Spin-Orbit Coupling-Induced Magnetic Phase in the $d$-Density-Wave Phase of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ Superconductors

We study the effects of spin-orbit coupling in the $d$-density wave (DDW) phase. In the low-temperature orthorhombic phase of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$, we find that spin-orbit coupling induces ferromagnetic moments in the DDW phase, which are polarized along the $[110]$ direction with a considerable magnitude. This effect does not exist in the superconducting phase. On the other hand, if the $d$-density wave order does not exist at zero field, a magnetic field along the $[110]$ direction always induces such a staggered orbital current. We discuss experimental constraints on the DDW states in light of our theoretical predictions.

Figure 1

(a) The lattice symmetry in the LTO phase. ${\mathrm{O}}^{+}$ (${\mathrm{O}}^{-}$) denotes the oxygen atom moving into (out of) the CuO plane. The arrow indicates the $[110]$ direction. (b) The anisotropic Dirac-cone-like dispersion relations around nodes $(\pm \pi /2,\pm \pi /2)$ for spin parallel ($\uparrow$) and antiparallel ($\downarrow$) to the $[110]$ direction.

Figure 2

Spin polarization in the LTO phase at doping $\delta =0.1$ with ${\lambda }_1=\lambda \cos \theta$, ${\lambda }_2=\lambda \sin \theta$ ($45°\le \theta \le 90°$). The results within the range $0°\le \theta \le 45°$ are symmetric to the case of $90°-\theta$.

Figure 3

(a) Spin polarizations $M$ vs doping levels $\delta$ at $\mathrm{T}=0\mathrm{K}$. (b) The DDW order parameter $\mathrm{Im}\chi$ vs doping levels $\delta$ at $\mathrm{T}=0\mathrm{K}$.

Figure 4

Spin polarization vs temperatures at $\delta =0.06$, 0.12. Temperatures are in the unit of $t_{\mathrm{eff}}$.

Figure 5

Staggered orbital moment in the unit of ${\mu }_B$ vs the Zeeman energy $E_z=g/2B{\mu }_B$, where $t_{\mathrm{eff}}=100\mathrm{meV}$, $\lambda /t_{\mathrm{eff}}=0.02$, $\Delta \lambda =0$, and the Lande factor $g=2$. $\overrightarrow{B}$ is along the $[110]$ direction.