Mesoscopic scattering dynamics under generic uniform SU(2) gauge fields: Spin-momentum relaxation and coherent backscattering

Abstract

We investigate the time- and momentum-resolved dynamics of matter waves undergoing elastic scattering from a disordered potential in the presence of spatially uniform SU(2) gauge fields. We derive the disorder-averaged density matrix as a function of time and momentum within the weak-localization regime. By accurately approximating the frequency dependence of the ladder and maximally crossed diagram series beyond the diffusive approximation, we describe short-time spin-momentum dynamics on timescales comparable to the scattering mean free time, for arbitrary strengths of the SU(2) gauge fields and disorder. We also present a cubic equation that determines the spin isotropization time, which gives accurate asymptotic forms in the limits where the spin-orbit length is much longer (Dyakonov-Perel spin relaxation regime) or much shorter than the scattering mean free path, as well as in the SU(2)-symmetric (persistent spin helix) limit. In comparison with numerical calculations, we reproduce both the relaxation of the momentum distribution and the transient backscattering peak with a momentum offset coexisting with the robust coherent backscattering dip, supporting the reliability of our calculations.

Figures (4)

• Figure 1: Cut of the energy branches (Fermi surface) at a fixed energy $E_0=25E_{\kappa}$ for $\eta=\pi/6$ and $\eta=\pi/24$. The color scale represents the overlap of the spin states with the $\ket{S_+(\theta\!=\!0)}$ state, see Eq. \ref{['eq:scattering_prob']}.
• Figure 2: Panel (a): Spin isotropization time $\tau_{\rm iso}$ (in units of the scattering time $\tau$) in the ($\kappa\ell$, $\eta$) plane with its associated color scale values, as obtained from the prescription Eq. \ref{['eq:def_tau_iso']} applied to the solutions of Eq. \ref{['eq:cubic_eq_transport_mean_free_time']}. The dashed lines mark the contours of $\tau_{\rm iso}/\tau = C$ for $C = 3, 10, 10^2, 10^3, 10^4,$ and $10^5$. This result holds regardless of $E_0$ as long as the conditions $k_\mathrm{F}\ell\gg1$ and $k_\mathrm{F}/\kappa\gg1$ are satisfied. Panel (b): $\kappa\ell$ dependence of $\tau_{\rm iso}$ for $\eta= \pi/6$ (blue curve) and $\eta=\pi/24$ (red curve). The dashed lines materialize the predicted asymptotic behavior $(\kappa\ell)^{-2}$ in Eq. \ref{['eq:tau_iso_small_kappal']}. Panel (c): $\eta$ dependence of $\tau_{\rm iso}$ for $\kappa\ell=0.5$ (blue curve) and $\kappa\ell=10$ (red curve). The dashed lines represent the analytical asymptotic estimates given by Eq. \ref{['eq:tau_iso_asymptotic']}. $r_{\eta}$ is defined in Eq. \ref{['eq:def_r_eta']}. For the behavior $\eta^{-2}$, see also Eq. \ref{['eq:tau_iso_small_kappal']} at small $\eta$.
• Figure 3: Disorder-averaged momentum distribution $n(\bm{k},t)$ in Eq. \ref{['eq:momentum_distrib']} at $t=8\tau$ (in units of $n_0$, see Eq. \ref{['eq:def_n0']}) as a function of $\bm{k}$. Panel (a): $\eta=\pi/12$. Panel (b): $\eta=\pi/24$. Panel (c): $\eta=\pi/48$. The left panels are obtained from numerical simulations, while the right panels are derived analytically using Eqs. \ref{['eq:Diffuson_matrix_branch_rep']}--\ref{['eq:momentum_distrib_analytic']}, Eqs. \ref{['eq:Gamma_1']}--\ref{['eq:Gamma_34']}, and Eqs. \ref{['eq:assumption_diagonal_Pi']} and \ref{['eq:approx_Gamma_C_w_dephasing_time']} for the transient peak. As one can see, the analytics reproduces well the numerics.
• Figure 4: Momentum distributions obtained at different (small) times along each branch of the Fermi surface around the backscattering direction. The parameter $\eta$ is fixed at $\pi/24$. The blue and red solid lines represent the analytical results for the $(+)$-branch and $(-)$-branch respectively. They reproduce well the numerical simulation data without any adjustable parameters.