Slow quenches in Chern insulator ribbons

We investigate slow quenches in Chern insulators in ribbon geometry. We consider the Qi-Wu-Zhang model and slowly ramp the parameters (large time of the quench $\tau$) from a nontopological (Chern number = 0) to a topological regime (Chern number $\ne$ 0). In contrast to the Haldane model considered in [L. Privitera and G. E. Santoro, Phys. Rev. B 93, 241406(R) (2016)], the in-gap state degeneracy point is pinned to an inversion symmetric momentum, which changes the behavior drastically. The density of excitations in the in-gap states scales with the quench time as ${\tau }^{-1/2}$ as the ramp becomes slow, and the Kibble-Zurek mechanism applies. Despite the slower scaling of the density of in-gap excitations with $\tau$, the Hall conductance after the quench deviates from that of the ground state of the final Hamiltonian by an amount that drops as ${\tau }^{-1}$.

Figure 1

Ribbon geometry with periodic and open boundary conditions along $y$ and $x$ directions, respectively. Homogeneous electric field $\mathit{E}$ is applied in $x$ direction to probe the Hall conductance.

Figure 2

(a) Occupancy of energy bands of the ribbon of width $N_x=20$ at $u=-1.2$ after the quench with $\tau =1$. The surface of a point is proportional to the probability of finding a particle in the corresponding eigenstate. (b) Energy levels of the whole Hamiltonian $\widehat{H}\left(k_y\right)$ (gray) and of the two-level Hamiltonian ${\widehat{H}}_2\left(k_y\right)$ (red) at $k_y=0$ as a function of $u$.

Figure 3

Number of excitations (a) in the bulk and (b) in the in-gap states as functions of $\tau$ for different ribbon widths, ranging from $N_x=10$ (light grey) to $N_x=50$ (black). Red lines denote (a) the exact result for the system with periodic boundary conditions in both directions which scales as ${\tau }^{-1}$ and (b) a fitted ${\tau }^{-1/2}$ line. Blue dots show results for a Hamiltonian with a broken inversion symmetry of magnitude $b=0.15$ and $N_x=20$.

Figure 4

(a) Momentum distribution of bulk excitations in a ribbon with $N_x=70$ after quenches with $\tau =10$ (dashed), $\tau =20$ (dotted), and $\tau =50$ (full). Analytical result of Eq. (10) is shown by the red line. (b) Momentum distribution of in-gap excitations in a ribbon with $N_x=20$ after quenches with $\tau =10$ (dashed), $\tau =100$ (dotted), and $\tau =1000$ (full). A rescaled ($\tau \to 2.6\tau$) analytical result of Eq. (B5) in Appendix pp2 is shown by the red line.

Figure 5

(a) Hall conductance of the ground state of the final Hamiltonian (black) and after a quench with $\tau =50$ (blue), and Hall conductivity after a quench with $\tau =50$ in a system with periodic boundary conditions in both spatial directions (gray), for $N_x=20, E_0=0.001$ and ${\tau }_E=5$. (b) Deviations from the ground-state Hall conductance as a function of $\tau$ for different ribbon widths, ranging from $N_x=10$ (light gray) to $N_x=50$ (black). The red line is a fitted ${\tau }^{-1}$ scaling. Blue dots show results for a Hamiltonian with a broken inversion symmetry of magnitude $b=0.15$ and $N_x=20$.

Figure 6

(a) Energy spectrum in the topologically nontrivial phase in absence (gray) and in presence of the electric field (blue and black). Black lines denote the occupied states of the system that adiabatically evolved due to insertion of the electric potential from the ground state without the electric potential. (b) Current densities $j_y^{\mathrm{gs}}$ in the ground state of the system with a static electric field (gray) and $j_y$ in the state which adiabatically evolved from the ground state of the system without the electric field at $t\gg {\tau }_E$ (blue). The deviation of the particle density $n-1$ is also shown (black). (c) Conductance $G_{yx}$ and the current density in $x$ direction in the middle of the ribbon $j_x(N_x/2)$ as a response to the adiabatically inserted electric potential. The system size is $N_x=20, N_y=201, u=-1$ and the electric field $E_0=0.04, {\tau }_E=5$.

Figure 7

Energy dispersion at different values of parameter $u$ during the quench.

Figure 8

Total number of excitations in the in-gap states (solid) and in the bulk (dashed) during the quench with $\tau =80$ and $N_x=50$.

Figure 9

${\mathrm{\Omega }}_L\left(k_y\right)$ (blue) and ${\mathrm{\Omega }}_R\left(k_y\right)$ (red) for a ribbon with $N_x=10$ at $u=-1.2$. ${\mathrm{\Omega }}_m\left(k_y\right)$ for bulk subbands are plotted in gray color.