Detection of topological states in two-dimensional Dirac systems by the dynamic spin susceptibility

We discuss dynamic spin susceptibility (DSS) in two-dimensional (2D) Dirac electrons with spin-orbit interactions to characterize topological insulators. The imaginary part of the DSS appears as an absorption rate in response to a transverse ac magnetic field, just as in an electron spin resonance experiment for localized spin systems. We found that when the system is in a static magnetic field, the topological state can be identified by an anomalous resonant peak of the imaginary part of the DSS as a function of the frequency of the transverse magnetic field $\omega$. In the absence of a static magnetic field, the imaginary part of the DSS becomes a continuous function of $\omega$ with a threshold frequency ${\omega }_{\mathrm{c}}$. In this case, the topological and the trivial phases can also be distinguished by the values of ${\omega }_{\mathrm{c}}$ and by the line shapes. Thus the DSS is an experimentally observable physical quantity to characterize a topological insulator directly from bulk properties, without observing a topological transition.

Figure 1

Imaginary part of the DSS $-\omega \mathrm{Im}{\chi }_{xx}\left(\omega \right)$ (in units of ${\hslash }^2/l^2$) with $\mu =0$ and $\mathrm{\Gamma }/\kappa =0.02$ for (a) BI ($\mathrm{\Delta }/\kappa =1.2$) and (b) TI ($\mathrm{\Delta }/\kappa =0.8$). The parameters are $B=0.1\mathrm{T}$, $v=5.4\times {10}^5\mathrm{m}/\text{s}$, and $\kappa =4.0\mathrm{meV}$ assuming a silicene. There are regular peaks for both cases which stem from transitions between $-n$ and $n$ Landau levels with $\left|n\right|\ge 1$. In addition to those, an anomalous peak due to $n=0$ Landau levels appears at $\omega =2\kappa /\hslash$ (independent of the strength of the magnetic field) only for TI. When the Zeeman effect exists, e.g., $g{\mu }_{\mathrm{B}}B/\kappa =0.1$, the regular peaks split into two, while the anomalous peak shifts only to one direction.

Figure 2

Landau level structures of Dirac fermions with alternating potential $\mathrm{\Delta }$ and spin-orbit interactions $\kappa$ for (a) the BI ($\left|\kappa \right|<\left|\mathrm{\Delta }\right|$) and (b) the TI ($\left|\kappa \right|>\left|\mathrm{\Delta }\right|$) states. The two phases are characterized by the configurations of the $n=0$ Landau levels around the Fermi level ($\mu =0$). For the BI, the $n=0$ Landau levels have an opposite sign in each spin, so that the quantized Hall conductivity canceled out when the magnetic field is tuned off, and the absorption rate does not show the anomalous peak, because the transitions between $n=0$ Landau levels are not allowed.

Figure 3

Imaginary part of the DSS, $-\omega \mathrm{Im}{\chi }_{xx}\left(\omega \right)$ (in units of ${\kappa }^2/v^2$), with $B=0$ and $\mathrm{\Gamma }/\kappa =0.02$ for several values of $\mathrm{\Delta }/\kappa =0.4–1.6$. For small $\mathrm{\Gamma }$, $-\omega \mathrm{Im}{\chi }_{xx}\left(\omega \right)$ is a discontinuous (continuous) function at the threshold frequency ${\omega }_{\mathrm{c}}=2\kappa /\hslash \left({\omega }_{\mathrm{c}}=2\mathrm{\Delta }/\hslash \right)$ for TI (BI).