Electronic origin of ferroic quadrupole moment under antiferroic quadrupole order and finite magnetic moment in $J_{\mathrm{eff}}=\frac{3}{2}$ systems

We study the electronic origin of parasitic ferroic quadrupole moments in antiferroic quadrupole orders by extending a model studied in G. Chen et al., Phys. Rev. B 82, 174440 (2010) with the effective angular momentum $J_{\mathrm{eff}}=3/2$ quartet ground states. Taking into account the first crystalline-electric-field (CEF) excited doublet, cubic anisotropy in the quadrupole moments emerges, which leads to the induced ferroic quadrupole moments in the antiferroic quadrupolar phases. The hybridization with the CEF excited quartet states also causes finite magnetic moments compatible to the observed size of the effective moment in typical $J_{\mathrm{eff}}=3/2$ systems, as opposed to the naive expectation of vanishing moments in the $J_{\mathrm{eff}}=3/2$ systems. These results suggest the importance of the corrections arising from the high-energy CEF excited states in the $J_{\mathrm{eff}}=3/2$ systems.

Figure 1

The matrix elements of $u$ and $q_u$ as a function of $\varepsilon =\lambda /B_4$. For the off-diagonal elements, $u$ and $q_u$ are identical: ${\left(u\right)}_{13}=-{\left(u\right)}_{24}={\left(q_u\right)}_{13}=-{\left(q_u\right)}_{24}$, while for the diagonal ones, they are different for finite $\varepsilon$: ${\left(u\right)}_{33}={\left(u\right)}_{44}=-{\left(u\right)}_{55}=-{\left(u\right)}_{66}$. ${\left(q_u\right)}_{33}={\left(q_u\right)}_{44}=-{\left(q_u\right)}_{55}=-{\left(q_u\right)}_{66}$. The approximated values for the matrix elements of $u$: $1-\delta$ and $(2-\delta )/\sqrt{2}$, are also plotted.

Figure 2

Temperature dependence of the quadrupole moments $Q_u^{\mathrm{F}}$ and $Q_v^{\mathrm{AF}}$ for $J^{\prime }/J=0.3,V/J=0.3$, and $\lambda /J={10}^4,500,100,50$, and 20. $Q_v^{\mathrm{F}}=Q_u^{\mathrm{AF}}=0$ and are not shown.

Figure 3

$T\text{−}\lambda$ phase diagram for $V/J=0.3$, and (a) $J^{\prime }/J=0.1$, (b) $0.3$, and (c) 0.5. The inset in (a) shows the AFM configurations ${\mathit{S}}^{\mathrm{AF}}$, ${\mathit{T}}_{\alpha }^{\mathrm{AF}}$, and ${\mathit{T}}_{\beta }^{\mathrm{AF}}$ in the AFM* phase.

Figure 4

$T\text{−}V$ phase diagram for $J^{\prime }/J=0.2$. The lines represent the phase boundaries. The color map shows $\sqrt{-Q_u^{\mathrm{F}}}$, where $Q_u^{\mathrm{F}}\le 0$ for any $V$ and $T$. The square-root of $-Q_u^{\mathrm{F}}$ is used just for visualization of the small value of $Q_u^{\mathrm{F}}$ in the AFQ phase.

Figure 5

Magnetization curve for $J^{\prime }/J=0.3$ and $V/J=0.27$. (a) $\mathit{h}\parallel \left[001\right]$, (b) $\mathit{h}\parallel \left[110\right]$, and (c) $\mathit{h}\parallel \left[111\right]$. The magnetization $M_h$ is defined as $M_h\equiv \langle \mathit{M}\rangle ·\mathit{h}/h$, with $h\equiv \left|\mathit{h}\right|$ and plotted by lines, while the absolute value $M\equiv |\langle \mathit{M}\rangle |$ is indicated by symbols. The domain with the lowest free energy is considered; (a) ${\mathit{k}}_1$ or ${\mathit{k}}_2$ domains, (b) ${\mathit{k}}_3$ domain, and (c) ${\mathit{k}}_1$, ${\mathit{k}}_2$, or ${\mathit{k}}_3$ domains. The temperatures shown correspond to the FM110 phase $(T/J=0.1<T_m/J\simeq 0.34$), the $\mathrm{AFQ}+f\mathrm{q}$ phase $(T/J=0.5$), and the normal phase $(T/J=1.0$).

Figure 6

$\gamma$ dependence of $M,M_h$ for $T/J=0.1$ K, $h/J=0.05$ along $\mathit{h}\parallel \left[001\right]$, $J^{\prime }/J=0.3$ and $V/J=0.27$. The effective moment ${\mu }_{\mathrm{eff}}$'s given by Eq. (31) for several values of $\varepsilon$ are also drawn.