Dynamics of edge currents in a linearly quenched Haldane model

In a finite-time quantum quench of the Haldane model, the Chern number determining the topology of the bulk remains invariant, as long as the dynamics is unitary. Nonetheless, the corresponding boundary attribute, the edge current, displays interesting dynamics. For the case of sudden and adiabatic quenches the postquench edge current is solely determined by the initial and the final Hamiltonians, respectively. However for a finite-time ($\tau$) linear quench in a Haldane nanoribbon, we show that the evolution of the edge current from the sudden to the adiabatic limit is not monotonic in $\tau$ and has a turning point at a characteristic time scale $\tau ={\tau }_0$. For small $\tau$, the excited states lead to a huge unidirectional surge in the edge current of both edges. On the other hand, in the limit of large $\tau$, the edge current saturates to its expected equilibrium ground-state value. This competition between the two limits lead to the observed nonmonotonic behavior. Interestingly, ${\tau }_0$ seems to depend only on the Semenoff mass and the Haldane flux. A similar dynamics for the edge current is also expected in other systems with topological phases.

Figure 1

(a) Three plaquettes of the hexagonal lattice with lattice vectors ${\mathbf{a}}_{\mathbf{1}}$ and ${\mathbf{a}}_{\mathbf{2}}$. Blue and red circles represent the two sublattices A and B. (b) The Haldane lattice, with locally broken time-reversal symmetry. Regions labeled “a” and “b” enclose flux in opposite directions, and arrows on the next-nearest bonds (dashed lines) denote the direction of positive phase hopping due to the locally broken time-reversal invariance. (c) Brillouin zone of a hexagonal lattice with reciprocal lattice vectors ${\mathbf{b}}_{\mathbf{1}}$ and ${\mathbf{b}}_{\mathbf{2}}$, with $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ representing the two inequivalent Dirac points; the color corresponds to sublattices A and B. (d) Chern phase diagram of the Haldane model in the on-site energy, $M$ (also called the Semenoff mass), and the staggered phase, $\varphi$, plane. The white region is the topologically trivial phase ($\nu =0$), while the colored region is the topologically nontrivial Chern phase ($\nu =\pm 1$). The arrow indicates a quenching scheme of varying $M$, which takes the model from point $P$ ($\varphi =\pi /3$, $M/t_2=6$) in the nontopological phase to point Q ($\varphi =\pi /3$, $M/t_2=0$) in the topological phase.

Figure 2

Haldane model on a nanoribbon, which is periodic along the armchair edge ($x$ direction) and finite along the zigzag direction ($y$ direction). The nanoribbon is $N$ cells wide and each unit cell is labeled with two indices, $m$ and $n$, and has two lattice sites ($A$ or $B$). The phase of the complex hopping $t_2$ is negative (positive) for hopping in a clockwise (counterclockwise) sense between next-nearest neighbors.

Figure 3

Energy spectrum of the Haldane nanoribbon described by the Hamiltonian in Eq. (4) in (a) the topologically trivial phase with $M=6$ and Chern number $\nu =0$ and (b) the topological phase with $M=0$ and $\nu =-1$. Blue and red lines represent the conduction and the valance bands, respectively. Note that unlike the topologically trivial phase, the topological phase has a single band crossing at $k_x=0$. We have chosen other parameters to be $t_1=1$, $t_2=1/3$, $\varphi =\pi /3$, and $N=20$.

Figure 4

Average equilibrium current in the $x$ direction along each row, $n=1,...,20$, of the Haldane nanoribbon (see Fig. 2) in the topological phase. Two counter-propagating current-carrying states appear at the two edges. On the other hand, the current remains 0 in the bulk. Here we have chosen $t_1=1$, $t_2=1/3$, $M=0$, $\varphi =\pi /3$, and $N=20$. (See also Fig. 4 of Caio et al. [51].)

Figure 5

Dynamics of the edge current from the nontopological phase ($M=6$, $\varphi =\pi /3$) to the topological phase ($M=0$, $\varphi =\pi /3$) following two quenching protocols. (a) A sudden quench, when the rate of quenching is so low that the system should ideally remain in the ground state of the initial Hamiltonian and should have no current. However, the finiteness of $\tau$ leads to some excited states, which give a finite contribution to the current (represented by the dashed horizontal line). (b) A linear slow quench (i.e., $\tau$ large) from the trivial phase to the topological. The system maintains the instantaneous ground state of the time-evolved Hamiltonian, at each instant of time, and postquench it reaches the actual ground-state current of the final Hamiltonian (represented by the dashed horizontal line). (See also the sudden quench case of Caio et al. [51, 52].)

Figure 6

(a) Variation of the postquench current for a linear sweep from a nontopological to a topological phase at different quenching rates $\tau$. Here $\varphi =\pi /3$, $t_1=1$, and $t_2=1/3$ are held fixed, while the Semenoff mass $M$ is varied concerning the time to quench to different phases. The curve (represented by the solid line with filled circles) shows a generation of excess current for smaller values of $\tau$ which reaches a minimum at ${\tau }_0=2.2$ (represented by the dashed vertical line), following which it increases again to reach the equilibrium current value of the final Hamiltonian (represented by the dotted horizontal line) for large $\tau$. (b) Zoomed-in version of the $\tau \in (0,20)$ region of (a), with the intraband (solid line with diamonds) and interband (solid line with triangles) contributions [see Eq. (6)] shown separately. Clearly the excited states play a significant role, since the current is dominated by the interband contribution.

Figure 7

Spatial $n$ (see Fig. 2) and $k_x$-resolved current in (a) the topological phase and (b) the nontopological phase. The size of the circle represents the amount of current at that point and the color denotes the sign of the current (blue-violet, negative; black, positive). The existence of a finite edge current in the topological phase (a) and no current in the nontopological phase (b) is evident.

Figure 8

(a) Spatial $n$ and $k_x$-resolved contribution to the edge current of the dynamical part [arising from the second term in Eq. (14)] for small $\tau$ values. (b) The unidirectional negative contribution in both edges leads to the huge initial dip in current at both edges. The size of the circle in (a) is proportional to the magnitude of the current. Here all the parameters are identical to those in Fig. 6. (b) Variation of the postquench current (calculated at $t=\tau$) at different quenching rates $\tau$ at the two edges of the system (represented by the dashed line with diamonds and the solid line with circles). Starting from the $\tau =0$ scenario of zero current, the current eventually goes to the respective equilibrium value (represented by dotted horizontal lines) for both edges at large $\tau$, though for small $\tau$ both edges show a large dip in current and have different values of the turning point ${\tau }_0$ (highlighted in the inset). The dot-dashed vertical line indicates the value of $\tau$ after which the current at the two edges propagates in opposite directions.

Figure 9

Dependence of ${\tau }_0$ (for a given edge) on various system parameters. (a) ${\tau }_0$ is independent of the width of the Haldane nanoribbon. However, ${\tau }_0$ seems to depend on the parameters which determine the topological phase of the system, i.e., on (b) the Semenoff mass $M$ and (c) the Haldane flux $\varphi$. The positions of ${\tau }_0$ for different parameters are represented by vertical lines.