Quantum paramagnet and frustrated quantum criticality in a spin-one diamond lattice antiferromagnet

Motivated by the proposal of a topological quantum paramagnet in the diamond lattice antiferromagnet ${\mathrm{NiRh}}_2{\mathrm{O}}_4$, we propose a minimal model to describe the magnetic interaction and properties of the diamond material with spin-one local moments. Our model includes the first- and second-neighbor Heisenberg interactions as well as a local single-ion spin anisotropy that is allowed by the spin-one nature of the local moment and the tetragonal symmetry of the system. We point out that there exists a quantum phase transition from a trivial quantum paramagnet when single-ion spin anisotropy is dominant to the magnetic ordered states when the exchange is dominant. Due to the frustrated spin interaction, the magnetic excitation in the quantum paramagnetic state supports extensively degenerate band minima in the spectra. As the system approaches the transition, extensively degenerate bosonic modes become critical at the criticality, giving rise to unusual magnetic properties. Our phase diagram and experimental predictions for different phases provide a guideline for the identification of the ground state for ${\mathrm{NiRh}}_2{\mathrm{O}}_4$. Although our results are fundamentally different from the proposal for topological quantum paramagnets, they represent interesting possibilities for spin-one diamond lattice antiferromagnets.

Figure 1

A diamond lattice with $J_1$ and $J_2$ interactions. Due to the tetragonal symmetry of the lattice, the $a$ and $b$ directions are not equivalent to the $c$ direction. The ${\mathrm{Ni}}^{2+}$ ion is in a tetrahedral environment, so the $e_g$ orbitals are lower in energy than the $t_{2g}$ levels. Tetragonal distortion further splits the two $e_g$ orbitals and the three $t_{2g}$ orbitals, but the degeneracy of the $xz$ and $yz$ orbitals remains intact under tetragonal distortion. To avoid the orbital degree of freedom, we here place the $xz$ and $yz$ orbitals above the $xy$ orbitals [31].

Figure 2

The phase diagram of the $J_1-J_2-D_z$ model. Because the powder sample Curie-Weiss temperature ${\mathrm{\Theta }}_{\text{CW}}^{\text{powder}}=-8(J_1+3J_2)/3$, we set the energy unit of the spin anisotropy $D_z$ to $J_1+3J_2$ in the plot. The transition from a quantum paramagnet to the ordered regions is continuous in mean-field theory. On the left of the (red) dashed line, the band minimum of the magnetic exciton is unique and appears at the $\mathrm{\Gamma }$ point. On the right side, the band minima form a degenerate surface in reciprocal space. Please refer to the main text for details.

Figure 3

The magnetic excitation ${\omega }_{2,\mathit{k}}$ in the $k_x-k_y$ plane of the quantum paramagnet. We have chosen the following parameters: (a) $J_2=0.05J_1, D_z=3J_1$; (b) $J_2=0.18J_1, D_z=1.5J_1$; (c) $J_2=0.4J_1, D_z=1.5J_1$; (d) $J_2=0.8J_1, D_z=2J_1$. In the figure, we set $k_z=0$, and an extended zone with $k_x\in [-4\pi ,4\pi ],k_y\in [-4\pi ,4\pi ]$ is used. Degenerate minima are marked with contours. One can observe the evolution of the band minima.

Figure 4

The degenerate surface of the band minima at (a) $J_2=0.18J_1$ and (b) $J_2=J_1/3$. The $(k_{t_1},k_{t_2})$ are the two tangential momenta and $k_{\perp }$ is the component normal to the degenerate surface.