Setting Boundaries with Memory: Generation of Topological Boundary States in Floquet-Induced Synthetic Crystals

When a $d$-dimensional quantum system is subjected to a periodic drive, it may be treated as a ($d+1$)-dimensional system, where the extra dimension is a synthetic one. This approach, however, affords only a limited level of control of the effective potential along the synthetic direction. In this work, we introduce a new mean for controlling the Floquet synthetic dimension. We show that arbitrary potentials, as well as edges in the synthetic dimension, could be introduced using a memory component in the system’s dynamics. We demonstrate this principle by exploring topological edge states propagating normal to synthetic dimensions. Such systems may act as an optical isolator which allows the transmission of light in a directional way. Also, we suggest an experimental realization of the memory effect in spins coupled to nanofabricated Weyl semimetal surface states.

Figure 1

(a) The equivalence between a TLS in the presence of a drive and a memory kernel and particles hopping on a 1D synthetic lattice. (b) In blue, the gap parameter ${\mu }_n$ [Eq. (5)], the synthetic electric field in units of $5\hslash \omega$ (red), and the absolute square of the two solutions of Eq. (7) with $\eta =0$ (green). Evidently, there is an electric-field-free zone in $n$ space in which the system hosts two localized solutions near the jump in the gap parameter ($n_0=110$, $n_1=50$, $\mu =10$, and $\delta =-9.9$).

Figure 2

(a) $1+1\mathrm{D}$ case: The absolute square of the solution to Eq. (10) with $\eta =0$, along with the synthetic field and the gap parameter. At low energies, the system supports QH-like edge states in the combined $x-n$ space ($n_0=48$, $n_1=35$, $\mu =10$, and $\delta =-9.9$). (b) The cycle that a system of spinful particles performs. Spin up (down) denotes a full (empty) site. The system evolves along the state which is depicted in (a). Along the left (right) real space edge the system emits (absorbs) photons, while along the top (bottom) $n$ space edge the system supports left (right) propagation.

Figure 3

(a) The system in Eqs. (14) and (15). $\varphi$ represents a pseudospin degree of freedom which is coupled to a chiral field $\psi$. The effective evolution of $\varphi$ follows Eq. (1). (b) The one-dimensional version of (a). Placing a 1D lattice of the system in (a) and introducing couplings between the different TLS yields an effective model for the $\varphi$ field that obeys Eq. (8).