Prediction for the singlet-triplet excitation energy for the spinel ${\mathrm{MgTi}}_2{\mathrm{O}}_4$ using first-principles diffusion Monte Carlo

The spinel ${\mathrm{MgTi}}_2{\mathrm{O}}_4$ undergoes a transition into a dimerized state at low temperatures that is expected to be a spin singlet. However, no signature of a singlet-triplet transition has been discovered, in part due to the difficulty of predicting the energy of the transition from theory. We find that the dimers of ${\mathrm{MgTi}}_2{\mathrm{O}}_4$ can be described by a Heisenberg model with very small interactions between different dimers. Using high-accuracy first-principles quantum Monte Carlo combined with a model-fitting approach, we predict the singlet-triplet gap of these dimers to be 350(50) meV, a higher energy than previous experimental observations have considered. The prediction is published in advance of experimental confirmation.

Figure 1

Spin density contour plot for the spin states considered in this work. Up- and down-spin densities are represented as blue and red. Orange atoms with spin are Ti atoms, of the spinless atoms, the smaller brown atoms are O, and the white atoms are Mg. Bonds are drawn between dimerized Ti atoms, and bonds that exit the unit cell have no Ti atom on the end outside the cell, which is drawn with black lines. The diffusion Monte Carlo energies of the dimer and triplet state determine the model, and the “interdimer” states are used to test the accuracy of the model.

Figure 2

Energies of each state relative to the dimer state of Fig. 1 according to DMC and PBE0 as well as models fit to DMC and PBE0 data. The triplet energy difference is sufficient to fit the parameter in (1), with $J=350\left(50\right)$ meV from DMC, for example. This makes the singlet-triplet gap 350(50) meV for this system. The interdimer states are predictions from the independent dimer model (1), and are used to test the predictability of the model. Because PBE0 reproduces the energy differences for the unpolarized, triplet, and interdimer 1 state, PBE0 energies should be sufficient for interdimer 2–5, which are only used to probe the relevance of interdimer interactions. Due to computational constraints we did not also provide DMC energies for interdimer 2–5. The model predictions for the interdimer states agree quite well with (1), suggesting that the independent-dimer model is an accurate effective model for the spins in this system.

Figure 3

Spin density (top row) and charge density ($\rho$) relative to the unpolarized charge density ${\rho }_{\mathrm{unpolarized}}$ (bottom row) through Ti pairs and their intermediate O. Dimer signifies the Ti atoms in the center form a nearest-neighbor dimer pair in the structure. Interdimer signifies that the Ti atoms are nearest neighbors in the $x\text{−}y$ plane. The dimer bond shows spin density typical for atoms interacting through superexchange, as well as an increase in hybridization between Ti and O within the dimer as compared to the unpolarized state. The interdimer bond shows no such signals of superexchange, in agreement with the model results.

Figure 4

Energy vs Brillouin zone sampling for the PBE0 calculations. The energies are shown relative to the largest number of $k$ points ($6\times 6\times 6$), so that the left point is zero by definition. The error in the total energies is converged to the micro-eV scale, which is well below the scale of other errors in our work.

Figure 5

Time-step extrapolation for the DMC calculations was performed with two points at 0.01 and 0.02 time step. These points were extrapolated to zero time step. The time-step error in the energy difference is well within our statistical error.