Mixed-partial-wave scattering with spin-orbit coupling and validity of pseudopotentials

We present exact solutions of a two-body problem for spin-$1/2$ fermions with isotropic spin-orbit (SO) coupling and interacting with an arbitrary short-range potential. We find that in each partial-wave scattering channel, the parametrization of a two-body wave function at short interparticle distance depends on the scattering amplitudes of all channels. This reveals the mixed-partial-wave scattering induced by SO couplings. By comparing with results from a square-well potential, we investigate the validity of original pseudopotential models in the presence of SO coupling. We find the $s$-wave pseudopotential provides a good approximation for low-energy solutions near $s$-wave resonances, given the length scale of SO coupling much longer than the potential range. However, near $p$-wave resonance the $p$-wave pseudopotential gives low-energy solutions that are qualitatively different from exact ones, based on which we conclude that the $p$-wave model can not be applied to the fermion system if the SO coupling strength is larger or comparable to the Fermi momentum.

Figure 1

Single-particle spectrum, ${\epsilon }_{\mathbf{k}}^{(\pm )}={\left(\right|\mathbf{k}|\pm \lambda )}^2/\left(2m\right)$, with isotropic SO coupling. For given energy ${\epsilon }_0=k_0^2/\left(2m\right)$, two magnitudes of momentum are available as $|k_1|$ and $k_2$, with $k_1=\lambda -k_0,k_2=\lambda +k_0$. $k_0<\lambda$ in (a) and $k_0>\lambda$ in (b).

Figure 2

Phase shifts of scattering states in a square-well potential with isotropic SO coupling strength $\lambda r_0=0.2$. Exact solutions are shown (black circles) in comparison with results from pseudopotential models with effective-range corrections [see Eq. (A3)] (orange dashed lines). (a) ${\delta }_1$ near $s$-wave resonance with (from bottom to top) $q{r}_0/\pi =0.25(a_s<0),0.5(a_s=\infty ),0.7(a_s>0)$. (b) ${\delta }_2$ near $p$-wave resonance with (from bottom to top) $q{r}_0/\pi =0.9(a_p<0),0.99(a_p<0),1(a_p=\infty ),1.02(a_p>0)$.

Figure 3

(a) Inverse of effective scattering lengths [$a_{\mathrm{eff}}\left(k\right)$] as functions of $k$ for different SO couplings $\lambda r_0=0,0.1,0.2,0.3$ (from top to bottom). The potential depth is $q{r}_0/\pi =0.25$. (b) Inverse of zero-energy scattering lengths ($a_{\mathrm{eff}}$) as functions of $\lambda$ at different $q{r}_0/\pi =0.25,0.5,0.7$(from bottom to top). The lines are fits to Eq. (50) with $C=-2.10,-1.00,-0.58$ (from bottom to top).

Figure 4

$C$ in Eq. (50) as functions of $r_0/a_s$ near and across the first $s$-wave resonance ($q{r}_0<\pi$) in the square-well potential. The red dashed line is the linear fit in the weak interaction limit as $C=-1.073+0.275r_0/a_s$. Inset shows the ratio of $p$-wave to $s$-wave scattering length, where the gray (light) lines denote the $s$-wave resonance.

Figure 5

Critical potential depth $(q_c{r}_0/\pi )$ for the emergence of each new bound state as a function of isotropic SO coupling strength. When $\lambda r_0\to 0$, the solid lines approach $q_c{r}_0/\pi =0,1.430,2.459,...$ with $a_s=0$; the dashed lines approach $q_c{r}_0/\pi =1,2,...$ with $a_p=\infty$ (see text).

Figure 6

Bound state solutions (solid lines) in the square-well potential as functions of the potential depths, in comparison with results from $s$-wave (a) and $p$-wave (b) pseudopotential models. The SO coupling is $\lambda r_0=0.2$. In (a), we have used $s$-wave scattering length without (green circles) or with (black triangles) energy dependence. In (b), we use the energy-dependent $p$-wave scattering length (blue triangles).

Figure 7

(a1) and (a2) Bound state solutions in a square-well potential as functions of isotropic SO coupling strengths at potential depths $q{r}_0/\pi =0.25[\left(a1\right),a_s<0]$ and $0.7[\left(a2\right),a_s>0]$, in comparison with results using $s$-wave scattering length without (red dashed line) or with (green dot line) energy dependence. (b) Relative deviations between exact solutions and results from the $s$-wave pseudopotential model, at the same potential depths as in (a1) and (a2). For $s$-wave pseudopotential, we use $s$-wave scattering length without (${}^{\prime }{\times }^{\prime }$) or with (${}^{\prime }{+}^{\prime }$) energy dependence.

Figure 8

Bound state solutions as functions of isotropic SO coupling strengths for given potential depth $q{r}_0/\pi =0.4$, in comparison with results using $s$-wave scattering length without (pink circles) or with (black triangles) energy dependence. The orange dashed line denotes the lower bound as $\kappa =q$.