Tracing the nonequilibrium topological state of Chern insulators

Chern insulators exhibit fascinating properties, which originate from the topologically nontrivial state characterized by the Chern number. How these properties change if the system is quenched between topologically distinct phases is, however, not fully understood. In this paper, we investigate the quench dynamics of the prototypical massive Dirac model for topological insulators in two dimensions. We consider both dissipationless dynamics and the effect of electron-phonon interactions, and ask how the transient dynamics and nonequilibrium steady states affect simple observables. Specifically, we discuss a time-dependent generalization of the Hall effect and the dichroism of the photoexcitation probability between left and right circularly polarized light. We present optimized schemes based on these observables, which can reveal the evolution of the topological state of the quenched system.

Figure 1

Top: Dispersion of the longitudinal acoustic (LA), transverse acoustic (TA), longitudinal optical (LO), and transverse optical (TO) phonon modes. The energy is measured in units of the hopping constant $T_0$. Bottom: Corresponding density of states (DOS) along with the occupation-weighted DOS for the LA and TA modes (filled curves).

Figure 2

Circular asymmetry $\mathrm{\Delta }\left(\mathbf{k}\right)$ of the transition probability for (a) the BI ($M=M_{\mathrm{BI}}=-5\left|T_0\right|$) and (b) the TI ($M=M_{\mathrm{TI}}=-3\left|T_0\right|$).

Figure 3

Population of the upper band after applying a pulse with central frequency ${\omega }_0$ to the BI (top) and TI (bottom). The dashed line marks the optimal pulse frequency ${\omega }_{\mathrm{opt}}$ where the difference between the BI and TI response is most pronounced.

Figure 4

Asymmetry signal analogous to Fig. 3, but for different EL-PH coupling strengths $\gamma >0$.

Figure 5

Real part of the optical conductivity for the BI (top) and the TI (bottom). The filled curve illustrates the power spectrum of the HCP used for the time-dependent calculations (see text). The small but finite value of ${\sigma }_{xx}(\omega =0)$ is due a small broadening of the Fermi surface.

Figure 6

(a) Time-dependent current $J_{\alpha }\left(t\right)$ induced by a HCP with $T_{\mathrm{p}}=10$, $t_0=10$, and $E_0={10}^{-4}$, polarized in the $x$ direction. The current has been normalized by the pulse strength. (b) Corresponding pulse-current correlation functions [Eq. (24)] as functions of the delay $\mathrm{\Delta }t$.

Figure 7

Band structure of (a) the BI and (b) the TI. The coloring of the lines indicates the orbital weight of the upper band underlying the Hamiltonian (1). Band structure of (c) the BI and (d) the TI, where the dots indicate that the occupation of the respective band after the quench is larger than 0.5. The energy is measured in units of $|T_0|$

Figure 8

(a) Time-dependent current induced by a HCP (sketched in the plot) after the quench from the BI state to the TI (illustrated by the shaded background). The current in the $y$ direction has been multiplied by 5 for better visibility. The normalization of the current is consistent with Fig. 6. (b) Corresponding pulse-current correlation functions.

Figure 9

Optical conductivity of the postquench state ${\sigma }_{\alpha \beta }^{\mathrm{B}}\left(\omega \right)$ according to the definition (26) for (a) the quench $M_{\mathrm{TI}}\to M_{\mathrm{BI}}$ and (b) $M_{\mathrm{BI}}\to M_{\mathrm{TI}}$.

Figure 10

The circular asymmetry of the absorbed energy $\mathrm{\Delta }{E}_{\mathrm{abs}}$ as a function of the delay $\mathrm{\Delta }t$ (bottom) between the quench and the probe pulse (illustrated in the top). The color gradient shading of the curves emphasizes the transition from positive (red) to negative (blue) asymmetry.

Figure 11

Top row: time-dependent occupation with respect to the postquench Hamiltonian (defined by $M=M_{\mathrm{BI}}$) for different time delays $\mathrm{\Delta }t$ relative to the quench time (at $\mathrm{\Delta }t=0$). Bottom row: corresponding tr-ARPES spectra according to Eq. (28). The full purple lines indicate the post-quench BI band structure, while the dashed orange lines is the lower band of the TI. The EL-PH coupling is set to $\gamma =0.5$.

Figure 12

Top row: Time-dependent occupation with respect to the postquench Hamiltonian (defined by $M=M_{\mathrm{TI}}$) for different time delays $\mathrm{\Delta }t$ as in Fig. 11. Bottom row: tr-ARPES spectra according to Eq. (28). The full orange lines indicate the postquench TI band structure, while the dashed purple line represents the lower band of the BI. The EL-PH coupling is set to $\gamma =0.5$.

Figure 13

Real part of the nonequilibrium optical conductivity of the postquench steady-state reached in the presence of EL-PH interactions, for (a) the transition $M_{\mathrm{TI}}\to M_{\mathrm{BI}}$, and (b) the switching $M_{\mathrm{BI}}\to M_{\mathrm{TI}}$.

Figure 14

Asymmetry of the absorbed energy $\mathrm{\Delta }{E}_{\mathrm{abs}}$ as a function of the delay $\mathrm{\Delta }t$ between the quench and the probe pulse for (a)–(b) weak EL-PH coupling $\gamma =0.1$, and (c)–(d) moderate EL-PH coupling $\gamma =0.3$. The color gradient filling of the curves is analogous to Fig. 10. The gray shaded area indicates the time when the system is switched from BI to TI (left panels) or TI to BI (right panels), respectively.