Interplay of spin-orbit coupling and Zeeman splitting in the absorption lineshape of fermions in two dimensions

We suggest that electron spin resonance (ESR) experiments can be used as a probe of spinon excitations of hypothetical spin-liquid state of frustrated antiferromagnet in the presence of asymmetric Dzyaloshinskii-Moriya interaction. We describe assumptions under which the ESR response is reduced to the response of two-dimensional electron gas with Rashba spin-orbit coupling. Unlike previous treatments, the spin-orbit coupling ${\Delta }_{\text{SO}}$ is not assumed small compared to the Zeeman splitting ${\Delta }_Z$. We demonstrate that ESR response diverges at the edges of the absorption spectrum for ac magnetic field perpendicular to the static field. At the compensation point ${\Delta }_{\text{SO}}\approx {\Delta }_Z$, the broad absorption spectrum exhibits features that evolve with temperature $T$, even when $T$ is comparable to the Fermi energy.

Figure 1

Energy dispersion of two subbands in the presence of Zeeman splitting and spin-orbit coupling are plotted from Eq. (6). The curves cross at $k_x={\Delta }_Z/2\alpha$. Yellow regions designate the momenta of the states taking part in absorption. For Fermi-level position $E_F={\Delta }_Z^2/4{\omega }_1$, where ${\omega }_1=2m{\alpha }^2$, there is no threshold for absorption.

Figure 2

(a) Two Fermi surfaces corresponding to $E_F^{\left(0\right)}={\Delta }_Z^2/4{\omega }_1$ and ${\omega }_1/{\Delta }_Z=0.1$ are plotted from Eq. (30), where the projections of momenta $k_x$ and $k_y$ are in the units ${\Delta }_Z/2\alpha$. (b) Evolution of the Fermi surfaces near the point $k_y=0$ for different $E_F$ [enclosed by the gray box in (a)]: (thin line) $E_F=0.95E_F^{\left(0\right)}$, (thick line) $E_F=E_F^{\left(0\right)}$, (dashed line) $E_F=1.1E_F^{\left(0\right)}$.

Figure 3

Smearing of the right edge of the chiral resonance at small ${\Delta }_Z\ll {\omega }_1$ is plotted from Eq. (23) as a function of dimensionless deviation $\eta \left(\omega \right)$ [Eq. (22)] from the boundary. ${\sigma }_{\parallel }$ and ${\sigma }_{\perp }$ are plotted in the units $e^2/16$.

Figure 4

Optical conductivity at zero temperature [Eqs. (26) and (27)] vs dimensionless frequency $\omega /{\Delta }_Z$ are plotted in the units $(e^2/\pi )({\omega }_1/{\Delta }_Z)$ for three values of the ratio ${\Delta }_{\text{SO}}/{\Delta }_Z$. (a) ${\Delta }_{\text{SO}}/{\Delta }_Z=0.5$, Zeeman splitting dominates. (b) Zeeman and SO splitting nearly “compensate” each other, ${\Delta }_{\text{SO}}/{\Delta }_Z=0.99$. (c) ${\Delta }_{\text{SO}}/{\Delta }_Z=10$, SO splitting dominates.

Figure 5

The lineshape of the chiral resonance is plotted from Eq. (33) in the units of $e^2/16$ for three dimensionless temperatures ${\Delta }_{\text{SO}}/T$ and spin-orbit coupling strength ${\Delta }_{\text{SO}}/E_F=0.01$. In the low-temperature limit ${\Delta }_{\text{SO}}/T=50$ (thick line), the lineshape is boxlike. At temperature ${\Delta }_{\text{SO}}/T=15$ (thin line), the edges of the box get smeared. Complete smearing corresponding to ${\Delta }_{\text{SO}}/T=2$ is illustrated with dashed line.

Figure 6

Evolution of optical conductivity with $T$ in the regime ${\Delta }_Z={\Delta }_{\text{SO}}$. Equations (45) and (46) are plotted in the units $(e^2/16\pi )({\Delta }_Z/E_F)$ for three values of $T/E_F$: (a) low-temperature limit $T/E_F=0.05$, (b) intermediate temperature $T/E_F=0.2$, (c) “high” temperature $T/E_F=0.5$. Inset in (a) shows the dimensionless functions $G_{\perp }\left(s\right)$ and $G_{\parallel }\left(s\right)$ calculated from Eqs. (49) and (50), respectively.