Spin Hamiltonian, Competing Small Energy Scales, and Incommensurate Long-Range Order in the Highly Frustrated ${\mathrm{Gd}}_3{\mathrm{Ga}}_5{\mathrm{O}}_{12}$ Garnet Antiferromagnet

Despite the availability of a spin Hamiltonian for the ${\mathrm{Gd}}_3{\mathrm{Ga}}_5{\mathrm{O}}_{12}$ garnet (GGG) for over 25 years, there has so far been little theoretical insight regarding the many unusual low temperature properties of GGG. Here we investigate GGG in zero magnetic field using mean-field theory. We reproduce the spin liquid-like correlations and, most importantly, explain the positions of the sharp peaks seen in powder neutron diffraction experiments. We show that it is crucial to treat accurately the long-range nature of the magnetic dipolar interactions to allow for a determination of the small exchange energy scales involved in the selection of the experimental ordering wave vector. Our results show that the incommensurate order in GGG is classical in nature, intrinsic to the microscopic spin Hamiltonian and not caused by weak disorder.

Figure 1

MFT $I(Q)$ of GGG with $J_2=-0.003\mathrm{K}$, $J_3=0.010\mathrm{K}$ [11] and $\tau ={10}^{-4}$ for $R_c=3$, 4, 5, 10, 1000. The $R_c=\infty$ profile is obtained using the Ewald method. The profiles are uniformly shifted to emphasize their strong dependence on $R_c$ for small values, as well as their saturation to a well-defined limit at large $R_c$.

Figure 2

The optimization neutron-scattering penalty function, $\mathcal{P}$, in the $J_2-J_3$ plane behaves discontinuously and shows minima in the black domains. Only for the couplings demarcated by a dotted trapezoid does the theoretical scattering profile reconstitute both the location of the sharp peaks and the overall shape of the broad and diffuse experimental scattering [9, 10]. The circle corresponds to the values of $J_2$ and $J_3$ suggested in Ref. 11. The upper left and bottom right white regions correspond to different ${\mathbf{q}}_{\mathrm{ord}}=0$ structures.

Figure 3

Scattering profile for regions of the $J_2-J_3$ plane where the positions of the sharp peaks have been optimized; $\tau =1.6\times {10}^{-6}$. Upper left and bottom right panels represent correspondingly located black domains in Fig. 2. The other two panels represent small domains in the central and bottom left part of Fig. 2. The overall shape of $I(Q)$ varies significantly, but all the profiles display sharp peaks agreeing with the experiment (shown by vertical lines).

Figure 4

Positions of the first three strongest peaks of the experimental scattering profile (pointed to by arrows in the left panel) as well as the overall shape of scattering profile are used as criteria for the optimization procedure in the $J_2-J_3$ plane. The resulting mean-field theory profile at dimensionless temperature $\tau =1.6\times {10}^{-6}$ (right panel) compares well with the experimental one. The positions of three experimental peaks are denoted by vertical lines. The experimental scattering profile has been obtained by digitizing Fig. 2 of Ref. 9. The theoretical profile is shown for $J_2=-0.005\mathrm{K}$, $J_3=0.010\mathrm{K}$, denoted by a box in Fig. 2.