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

TL;DR

This paper investigates the effects of various driving protocols on the Su-Schrieffer-Heeger model using sequences of two unitaries, revealing complex behaviors of end modes and Loschmidt echoes under different temporal arrangements.

Contribution

It introduces a comprehensive analysis of the SSH model driven by two unitaries in periodic, quasiperiodic, aperiodic, and random sequences, highlighting novel dynamical phenomena.

Findings

End modes appear in periodic driving but do not always match topological invariants.

Loschmidt amplitude shows oscillations with peaks at end mode quasienergies.

Loschmidt echo decays rapidly under random driving sequences.

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, aperiodically and randomly. For a periodic protocol, when $U_1 = e^{-i H_1 T/2}$ and $U_2 = e^{-i H_2 T/2}$ 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 amplitude ($LA$) starting with a initial state which is an end mode of $H_1$. We find that the $LA$ exhibits pronounced oscillations whose Fourier transform has a peak at a frequency which is equal to the quasienergy of an end mode of $U$. Next, when $U_1$ and $U_2$ are applied in a quasiperiodic or aperiodic way (we consider the Fibonacci and Thue-Morse protocols as examples), we study the Loschmidt echo ($LE$) starting with an initial state which is an end mode of the Hamiltonian $H_1$. When the inter-cell hoppings differ by a small amount denoted by $\epsilon$, and the time period $T$ of each unitary is also small, the distance between the unitaries is found to be proportional to $\epsilon T$. 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 $\epsilon^2$ for a fixed value of $T$, while the time after which the $LE$ starts decaying to zero has an interesting dependence on $\epsilon$ 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.