Intrinsic Damping of Collective Spin Modes in a Two-Dimensional Fermi Liquid with Spin-Orbit Coupling

A Fermi liquid with spin-orbit coupling (SOC) is expected to support a new set of collective modes: oscillations of magnetization in the absence of the magnetic field. We show that these modes are damped by the electron-electron interaction even in the limit of an infinitely long wavelength ($q=0$). The linewidth of the collective mode is on the order of ${\overline{\mathrm{\Delta }}}^2/E_F$, where $\overline{\mathrm{\Delta }}$ is a characteristic spin-orbit energy splitting and $E_F$ is the Fermi energy. Such damping is in stark contrast to known damping mechanisms of both charge and spin collective modes in the absence of SOC, all of which disappear at $q=0$, and arises because none of the components of total spin is conserved in the presence of SOC.

Figure 1

Left: The Silin-Leggett mode in a partially spin-polarized FL. Right: The chiral-spin modes in a FL with Rashba spin-orbit coupling. The shaded regions denote the particle-hole continua, ${\mathrm{\Delta }}_B$ is the Larmor frequency, ${\tilde{\mathrm{\Delta }}}_B$ is the quasiparticle Zeeman energy, and ${\mathrm{\Delta }}_{\min (\max )}$ is the lower (upper) boundary of the continuum at $q=0$.

Figure 2

Top: The ladder (RPA) series for the spin susceptibility. The boxed wavy line is the static effective interaction $U_{\mathit{x}}$. Bottom: Diagrams contributing to damping of the collective modes. The wavy line in diagrams (a)–(e) denotes a dynamic interaction, $V_{\text{eff}}(P)$.