Incipient magnetic instability in RuO$_2$ with random phase approximation

Abstract

We study the instability in RuO$_2$ using the Hartree-Fock approximation followed by the random phase approximation. We employ a three-orbital Hubbard model without spin-orbit coupling. An analysis of the eigenvalues and eigenvectors of the static susceptibility in the non-magnetic phase for various local interaction parameters $U$, $J_H$, and hole doping $n$ shows that the spin susceptibility is the dominant response channel. In the stoichiometric system without spin-orbit coupling, commensurate altermagnetic order is identified as the leading instability at sufficiently low temperatures, whereas at higher temperatures or finite hole doping, incommensurate wave vectors emerge. To elucidate the origin of the magnetic instability, we analyze the band spitting by the staggered Weiss field and discuss the qualitative difference between altermagnets and antiferromagnets.

Figures (12)

• Figure 1: (a) Crystal structure of RuO$_{2}$. Red spheres represent oxygen ions, and blue spheres represent ruthenium ions. The visualization was created using VESTA3 vesta. (b) First Brillouin zone with highlighted planes where $\eta(\mathbf{k})$ is shown and paths along which the band structures are analyzed. (c) Schematic illustration of the direct hopping path between $d_{x^2 - y^2}$ orbitals on the same sublattice. Orange arrows indicate the local coordination, while $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ denote the global axes. Red and black dots represent oxygen and ruthenium ions, respectively.
• Figure 2: (a) Leading eigenvalue $\Lambda(\mathbf{q})$ of the static susceptibility $\chi(\mathbf{q})$ ($U = 1.24\,\mathrm{eV}$, $J_{\mathrm{H}} = 0.3\,\mathrm{eV}$, $n = 0$ and $T = 232\,\mathrm{K}$) and (b) unfolded spin susceptibility $\tilde{\chi}(\mathbf{q})$ shown for various cuts in $q_z$ within the first Brillouin zone.
• Figure 3: The position of the instability along the $q_z$ axis in $\mathbf{Q} = (0, 0, Q_z)$ as a function of the interaction parameters $U$, $J_{\rm H}$, and the absolute value of the hole doping $|n|$, expressed in electrons per atomic site.
• Figure 4: The temperature dependence of the largest eigenvalue $\Lambda(\mathbf{q})$ of the static susceptibility $\chi(\mathbf{q})$ for $\mathbf{q} = (0, 0, q_z)$ in the NM phase. Different interaction parameters are compared: (a) $U = 1.35\,\mathrm{eV}$, $J_{\rm H} = 0.28\,\mathrm{eV}$; (b) $U = 1.27\,\mathrm{eV}$, $J_{\rm H} = 0.3\,\mathrm{eV}$; (c) $U = 1.3\,\mathrm{eV}$, $J_{\rm H} = 0.3\,\mathrm{eV}$; (d) $U = 1.35\,\mathrm{eV}$, $J_{\rm H} = 0.3\,\mathrm{eV}$.
• Figure 5: Comparison of (a) the static spin bubble $\tilde{\chi}_0(\mathbf{q})$ and (b) the static spin susceptibility $\tilde{\chi}(\mathbf{q})$ after the unfolding for $\mathbf{q} = (0, 0, q_z)$ in the NM phase for different temperatures. The corresponding temperatures interaction parameters are $U = 1.24\,\mathrm{eV}$, $J_{\rm H} = 0.3\,\mathrm{eV}$ and no doping.