2t$); $X_0^2=0.1$ (yellow, $U=1.1t$); $X_0^2=0.25$ (green, $U=0.8t$); $X_0^2=0.5$ (blue, $U=0.3t$); $X_0^2=0.75$ (purple, $U=0.08t$); and $X_0^2=1$ (gray, $U=0$). Note that the distance from the Mott transition increases the effective fermionic bandwidth. (b) Bosonic transverse current-current correlator, ${\Pi }_X\left(\mathbf{q}\right)$, on the pyrochlore lattice for varying proximity to the Mott transition (same color scheme as above). Note that for large $X_0^2$, the correlator approaches a constant in $q$ space, just as expected in the Fermi liquid.

Figure 6

Chiral coupling $J_{\kappa }\left(r\right)$ in real space for several $X_0^2$ (color scheme explained in Fig. 5). $J_{\kappa }\left(r\right)$ is ferrochiral for all $r$ and $X_0^2$, and increases in magnitude as the Mott transition is approached. The coupling constant is given in kelvins, and even for $U=0$, the chiral couplings between hexagons, $r=a/\sqrt{2}$, and between layers, $r=\sqrt{11}/3a$, are both on the order of 10 K.

Figure 7

Classical versus quantum chiral spin liquids. (a) A generic phase diagram. The classical CSL* (cCSL*) occurs above a magnetically ordered state with net chirality (CMO), while the quantum CSL* (qCSL*) is only revealed when quantum fluctuations kill the magnetic order. (b) A rough sketch of the frequency dependence of the spin structure factor to illustrate the difference between classical (solid line) spin liquids, whose fluctuations are mainly confined to $\omega <T$ and quantum (dashed line) chiral spin liquids, whose fluctuations have a much broader distribution at finite frequencies.