Electron scattering from domain walls in ferromagnetic Luttinger liquids

We study the properties of interacting electrons in a one-dimensional conduction band coupled to bulk noncollinear ferromagnetic order. The specific form of noncollinearity we consider is that of an extended domain wall. The presence of ferromagnetic order breaks spin-charge separation and the domain wall introduces a spin-dependent scatterer active over the length of the wall $\lambda$. Both forward and backward scattering off the domain wall can be relevant perturbations of the Luttinger liquid and we discuss the possible low-temperature phases. Our main finding is that backward scattering, while determining the ultimate low-temperature physics, only becomes important at temperatures $T/J<\exp (-\lambda /{\lambda }_{+})$, with $J$ being the magnetic exchange and ${\lambda }_{+}$ the backward scattering length scale. In physical realizations, $\lambda \gg {\lambda }_{+}$ and the physics will be dominated by forward scattering, which can lead to a charge-conducting but spin-insulating phase. In a perturbative regime at higher temperatures we furthermore calculate the spin and charge densities around the domain wall and quantitatively discuss the interaction-induced changes.

Figure 1

A schematic view of the magnetization orientation in the wire. $\lambda$ is the length of the domain wall.

Figure 2

A schematic view of the spin-split bands. One of the $g_{1\perp }$ processes is shown, which involve spin-flip scattering between left- and right-moving electrons.

Figure 3

${\gamma }_{f,b}$ and ${\gamma }_{b\sigma }$ for forward and backward, spin flip, and potential, scattering from an extended domain wall obtained for a specific $U$-$V$ model (see text for details) as a function of $\xi =J/2{\varepsilon }_F$. The regions where the operators are relevant or irrelevant for $d=1$ are marked. The condition $K_s=1$ has been imposed by hand at the SU(2) symmetric point ($J=0$) (see Appendix).

Figure 4

The spin density $S_z\left(z\right)/S$, showing the zeroth-order (dashed line) term and the total value (blue solid line). The inset shows a comparison of the the zeroth-order (dashed line) and first-order (red solid line) terms. Zeroth and first order refer to an expansion in the scattering potential. All the spin figures are normalized to the average value per conduction electron, $S=\frac{1}{2}$. The domain-wall length is $\lambda =10{\lambda }_F$ with (a) $\frac{J}{2}=0.112$ eV and $T=25$ K and (b) $\frac{J}{2}=2.23$ eV and $T=50$ K.

Figure 5

Same as Fig. 4 but for the spin density $S_y\left(z\right)/S$.

Figure 6

The charge-density correction, $\Delta \rho \left(z\right)$, per electron per unit cell: ${\rho }_0=2e/\alpha$Cm${}^{-1}$. The domain-wall length is $\lambda =10{\lambda }_F$ with parameters for panels (a) and (b) as given in Fig. 4.

Figure 7

Same as Fig. 6b where we have zoomed in on a shorter length making the oscillations due to backscattering clearly visible.