Anomalous levitation and annihilation in Floquet topological insulators

Anderson localization in two-dimensional topological insulators takes place via the so-called levitation and pair annihilation process. As disorder is increased, extended bulk states carrying opposite topological invariants move towards each other in energy, reducing the size of the topological gap, eventually meeting and localizing. This results in a topologically trivial Anderson insulator. Here, we introduce the anomalous levitation and pair annihilation, a process unique to periodically driven, or Floquet, systems. Due to the periodicity of the quasienergy spectrum, we find it is possible for the topological gap to increase as a function of disorder strength. Thus, after all bulk states have localized, the system remains topologically nontrivial, forming an anomalous Floquet-Anderson insulator (AFAI) phase. We show a concrete example for this process, adding disorder via on-site potential “kicks” to a Chern insulator model. By changing the period between kicks, we can tune which type of (conventional or anomalous) levitation and annihilation occurs in the system. We expect our results to be applicable to generic Floquet topological systems and to provide an accessible way to realize AFAIs experimentally, without the need for multistep driving schemes.

Figure 1

When increasing disorder, the extended states separating topological and trivial quasienergy gaps levitate towards each other, eventually annihilating pairwise (white star). The conventional form of this process leads to the elimination of the topological gap, resulting in a trivial Anderson insulator (AI, left panel). The anomalous levitation and pair annihilation, in which the trivial gap shrinks, leaves behind an anomalous Floquet-Anderson insulator (AFAI, right panel). The background colors are numerically obtained by computing the transmission, as explained in the main text [35].

Figure 2

(a) The model Hamiltonian: One unit cell (gray square) contains two sites, belonging to the $A$ and $B$ sublattice (solid and open circle, respectively). Bravais vectors ${\mathbf{a}}_{x,y}$ are indicated as blue arrows. Nearest-neighbor hopping amplitudes along and against the arrows are $v_1(1+i)$ and $v_1(1-i)$, respectively. Next-nearest-neighbor hopping amplitudes are $+v$ along the dashed lines and $-v$ along the dotted lines. (b) Dispersion relation computed for $v_1=v$ in a ribbon geometry, infinite along ${\mathbf{a}}_x$, and consisting of 20 unit cells in the ${\mathbf{a}}_y$ direction. Top and bottom edge modes are shown as dashed and solid black lines, respectively.

Figure 3

Floquet spectrum of Eq. (4) with $v_1=v$ in the clean limit, $w=0$, in a repeated Floquet zone scheme. Tuning the time period $T$ has no physical effects here, as it only changes our description of the same physics, described by the constant Hamiltonian ${\widehat{H}}_0$—as in the first step of the nearly-free-electron approximation of crystalline solids. Hatching indicates the presence of edge states in a gap. Increasing the period $T$, bands cross over between Floquet zones, delimited by thick horizontal lines. This results in a sequence of topological phase transitions: Gaps are closed and reopened (even if in this undriven case no transitions between bands happen), with the number of edge states in them increasing by 2 each time [37]. Vertical dashed lines indicate periods for which the gaps are equal, $T=n{T}_c$ with $n=1,3,5$.

Figure 4

Phase diagram of the transmission $G$ depending on disorder $w$ and period $T$, at quasienergy (a), (b) $\varepsilon =0$ and (c), (d) $\varepsilon =\pi$. The boundary conditions along $x$ are periodic in (a) and (c), and open in (b) and (d). Transmissions are calculated for a system size of $20\times 20$, with $v_1=v$, and by averaging over 50 disorder realizations. Dashed lines show the analytically predicted phase transition points, as described in the main text.

Figure 5

Phase diagram of the transmission with maximal disorder, $w=\pi$. With (a) periodic boundary conditions, white regions of low transmission, separated by ridges, are the AI and AFAI phases. The phase boundaries are well approximated by the analytical predictions (dotted lines), where $w=0$ gaps have equal sizes. With (b) open boundary conditions, we can read off the topological invariants of the Anderson localized phases via the quantized value of transmission. The system size is $20\times 20$, and each point is obtained by averaging over 50 disorder realizations.