Su-Schrieffer-Heeger model driven by sequences of two unitaries: periodic, quasiperiodic and random protocols

TL;DR

This work analyzes the SSH model under driving by two unitaries $U_1$ and $U_2$ with different inter-cell hoppings across periodic, Fibonacci, Thue-Morse, and random protocols. Through Floquet spectra, winding analysis, and Loschmidt-echo measurements, it reveals that periodic driving can create end modes not always tied to the winding invariant, while quasi-periodic drives can preserve end-mode memory for long times when $εT$ is small; random driving leads to rapid LE decay. A key finding is that the distance between the unitaries scales as $Δ\sim εT$ and that LE saturation scales as $d\sim α ε^2$, with long-time dynamics governed by a BCH-type effective Hamiltonian for small $εT$ and breakdown via higher-order commutators when this product grows. Overall, the paper demonstrates controllable manipulation of boundary modes via tailored drive sequences, with implications for dynamical topology and robust edge-state dynamics under time-dependent protocols.

Abstract

We study the effect of driving the Su-Schrieffer-Heeger model using two unitary operators $U_1$ and $U_2$ in different combinations; the unitaries differ in the values of the inter-cell hopping amplitudes. Specifically, we study the cases where the unitaries are applied periodically, quasiperiodically and randomly. For a periodic protocol, when $U_1$ and $U_2$ are applied alternately, we find that end modes may appear, but the number of end modes does not always agree with the winding number which is a $Z$-valued topological invariant. We then study the Loschmidt echo ($LE$) starting with a random initial state. We find that the $LE$ exhibits pronounced oscillations whose Fourier transform has peaks at frequencies which agree with the most prominent gaps between pairs of quasienergies. Next, when $U_1$ and $U_2$ are applied in a quasiperiodic way (we consider Fibonacci and Thue-Morse protocols), we study the $LE$ starting with an initial state which is an end mode of one of the unitaries. When the inter-cell hoppings differ by a small amount denoted by $ε$, and the time period $T$ of each unitary is also small, the distance between the unitaries is found to be proportional to $εT$. We then find that the $LE$ oscillates around a particular value for a very long time before decaying to zero. The deviation of the value of the $LE$ from 1 scales as $ε^2$ for a fixed value of $T$, while the time after which the $LE$ starts decaying to zero has an interesting dependence on $ε$ and $T$. Finally, when $U_1$ and $U_2$ are applied in a random order, the $LE$ rapidly decays to zero with increasing time. We have presented a qualitative understanding of the above results.

Figures (18)

• Figure 1: Schematic picture of the hopping amplitudes of the SSH model. There are two types of hopping denoted as $J_1$ (intra-cell hopping) and $J_2$ (inter-cell hopping).
• Figure 2: Plots (a) and (b) show the imaginary part vs real part of the eigenvalues $e^{i \theta_n}$ of the Floquet operator for periodic driving with $T = 2\pi$ and $T=\pi/2$ respectively. Plot (c) shows the probability $|\psi_i|^2$ versus the site index $i$ of an eigenvector of the Floquet operator which is localized at the left edge of the system, for $T = 2\pi$. (The probability $|\psi_i|^2$ for the left-localized eigenvector of the Floquet operator for $T=\pi/2$ is similar to plot (c) and is not shown here). The parameters are $J_1 = 1.1$, $J'= 1$, $\alpha = 0.5$, and $L = 1600$.
• Figure 3: Plot of $\phi_k$ versus $k$ for periodic driving with (a) $T=2 \pi$ and (b) for $T=\pi/2$ showing $\phi_k$ as a function of $k$. The parameters used are $J_1 = 1.1$, $J'=1$, $\alpha = 0.5$, and $L = 1600$. Plot (a) shows that $W=0$, while plot (b) shows that $W=-1$.
• Figure 4: Plot (a) shows the imaginary part versus real part of the eigenvalues of the Floquet operator for periodic driving. The points marked 1, 2 and 3 respectively show the Floquet eigenvalues of an end mode at $e^{i \theta} = -1$, a state at the end of the bulk states, and an end mode with $e^{i \theta} = 1$. Plot (b) shows the Loschmidt echo for an initial state which is chosen randomly. Plot (c) shows the Fourier transform of the Loschmidt echo. We see peaks at $\Omega \simeq 0.2$ and 1. The parameters used for these plots are $J_1 =1.1$, $J' = 1$, $\alpha = 0.5$, $T = \pi$, and $L=200$.
• Figure 5: Plots of the real versus imaginary parts of the Floquet eigenvalues of (a) $U_1$ and (b) $U_2$. We have taken $J_1 = 1.1$ and $T=0.1$ in both cases, while $J_2 = 1.5$ and $J_2 = 1.5 + \epsilon$ for $U_1$ and $U_2$ respectively, with $\epsilon = 0.1$. Both $U_1$ and $U_2$ have one mode localized et each end of an open system, with eigenvalue equal to 1 as shown.