Slow quenches in two-dimensional time-reversal symmetric ${\mathbb{Z}}_2$ topological insulators

We study the topological properties and transport in the Bernevig-Hughes-Zhang model undergoing a slow quench between different topological regimes. Due to the closing of the band gap during the quench, the system ends up in an excited state. We prove that for quenches that preserve the time-reversal symmetry, the ${\mathbb{Z}}_2$ invariant remains equal to the one evaluated in the initial state. On the other hand, the bulk spin Hall conductivity does change, and its time average approaches that of the ground state of the final Hamiltonian. The deviations from the ground-state spin Hall conductivity as a function of the quench time follow the Kibble-Zurek scaling. We also consider the breaking of the time-reversal symmetry, which restores the correspondence between the bulk invariant and the transport properties after the quench.

Figure 1

(a) Phase diagram of the BHZ model in $P=(u,c)$ parameter space. The gray colored area marks topological insulator parameter regimes where $N_{\mathrm{bulk}}=1$, the white area marks the trivial insulator regime with $N_{\mathrm{bulk}}=0$, and the blue area marks the Dirac semimetal regime. Points $P_0=(-3,0),{\tilde{P}}_0=(-3,0.3),P_1=(-1,0),{\tilde{P}}_1=(-1,0.3)$, and ${\tilde{P}}_2=(1,0.3)$ mark initial and final points between which parameters of the Hamiltonian are quenched (dashed). Several band gap closing points are marked as $P_{01},{\tilde{P}}_{01}$, and ${\tilde{P}}_{12}$. (b) Wannier center flows for the ground state of the Hamiltonian at $P_0$ and $P_1$. The dashed black line presents a chosen $\tilde{\theta }$ used for the determination of the ${\mathbb{Z}}_2$ invariant. (c) Spin Hall conductivities at $P_0, {\tilde{P}}_0, P_1$, and ${\tilde{P}}_1$, where ${\sigma }_0^{\mathrm{spin}}=e/2\pi$.

Figure 2

(a) Wannier center flows of the ground state at $P_1$ (black, dashed) and of nonequilibrium states (solid lines) at various times during the quench from $P_0$ to $P_1$ with ${\tau }_u=15$. (b) Population of the first energy level after the $P_0\to P_1$ quench. (c) Band-gap closing in $k$ space. For $P_0\to P_1$ and ${\tilde{P}}_0\to {\tilde{P}}_1$ quenches the band gap closes at $P_{01}$ and ${\tilde{P}}_{01}$, respectively. However, for the ${\tilde{P}}_1\to {\tilde{P}}_2$ quench, it closes gradually starting at the blue points and moving to the red ones as indicated with arrows.

Figure 3

(a) Topological invariant (dashed) and the spin Hall conductivity (solid lines) of the nonequilibrium states, resulting from the quench from $P_0$ to $P_1$. (b) Time-averaged spin Hall conductivity after the $P_0\to P_1$ quench (solid) converges for ${\tau }_u\to \infty$ to the ground-state value of $\widehat{H}\left(P_1\right)$ (thin, solid) and for systems after the ${\tilde{P}}_0\to {\tilde{P}}_1$ (dashed) to the ground-state value of $\widehat{H}\left({\tilde{P}}_1\right)$ (thin, dashed). Time-averaged spin Hall conductivity after the ${\tilde{P}}_1\to {\tilde{P}}_2$ quench (dotted) is multiplied by a factor of $-1$ and converges to the ground-state value of $\widehat{H}\left(\widetilde{P_2}\right)$. (c) Time at which the amplitude of oscillations in the spin Hall response after the quench $P_0\to P_1$ increases for 10% above the minimal amplitude.

Figure 4

Properties of the system quenched from $P_0$ to $P_1$ with a TRS-breaking term of amplitude $b_0=0.4$. (a) Population of the first energy-level $n_1(0,k_y)$ after the quench. (b) Wannier center flows of the nonequilibrium state at different times during the quench with ${\tau }_u=15$. (c) Spin Hall conductivity after the quench. The graphs with ${\tau }_u=30$ and ${\tau }_u=60$ overlap as the quench with such ${\tau }_u$ and $b_0$ is already adiabatic.

Figure 5

Band dispersions at (a) $P_{01}$, (b) ${\tilde{P}}_{01}$, and (c) ${\tilde{P}}_{12}$.

Figure 6

Occupancy of the upper valence band of the BHZ model quenched among (a) $P_0\to P_1$, (b) ${\tilde{P}}_0\to {\tilde{P}}_1$, and (c) ${\tilde{P}}_1\to \widetilde{P_2}$ for ${\tau }_u=60$.

Figure 7

Spin Berry curvature of the upper valence band of the BHZ model at parameters (a) $P_1$, (b) ${\tilde{P}}_1$, and (c) ${\tilde{P}}_2$. The value of the spin Berry curvature at $P_1$ is multiplied by 10. Black lines denote momenta at which the band gap closes during quenches (a) $P_0\to P_1$, (b) ${\tilde{P}}_0\to {\tilde{P}}_1$, and (c) ${\tilde{P}}_1\to {\tilde{P}}_2$. White regions correspond to the values out of the color code scale.

Figure 8

(a) Diagonal (diag), off-diagonal (offdiag), all terms (all) in ${\sigma }_{xy}^{\mathrm{spin}}\left(t\right)$ according to Eq. (C1), and ${\sigma }_{xy}^{\mathrm{spin}}\left(t\right)$ resulting from the full evaluation of Eq. (7) (dotted) for the system after the $P_0\to P_1$ quench for ${\tau }_u=15$. (b) Imaginary part of the time-dependent element $f_{n{n}^{\prime }m}(t,\mathrm{\Gamma })$ at the $\mathrm{\Gamma }$ point for the system at parameters $P_1$ is shown for different index combinations. In both figures the electric field is turned on as $E_x\left(t\right)=E_0\{1-exp[-(t-t_E)/{\tau }_E]\}$.