Topological properties of curved spacetime extended Su-Schrieffer-Heeger model

Abstract

The Su-Schrieffer-Heeger (SSH) model, a prime example of a one-dimensional topologically nontrivial insulator, has been extensively studied in flat space-time. In recent times, many studies have been conducted to understand the properties of the low-dimensional quantum matter in curved spacetime, which can mimic the gravitational event horizon and black hole physics. However, the impact of curved spacetime on the topological properties of such systems remains unexplored. Here, we investigate the curved spacetime (CST) version of the extended SSH model, which supports distinct topological phases characterized by different winding numbers, by introducing a position-dependent hopping parameter. The extended SSH model already exhibits topological phases and the associated phase transitions. Different topological markers suggest that for the same choice of parameters, the CST version of the model retains the imprint of the same topological phases and transitions. Furthermore, the topologically non-trivial phase of the CST model hosts zero-energy edge modes, which are spatially asymmetric in contrast to those of the conventional SSH model. We find that at the topological transition points between phases with different winding numbers, a critical slowdown takes place for zero-energy wave packets near the boundary, indicating the presence of a horizon, and interestingly, if one moves even a slight distance away from the topological transition points, wave packets start bouncing back and reverse direction before reaching the horizon. Moreover, we have also quantified the time scale of the critical slowdown of the wavepacket across different winding-number transition phases. A semiclassical description of the wave packet trajectories also supports these results.

Figures (13)

• Figure 1: Time evolution of Gaussian wave packet in the Extended CST-SSH Hamiltonian of system size $2N=1000$ (500 unit cells) with $\sigma=1, \omega=50, x_0=750$: (a) obeying the relation $t_1 - (t_2 + t_3) + t_4 = 0$ with hopping $(t_1,t_2,t_3,t_4) =(3,-2,1,-4)$, with $|p_0|=\pi/2$, (b) obeying the relation $t_1 +t_2 + t_3 + t_4 = 0$ with hopping $(t_1,t_2,t_3,t_4) = (0,1,1,-2)$, at $|p_0|=\pi$ (c) extended CST-SSH having winding no 2, with hopping parameters $(t_1,t_2,t_3,t_4) = (0,1,1,-3)$ at $|p_0|=\pi$ (d) peak position vs. time for different combinations of hopping and initial momentum$(t_1,t_2,t_3,t_4,p_0)$ : (i) $(0,1,1,-2,\pi)$, (ii) $(3,-2,1,-4,\pi/2)$, (iii) $(3,1,1,3,\pi/4)$ (iv) $(3,-2,1,-3,\pi/2)$, and, (v) $(0,1,1,-3,\pi)$. (i), (ii) and (iii) suffer eternal slowdown; whereas, (iv) and (v) return before reaching the origin. For $\sigma=1, \omega=25, x_0=750$: (e) peak position of the Gaussian wave packet vs. time for different $p_0$, at $\sigma=1$, having $(t_1,t_2,t_3,t_4)=(1,1.9,0,0)$; (f) peak position vs. time for different $t_1/t_2$ ratios with $(t_3,t_4)=(0,0)$ at $p_0=-\frac{\pi}{2}$. Dotted lines are numerical; solid lines are semiclassical predictions.
• Figure 2: Plotting of turning point by numerical calculation ( black dotted lines) and from semiclassical formula \ref{['eq: turning point']} (colored dashed lines) at different initial momenta $p_0$ for different values of $\sigma$, at $(t_1,t_2,t_3,t_4)=(1,1.5,0,0)$, for system size $2N=1000$ (500 unit cells),
• Figure 3: Closing of the energy band gap and variation of zero energy eigen states variation with $\sigma$: for a system size $2N=1000$ (500 unit cells) with $(t_1,t_2,t_3,t_4)=(0.5,1,0,0)$ . Top row (a)--(c): Eigenvalue spectra for $\sigma=0,0.5$, and $1$ respectively. Bottom row (d)--(f): Corresponding zero-energy eigenstates profiles.
• Figure 4: Ratio of right-to-left peak of probability densities vs $\sigma$ for a system size $2N=1000$ (500 unit cells) with $(t_1,t_2,t_3,t_4)=(0.5,1,0,0)$. Here the dotted lines represent the numerical calculations and the dashed lines are from the analytical expressions of the zero energy eigenstates \ref{['eq:left edge zes']},\ref{['eq:right edge zes']}
• Figure 5: Energy gap $\Delta E$ at various system size $N$ for different $\sigma$, for the Extended CST-SSH with $(t_1,t_2,t_3,t_4)=(2,1,-1,-4)$ (top), and, for $(t_1,t_2,t_3,t_4)=(0.5,1,0,0)$ ( bottom)