Role of orbital off-diagonal spin and charge condensates in a three orbital model for $\mathrm{C}{\mathrm{a}}_2\mathrm{Ru}{\mathrm{O}}_4$ -- Coulomb renormalized spin-orbit coupling, orbital moment, and tunable magnetic order

Abstract

Strongly anisotropic spin-orbit coupling (SOC) renormalization and strongly enhanced orbital magnetic moments are obtained in the fully self consistent approach including the orbital off-diagonal spin and charge condensates. For moderate tetragonal distortion as in $\mathrm{C}{\mathrm{a}}_2\mathrm{Ru}{\mathrm{O}}_4$, dominantly planar antiferromagnetic (AFM) order with small canting of moments in and about the crystal $c$ axis are obtained. For reduced tetragonal distortion, we find a tunable regime wherein the magnetic order can be tuned (AFM or FM) by the bare SOC strength and octahedral tilting magnitude. In this regime, with decreasing tetragonal distortion, AFM order is maintained by progressively decreasing octahedral tilting, as observed in $\mathrm{C}{\mathrm{a}}_{2-\mathrm{x}}\mathrm{S}{\mathrm{r}}_{\mathrm{x}}\mathrm{Ru}{\mathrm{O}}_4$. For purely planar order, the only self consistent solution is FM order along crystal $b$ axis, which is relevant for the bilayer ruthenate compound $\mathrm{C}{\mathrm{a}}_3\mathrm{R}{\mathrm{u}}_2{\mathrm{O}}_7$.

24 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:2006.02114

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