Multiple topological transitions and spectral singularities in non-Hermitian Floquet systems

Floquet systems, whose Hamiltonians are time-periodic, have attracted great attention over the past decades and played important roles in the study of quantum dynamics under external driving fields. Counterintuitive phenomena, such as dynamical localization [1, 2, 3, 4, 5] and stabilization [6, 7, 8, 9, 10, 11] have been revealed in Floquet systems. More recently, time-periodic drivings have been applied to engineer Floquet topological matter  [12, 13, 14, 15]. Irradiated by electromagnetic fields, semimetals and normal insulators could be driven into topological nontrivial phases  [14, 16, 17, 18], which may support novel topological states that are absent in static systems, such as Floquet winding metals [19, 20, 21], first-order  [22, 23, 24, 25, 26, 27] and higher-order [28, 29, 30, 31, 32, 33, 34] anomalous Floquet topological insulators. These nonequilibrium topological phases have also been widely explored in photonic, acoustic and cold atom experiments [24, 25, 26, 31, 35, 21, 36, 37].

Non-Hermitian effects, which are ubiquitous in open systems, have been recently employed as efficient means to actively control the physical properties of photonic, acoustic and cold atom setups [38, 39, 40, 41, 42]. With complex energies, non-Hermitian systems may support richer topological structures that originate from the windings of their eigenspectra [39, 40, 41, 42, 43], rather than the geometric phase of eigenfunctions like in Hermitian systems. Exceptional points (EPs) [44, 45, 46, 47] and non-Hermitian skin effects (NHSEs) [48, 49, 50, 51] are two representative examples that exhibit topological features unique to non-Hermitian systems.

The marriage between Floquet engineering and non-Hermitian control forms a natural extension in the study of driven open systems. In recent years, interesting discoveries have been made on non-Hermitian Floquet topological matter [52, 53, 54, 55, 56, 57, 58, 59, 60], including Floquet EPs and Floquet NHSEs. In experiments, gain and loss can be utilized as efficient means to achieve the non-Hermitian control of Floquet systems. Topological funneling of light [61], delocalized topological edge modes [62], Floquet NHSEs [63, 64], non-Hermitian topological phase transitions [65, 66], higher-order NHSEs [67], Floquet hybrid NHSEs [68, 69, 70] and transient NHSEs [71] have been theoretically proposed or experimentally realized along this line of thought. Considering the rich physics of Floquet and non-Hermitian systems, the interplay between Floquet engineering and gain/loss effects should bring about intriguing new phenomena that are awaited to be further unveiled.

In this paper, we uncover two unique phenomena not reported before in non-Hermitian Floquet systems, i.e., gain/loss-induced multiple topological transitions and anomalous wave transmissions due to the singularity of Floquet spectrum. We introduce spatially modulated gain and loss into a Floquet bipartite lattice, which supports anomalous Floquet second-order topological insulators (AFSOTI) under suitably chosen parameters [29]. With the increase of gain/loss strengths, we identify two topological transition points, across which the gaps between Floquet bands close and reopen. The first point is associated with a transition from AFSOTI to anomalous Floquet topological insulators (AFTI). The second point corresponds to a transition from AFTI to normal insulators (NI). Due to the existence of gain and loss, the AFTI also possesses hybrid-skin topological effects [72], and the boundary modes of such states are found to be geometry-dependent. For a horizontal square, the boundary modes are localized at the left-down corner for higher-order skin effects. For a square at $45^{\circ}$ to the horizontal line, the boundary modes are instead localized to different edges at different time slices and traversing all edges in one driving period. Such geometry-dependent effect can be understood by tracing the evolution of topological edge modes in coupled ring resonators. We further study the scattering properties of the system by the transfer-matrix method, and discover anomalous transmissions in certain parameter regions, even though the Floquet bands of the system are flat. These anomalous transmissions are found to originate from the Floquet-spectral singularities, whose locations in the parameter space are further sensitive to the system size.

This paper is organized as follows. In Sec. II, we introduce our model and outline its theoretical descriptions including the tight-binding approach and the transfer matrix method. In Sec. III, we reveal the multiple topological transitions through examining the changes in the Floquet bands and provide topological characterizations for different existing phases. These results are verified by quasienergy spectrum calculations under different boundary conditions. The geometry-dependent effects of gain/loss-induced AFTI are further discussed. In Sec. IV, we unveil the transmission properties of our system, which are consistent with quasienergy-band calculations in most cases. However, in a small parameter region where the quasienergy band is flat, the transmission is found to be unexpectedly large, which is due to the singularities in the Floquet spectrum. In Sec. V, we conclude our study, discuss potential applications of the corner skin modes and consider their possible experimental realizations.

We start with a time-dependent, tight-binding lattice model, whose driving protocol is illustrated in Fig. 1(a). One driving period contains four steps, with dimerized intersite couplings introduced in each step. Each site is sequentially coupled with one of its four neighboring sites in a counterclockwise direction. Every unit cell has two sublattices with balanced gain and loss. Under the periodic boundary condition, the time-dependent, four-step Hamiltonian in momentum space is given by

$\displaystyle H(\mathbf{k},t)=\left\{\begin{array}[]{cc}H_{1}(\mathbf{k})&\ell T<t\leq\ell T+T/4\;,\\ H_{2}(\mathbf{k})&\ell T+T/4<t\leq\ell T+T/2\;,\\ H_{3}(\mathbf{k})&\ell T+T/2<t\leq\ell T+3T/4\;,\\ H_{4}(\mathbf{k})&\ell T+3T/4<t\leq\ell T+T\;,\end{array}\right.$

(5)

where $\ell\in\mathbb{Z}$ and

$$H_{m}(\mathbf{k})=\theta(e^{i{\bf b}_{m}\cdot\mathbf{k}}\sigma^{+}+{\rm h.c.})+ig\sigma_{z}$$

(6)

for $m=1,2,3,4$. Here, $\theta$ is a coupling parameter, $g$ is the strength of gain and loss, $\sigma_{x,y,z}$ are Pauli matrices, and $\sigma^{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$. $T$ is the driving period and we set $T=4$ in dimensionless units. The vectors $\mathbf{b}_{m}$ are given by $\mathbf{b}_{1}=(0,0)$, $\mathbf{b}_{2}=(0,1)$, $\mathbf{b}_{3}=(-1,1)$ and $\mathbf{b}_{4}=(-1,0)$. The Hamiltonian possesses $\mathcal{PT}$ symmetry $\mathcal{PT}H(\mathbf{k},t)(\mathcal{PT})^{-1}=H(\mathbf{k},t)$, non-Hermitian particle-hole symmetry (NHPHS) $\mathcal{CK}H(\mathbf{k},t)(\mathcal{CK})^{-1}=-H(-\mathbf{k},t)$, and pseudo-inversion symmetry (PIS) $\mathcal{I}H(\mathbf{k},t)\mathcal{I}^{-1}=H^{{\dagger}}(-\mathbf{k},t)$. Here $\mathcal{P}=\mathcal{I}=\sigma_{x}$, $\mathcal{C}=\sigma_{z}$, $K$ represents complex conjugate and $\mathcal{T}=K$ is the (spinless) time-reversal operator.

The quasienergy dispersion $\varepsilon({\bf k})$ can be obtained by solving the Floquet eigenvalue equation,

$$U_{T}(\mathbf{k})\,|\Psi(\mathbf{k})\rangle=e^{-i\varepsilon(\mathbf{k})}|\Psi(\mathbf{k})\rangle\;,$$

(7)

where $U_{T}(\mathbf{k})\equiv\mathfrak{T}\mathrm{exp}\big{[}\!-\!i\!\int_{0}^{T}H(\mathbf{k},\tau)\,d\tau\big{]}$ is the Floquet operator. $\mathfrak{T}$ is the time-ordering operator. In the $\mathcal{PT}$-invariant region, the quasienergy bands are real. The $\mathcal{PT}$ symmetry further enforces the quasienergy bands to satisfy $\varepsilon_{n}(\mathbf{k})=\varepsilon^{*}_{m}(\mathbf{k})$ for $n\neq m$, where $n$ and $m$ are band indices. In the $\mathcal{PT}$-broken region, the Floquet bands thus form complex conjugate pairs. Besides, the Floquet bands are also constrained by the NHPHS [PIS] and satisfy $\varepsilon_{n}(\mathbf{k})=-\varepsilon^{*}_{m}(-\mathbf{k})$ [$\varepsilon_{n}(\mathbf{k})=\varepsilon^{*}_{m}(-\mathbf{k})$]. The PIS requires the band inversion to happen at high-symmetry momentum points ($\Gamma$ points in our model), where topological phase transitions happen.

The four-step Hamiltonian in Eq. (5) is equivalent to coupled ring resonators array [73, 74, 75], as illustrated in Fig. 1(b). In each ring resonators, only anticlockwise modes are considered. Red (blue) ring resonators correspond to gain (loss) sites. Each gain (loss) ring resonators is only coupled with its nearest-neighbor resonators. The transport properties of coupled ring resonators array can be described by the transfer matrix method (under periodic boundary condition along $y$ with $a_{m,n}=a_{m}e^{ik_{y}n}$, $b_{m,n}=b_{m}e^{ik_{y}n}$, $c_{m,n}=c_{m}e^{ik_{y}n}$, $d_{m,n}=d_{m}e^{ik_{y}n}$), i.e.,

$$\left(\begin{array}[]{c}a_{m+1}\\ b_{m+1}\end{array}\right)=M^{\prime}\left(\begin{array}[]{c}c_{m}\\ d_{m}\end{array}\right)=M^{\prime}M\left(\begin{array}[]{c}a_{m}\\ b_{m}\end{array}\right),$$

(8)

where $M$ and $M^{\prime}$ are two by two matrices. They can be constructed from the scattering matrices $S$ and $S^{\prime}$ by

	$\displaystyle M_{11}=\frac{S_{12}S_{21}-S_{11}S_{22}}{S_{12}},\ \ M_{12}=\frac{S_{22}}{S_{12}},$		(9)
	$\displaystyle M_{21}=-\frac{S_{11}}{S_{12}},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M_{22}=\frac{1}{S_{12}},$		(10)
	$\displaystyle M^{\prime}_{11}=\frac{S^{\prime}_{12}S^{\prime}_{21}-S^{\prime}_{11}S^{\prime}_{22}}{S^{\prime}_{21}},\ \ M^{\prime}_{12}=\frac{S^{\prime}_{11}}{S^{\prime}_{21}},$		(11)
	$\displaystyle M^{\prime}_{21}=-\frac{S^{\prime}_{22}}{S^{\prime}_{21}},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M^{\prime}_{22}=\frac{1}{S^{\prime}_{21}},$		(12)

with

	$\displaystyle S=$	$\displaystyle\begin{pmatrix}e^{i\varepsilon/4}&0\\ 0&e^{i(\varepsilon/4-k_{y})}\end{pmatrix}s\begin{pmatrix}e^{i\varepsilon/4}&0\\ 0&e^{i(\varepsilon/4+k_{y})}\end{pmatrix}s,$		(13)
	$\displaystyle S^{\prime}=$	$\displaystyle s\begin{pmatrix}e^{i\varepsilon/4}&0\\ 0&e^{i(\varepsilon/4-k_{y})}\end{pmatrix}s\begin{pmatrix}e^{i\varepsilon/4}&0\\ 0&e^{i(\varepsilon/4+k_{y})}\end{pmatrix},$		(14)

where $(b_{m},c_{m})^{T}=S(a_{m},d_{m})^{T}$, $(a_{m+1},d_{m})^{T}=S^{\prime}(b_{m+1},c_{m})^{T}$, and $s\equiv e^{i(\theta\sigma_{x}+ig\sigma_{z})}$.

As mentioned in Sec. II, the time-dependent Hamiltonian is equivalent to the coupled ring resonators array in Fig. 1(b). So we can naturally study the transport properties of the system by the transfer matrix method. The transport porperties are hopefully reflective of some properties of the Floquet bands studied in the previous sections. In particular, the transmission coefficient of the system can be obtained from

$$t_{N}=\frac{1}{T_{22}^{N}}.$$

(17)

The reflection coefficient of the system is given by

$$r_{N}=-\frac{T_{21}^{N}}{T_{22}^{N}}.$$

(18)

Here, $T_{22}^{N}$ is the second row and second column element of matrix $T^{N}$. $T_{21}^{N}$ is the second row and first column element of matrix $T^{N}$. $T^{N}$ is the transfer matrix for $N$ unit cells. It is obtained by taking the $N$th power of the matrix $M^{\prime}M$, i.e.,

$$T^{N}=(M^{\prime}M)^{N}.$$

(19)

Figs. 6(a1)–6(a10) show the bulk dispersion of Floquet bands vs $k_{y}$. Along the $x$-axis, we also choose the PBC with eight unit cells. Figs. 6(b1)–6(b10) show the transmission coefficients as functions of the quasienergy $\varepsilon$ and quasimomentum $k_{y}$. In our calculations, we set $N=5$. From the band structure and transmission coefficients, we find that that Floquet band gap close at $\pm\pi$ and hence topological phase transitions happen at $g\approx 1.56$ [Figs. 6(a3) and 6(b3)] and $g\approx 2.716$ [Figs. 6(a9) and 6(b9)]. These results are consistent with our previous analysis. As such, in general the transmission properties will be able to detect various topological phase transitions induced by gain and loss.

Comparing the results of band-structure calculations and transmission coefficients, as expected from above, they are mostly consistent with each other. That is, the dispersive bulk bands lead to nonzero transmission coefficients, whereas the flat bands and band gaps generally yield zero transmissions. Nevertheless, there are remarkable exceptions in certain cases. One obvious example is shown in Figs. 6(a5) and 6(b5) ($g=2.2$), where the transmission coefficient is very large even though the Floquet bands are partially flat. Another example is given in Figs. 6(a6) and 6(b6) ($g\approx 2.35$), where the transmission coefficient is unity even though the entire Floquet bands are flat. These anomalous transmission properties can be explained by the so-called spectral singularity or unit cell resonance [76].

Let us now focus on the spectral singularity of the system. Spectral singularities are special points that spoil the completeness of eigenfunctions of certain non-Hermitian operators [76]. It has been shown to happen at $T_{22}^{N}=0$ for complex scattering potentials, where the reflection and transmission coefficients tend to infinity. Fig. 7(a) shows the logarithm of reflection coefficient $\log|r_{N}|$ as a function of $k_{y}$ and $g$ at $\varepsilon=0$. We notice that there are several red lines corresponding to infinite reflection coefficients. The left four lines are related to multiple scatterings between different unit cells. The line marked by star is mainly determined by the property of unit cells. Different from other lines, here the spectral singularity exists at $k_{y}=\pi/d$ with $g\approx 2.2$. It hence explains the large transmission coefficient at $k_{y}=\pi/d$ and $\varepsilon=0$ in Fig. 6(b5), in the presence of a flat quasienergy band in that regime. Besides, the positions of singularity points are size-dependent. Fig. 7(b) shows the logarithm of transmission coefficient $\log|t_{N}|$ as a function of $g$ with a fixed $k_{y}=\pi/d$ for different numbers of unit cells. We find that the positions of singularity points shift to right with the increase of the unit-cell numbers. With a large number of unit cells, the positions of singularity points tend to coincide with the locations of unit cell resonance at $|t_{N}|=1$.

From Fig. 7(a), we observe another line marked by square, which corresponds to vanishing reflection coefficients. This line well explains the unit transmission at $\varepsilon=0$ in Fig. 6(b6), although the Floquet band is completely flat.

In this work, we revealed two unique phenomena in Floquet systems induced by non-Hermitian gain and loss, i.e., the multiple topological transition and the spectral singularity. Because of the multiple Floquet topological phase transitions, we discover Floquet hybrid skin topological modes by adding appropriate gain and loss to anomalous Floquet higher-order topological insulators. Interestingly, the profile of the associated Floquet hybrid skin topological modes is found to depend on the geometry of the lattice. Though in general we see a direct link between Floquet band features and transmission properties, we also discover that significant wave transmissions could appear even at flat bands due to the physics of spectral singularity. This represents another interplay between periodic driving and gain/loss effects. These intriguing properties might be observable in optical and acoustic experiments. The Floquet model proposed in this work can be realized in coupled waveguide system or coupled ring resonator array system and the non-Hermitian gain/loss can be replaced by loss difference [24, 25, 31, 35, 45].

W. Z. acknowledges support from the Start up Funding from Ocean University of China. L. Z. is supported by the Fundamental Research Funds for the Central Universities (Grant No. 202364008), the National Natural Science Foundation of China (Grants No. 12275260, No. 12047503, and No. 11905211), and the Young Talents Project of Ocean University of China. L. L. is supported by National Natural Science Foundation of China (Grant No. 12104519) and the Guangdong Project (Grant No. 2021QN02X073). J. G. acknowledges support by the National Research Foundation, Singapore and A*STAR under its CQT Bridging Grant.