Out of equilibrium higher-order topological insulator: Floquet engineering and quench dynamics

Higher-order topological~(HOT) states,~hosting topologically protected modes on lower-dimensional boundaries,~such as hinges and corners, have recently extended the realm of the static topological phases.~Here we demonstrate the possibility of realizing a two-dimensional \emph{Floquet} second-order topological insulator, featuring corner-localized zero quasienergy modes and characterized by quantized Floquet qudrupolar moment $Q^{\rm Flq}_{xy}=0.5$, by periodically kicking a quantum spin Hall insulator (QSHI) with a discrete fourfold ($C_4$) and time-reversal (${\mathcal T}$) symmetry breaking Dirac mass perturbation.~Furthermore, we show that $Q^{\rm Flq}_{xy}$ becomes independent of the choice of origin as the system approaches the thermodynamic limit.~We also analyze the dynamics of a corner mode after a sudden quench, when the $C_4$ and ${\mathcal T}$ symmetry breaking perturbation is switched off, and find that the corresponding survival probability displays periodic appearances of complete, partial and no revival for long time, encoding the signature of corner modes in a QSHI.~Our protocol is sufficiently general to explore the territory of dynamical HOT phases in insulators and gapless systems.

Introduction. Topological states of matter in equilibrium are characterized by the bulk-boundary correspondence: A nontrivial topological phase in two (three) dimensions supports gapless edge (surface) modes of codimension one, protected by the bulk topological invariant [1,2]. This principle is operative for gapped, such as insulators and superconductors, as well as to the gapless topological states, which for instance include Dirac, Weyl and nodal loop semimetals [3]. The realm of topological states also encompasses the systems out of equilibrium, with Floquet topological insulators realized in the periodically driven systems standing as its paradigmatic representative [4][5][6][7][8][9][10][11]. In such systems, the bulk-boundary correspondence is more subtle, since bands featuring a trivial static topological invariant may sustain topological boundary modes, due to the nontrivial winding of the wavefunctions in the time direction.
In this paper, we promote a general mechanism of engineering Floquet HOT phases by periodically kicking [ Fig. 1(a)] a first-order topological phase with a suitable symmetry breaking mass perturbation. As a demonstrative example, we show that a two-dimensional second-order Floquet topological insulator, supporting , with a C 4 and T symmetry breaking mass perturbation. Our proposed general protocol for engineering Floquet HOT phases is distinct from the threestep driving procedure in a QSHI [38] or a C 4 and T symmetry breaking trivial insulator [41]. The resulting higher order topological insulator (HOTI) is characterized by quantized Floquet quadrupolar moment Q Flq xy = 0.5 (within numerical accuracy, see Fig. 3(a)), which is independent of system size (L), see Fig. 3(b). Furthermore, a minor origin dependence of Q Flq xy inside the topological phase [ Fig. 3(c)] disappears as the system approaches the thermodynamic limit (L → ∞), as shown in Fig. 3

(d).
We also study the dynamics of a zero-energy corner mode following a quench [43][44][45][46][47][48][49], such that the final state is first order. In particular, we compute its survival probability for time t > 0 [ Fig. 4], after suddenly switching off the C 4 and T symmetry breaking perturbation at t = 0 [ Fig. 1(b)]. Due to the edge propagation of the corner mode (see Fig. 5) in the postquench QSHI phase, the survival probability displays periodic appearances of complete, partial and no revival for long time. Therefore, our results should open up a route to study the Floquet HOT phases and quenching dynamics of HOT states in different setups (such as semimetals) and dimensions.
If we neglect the time dependence of V (t), then the Hamiltonian H Stat = H SHI + Γ 4 V 12 describes a static HOTI. Since {H SHI , Γ 4 } = 0, the term proportional to ∆ acts as a mass for chiral edge states, and breaks the C 4 as well as time-reversal (generated by T = iσ 2 τ 0 K, where K is the complex conjugation) symmetries. It changes sign four times across the corners of a square lattice system. Then a generalized Jackiw-Rebbi index theorem [31,51] ensures the existence of four zero-energy corner modes, and we realize a static second order topological insulator.
The robustness of such zero modes can be ensured from the spectral or particle-hole symmetry of H Stat , generated by the unitary operator Γ 5 , as {H Stat , Γ 5 } = 0, where Γ 5 = σ 2 τ 1 . The zero-energy corner modes are eigenstates of Γ 5 with eigenvalues +1 and −1. We also identify an antiunitary operator A = Γ 1 K, such that {H Stat , A} = 0 [52]. In the following discussion on Floquet HOTI, this antiunitary operator plays an important role, about which more in a moment.
To demonstrate the possible realization of a Floquet HOTI in the presence of a periodic kick [see Fig. 1(a) and Eq. (2)], we focus on the corresponding Floquet operator where 'TO' stands for the time-ordered product. The We always compute Q Flq xy in a periodic system. Similar behavior of the quadrupolar moment (Qxy) has already been reported for static crystalline and amorphous HOTI [31].
Floquet operator after a single kick takes a compact form . In the highfrequency limit (T → 0), one can neglect the higher-order terms in T , and comparing with Eq. (4), we find [53] While arriving at the final expression we assumed that T, ∆ → 0, but ∆/T is finite. Notice that H Flq looses the spectral symmetry with respect to the unitary operator Γ 5 , but still satisfies {H Flq , A} = 0. This observation ensures the spectral symmetry among Floquet quasienergy modes, and suggests possible realization of corner localized zero quasienergy modes and a Floquet HOTI. We now anchor this anticipation by numerically diagonalizing the Floquet operator from Eq. (3), satisfying where |φ n is the Floquet state with quasienergy µ n , with open boundaries in both directions. In Fig. 2(d), we show the local density of states associated with the (almost) zero [O(10 −6 )] quasienergy Floquet states, which depicts a strong corner localization. Therefore, by means of a periodic kick, a Floquet HOTI can be generated from a first-order topological insulator (QSHI in this case), when the driving perturbation breaks the desired symmetries (C 4 and T here) and satisfies a specific algebraic relation ({H SHI , Γ 4 } = 0 in this case). The topological robustness of the corner modes can be tested by computing the associated topological charge measuring the overlap of a zero quasienergy state (eigenstate of A with eigenvalue +1 or −1) with the states within the subspace of four zero quasienergy modes {µ 0 }, after acted by A from left. In the HOTI phase Q = 1 by construction, since the spectral symmetry of H Flq generated by the antiunitary operator A, leaves {µ 0 } invariant and A|φ n is characterized by quasienergy −µ n . We indeed find Q = 1 (within numerical accuracy), confirming that the zero quasienergy corner modes are stable, eigenstates of A, separated from bulk states with |µ n | > 0, and they are topologically protected. Floquet quadrupolar moment (Q Flq xy ). A static twodimensional HOTI possesses a quantized quadrupolar moment Q xy = 1/2 in both crystalline [12,54,55] and amorphous [31] systems. We now compute the quadrupolar moment for Floquet HOTI in the following way. Notice that the Floquet modes reside within a quasienergy window (−ω/2, ω/2), where ω = 2π/T is the kick frequency. We work in the high frequency regime such that ω t 0,1 , and compute the following quantity where n al Flq = 1 2L 2 r (xy) is the value of n Flq in the atomic limit. Inside the Floquet HOTI phase (when |m| < 2) we find Q Flq xy = 0.5, whereas Q Flq xy = 0 for a trivial insulator (when |m| > 2), see Fig. 3(a). These features are insensitive of the system size, see Fig. 3

(b).
A slight origin dependence of Q Flq xy inside the Floquet HOTI [see Fig. 3(c)] disappears as the system approaches the thermodynamic limit L → ∞, see Fig. 3(d). Moreover, for trivial insulator, Q Flq xy always stays at 0, irrespective of the choice of origin. Therefore, quantized Floquet quadrupolar moment serves as the indicator for a twodimensional Floquet HOTI [56].
Quench dynamics. Upon Floquet engineering a HOTI, we now investigate the survival probability of a corner mode at time t > 0, after suddenly switching off the C 4 and T symmetry breaking perturbation at t = 0 [see Fig. 1(b)]. This process is parametrized by where Θ is the heaviside step function of its argument.
The survival probability at time t is defined as [43,[47][48][49] P s (t) = where |Ψ initial corner is one of the zero-energy corner modes of the initial Hamiltonian H Ini = H SHI + Γ 4 V 12 for t < 0, |Φ final n is a wavefunction of the final Hamiltonian (after the sudden quench at t = 0) H Fin = H SHI with energy E n , and L 2 is the total number of lattice sites in the real space. The complete, partial and no revival respectively correspond to the situations when the survival probability acquires maximum (close to unity), intermediate (but finite) and very small (close to zero) values, see Fig. 4.
The dynamics of the survival probability can be understood from the density profile of the time evolved state |Ψ(t) = exp(−iH Fin t)|Ψ initial corner at various instances after the quench, see Fig. 5. For concreteness, we select one of the four near (due to finite system size) zeroenergy modes, dominantly localized near the corner at (L, L) [identified as |Ψ initial corner in Eq. (11)], see Fig. 5(a). As the final Hamiltonian (H SHI ) accommodates gapless one-dimensional edge modes [see Fig. 2(b)], the initial corner state predominantly diffuses along the edges of the system at t > 0. Since the C 4 symmetry is restored for t > 0, v x = v y ≡ v, where v i = ∂|E(k)|/∂k i is the group velocity in the i-direction, with i = x, y and ±E(k) are the eigenenergies of H SHI . The maximal value of the group velocity is v max = (1+m)t 2 0 / t 2 1 + t 2 0 (1 + m) 2 ≈ 1 for t 1 = t 0 = m = 1, which sets the velocity of the corner mode following the sudden quench [57]. After a time t = L/v max ≈ L, the most dominant peak of the initial corner mode reaches (1, L), see Fig. 5(b). The spectral density then exhibits a weaker peak at (L, L). As a result, the survival probability shows an intermediate revival when t ≈ L, see Fig. 4.
At time t ≈ 2L, the corner mode encounters a substantial reduction of spectral weight at (L, L) and concomitantly P s shows (almost) no revival, see Fig. 4. Such behavior arises from the fact that |Ψ(t ≈ 2L) is (almost) orthogonal to |Ψ initial corner , compare Figs. 5(a) and 5(c). The next partial revival occurs at t ≈ 3L, and a complete revival takes place at t ≈ 4L, when the initial state (almost) returns to itself. These features in the survival probability are impervious to the system size, group velocity (v max ), obtained by tuning the hopping parameters (t 1 and t 0 ) [57]. At later times this pattern continues to repeat itself. But, after each such cycle the amplitudes of the revival become weaker. Therefore, even after a sudden quench from a HOTI to a first-order topological insulator, the corner mode leaves its fingerprint in the survival probability in future time. By contrast when the system is quenched into a HOT insulator from a QSHI, the survival probability of the edge mode does not reveal any specific structure, possibly due to the absence of any extended gapless mode for t > 0 [57].
Discussion. To summarize, we demonstrate a possible realization of a two-dimensional Floquet HOTI, supporting anitiunitary symmetry protected zero quasienergy corner modes [see Fig. 2(d)], by periodically kicking a QSHI (first-order topological insulator) with a discrete  Fig. 1(b)] in a system with linear dimension L = 16 in both directions. Here, |Ψ initial corner represents a corner mode of the initial Hamiltonian HIni = HSHI + Γ4V12. Since, the final Hamiltonain HFin = HSHI accommodates onedimensional chiral edge modes, propagation of the corner state at time t > 0 predominantly takes place through the boundaries of the system, leading to the observed dynamics of the survival probability, see Fig. 4. symmetry breaking mass perturbation. This mechanism can be generalized to three-dimensional topological insulators and semimetals [23]. Our proposed protocol for generating dynamic HOT phases can in principle be realized in cold atomic systems, in the presence of dynamic strain that can be generated by gluing the sample with a piezoelectric material, vibrating at high frequency [58], and in acoustic [35,59] and photonic [60] systems.
Finally, we show that the signature of the corner modes can persist for a long time after a sudden quench into a QSHI, which manifests through periodic appearances of partial and complete revival in the dynamics of the survival probability [see Figs. 4 and 5]. The predicted quench dynamics can possibly be observed in cold atomic systems [61], and generalizations of this scenario to higher-dimensional HOT phases are left for a future investigation. We hope that present discussion will motivate future theoretical and experimental works exploring the dynamical properties of HOT phases.