Edge-state dynamics in coupled topological chains

We study time-dependent electrical properties of the Su-Schrieffer-Heeger (SSH) chain and coupled SSH chains on a substrate. Focusing on the midgap edge state dynamics we consider the abrupt transition from the normal to the SSH chain and determine characteristic timescale needed for topological states to develop. We have found that the midgap state is formed from the inside peaks of the normal chain density of states. For a ladderlike system we show that the edge SSH state vanishes in time or oscillates between neighboring sites. Moreover, for nonadiabatical time-dependent perturbations the midgap state can partially leak to other sites leading to induced topological states inside the trivial chain. We also analyze the mean-field correlation effects between the coupled chains revealing the induced Friedel oscillations in nontrivial chains.

Figure 1

Model of two coupled atomic chains of the length $N$ that can be in different topological phases with the couplings $W,V$ and $W^{\prime },V^{\prime }$, respectively. $V_x$ is the chain-chain coupling, and $U^C$ stands for the Coulomb repulsion between charged sites.

Figure 2

LDOS time evolution of the normal chain ($V=W=2$) composed of $N=8$ sites on the surface for sudden change of the couplings to the SSH parametrization in the nontrivial phase ($W=0.5,V=2.0$) at $t=20$. Other parameters are ${\varepsilon }_0=0,{\mathrm{\Gamma }}_i=\mathrm{\Gamma }=1,U^C=0$. Red, yellow, green, and blue colors represent the LDOS values equal to 0.6, 0.4, 0.2, and 0.1, respectively.

Figure 3

Time-dependent charge occupancies, currents, and LDOS at the Fermi level (upper, middle, and bottom panels, respectively) for first four sites of the normal chain of length $N=8(V=W=2)$ switched at $t=20$ to the ${\mathrm{SSH}}_1$ nontrivial chain with the couplings $W=0.5,V=2.0$ for different on-site energies ${\varepsilon }_0=0$ (left panels) and ${\varepsilon }_0=1$ (right panels).

Figure 4

Electron occupancies (upper panels) and LDOS at the Fermi level (bottom panels) of the first four sites in the normal chain ($V=W=2$) switched to the nontrivial topological chain ($W=0.5,V=2.0$) at $t=100$ for $N=8,{\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_N=1,{\mathrm{\Gamma }}_i=0$ for ${\varepsilon }_0=0$ (left panels) and ${\varepsilon }_0=1$ (right panels). The insets in the upper panels show the left current $j_1\left(t\right)$.

Figure 5

LDOS time evolution at the first, second, seventh, and eighth sites for the nontrivial chain composed of $N=16$ sites on the surface ($W=0.5,V=2.0$) broken at $t=20$ between the eighth and the ninth sites as is indicated in the upper scheme ${\varepsilon }_0=0,{\mathrm{\Gamma }}_i=1$.

Figure 6

LDOS time evolution of two first sites ${\mathrm{LDOS}}_1,{\mathrm{LDOS}}_2$ (left and right panels) for ${\mathrm{SSH}}_0\text{−}{\mathrm{SSH}}_0$ and ${\mathrm{SSH}}_1\text{−}{\mathrm{SSH}}_1$ coupled chains with $V=2,W=4\left({\mathrm{SSH}}_0\right)$ and $V=4,W=2\left({\mathrm{SSH}}_1\right)$. The coupling between chains is switched on linearly from zero (at $t=10$) to $V_x=10$ (at $t=11$), ${\mathrm{\Gamma }}_i=1,{\varepsilon }_0=0,N=10$.

Figure 7

Quasienergy spectra of two coupled atomic chains as a function of the chain-chain hopping $V_x$ for two ${\mathrm{SSH}}_0$ chains (upper panel), two ${\mathrm{SSH}}_1$ chains (middle panel), and for ${\mathrm{SSH}}_0\text{−}{\mathrm{SSH}}_1$ chains of the length $N=24$. The other parameters as the same as in Fig. 6.

Figure 8

Time evolution of the LDOS at two sites $i=1$, 2 of initially decoupled chains of the length $N=10$ in different topological phases (${\mathrm{SSH}}_0$ with $V=2,W=4$ and ${\mathrm{SSH}}_1$ with $V=4,W=2$). The chains are coupled at $t=10$ and $V_x\left(t\right)$ changes in time as is indicated in the upper inset according to the (a), (b), or (c) curve, respectively.

Figure 9

LDOS at the Fermi energy at two sites ($i=1$ and $i=2$) for the same system as in Fig. 8: ${\mathrm{SSH}}_0\text{−}{\mathrm{SSH}}_1$ chains coupled together at $t=10$ for different $V_x$ changes (a), (b), and (c), respectively. All parameters are the same as in Fig. 8.

Figure 10

Time evolution of the LDOS at $i=1$ and $i=2$ sites of atomic chains composed of $N=8$ sites each being in different topological phases (${\mathrm{SSH}}_0$ and ${\mathrm{SSH}}_1$). At $t=10$ chains are immediately connected [similarly as in Fig. 8, (c)] $V_x=3$ and all other parameters are the same as in Fig. 8. Yellow, green, and blue colors represent the LDOS values equal to 0.4, 0.2, and 0.1, respectively.

Figure 11

Electron occupancies along the normal chain (broken green curves) and along the ${\mathrm{SSH}}_1$ chain (solid red curves) of the length $N=10$ for $U^C=0$ (thin curves) and $U^C=5$ (thick curves)—upper panel. The bottom panel shows time-dependent occupancies at $i=1\text{–}3$ sites in both chains for a sudden change of the Coulomb repulsion $U^C=0$ for $t<18$ and $U^C=5$ for $t\ge 18$. The other parameters are $V=4,W=1,{\varepsilon }_i=0$ (topological chain) and $V,W=4,{\varepsilon }_i=-V$ (normal chain). The lines in the upper panel are plotted for better visualization.