Floquet Topological Phases of Non-Hermitian Systems

Topological phases in non-Hermitian systems have attracted much attention both theoretically Lee (2016); Leykam et al. (2017); Kunst et al. (2018); Yao and Wang (2018); Yao et al. (2018); Jin and Song (2019); Shen et al. (2018); Yokomizo and Murakami (2019); Gong et al. (2018); Kawabata et al. (2019); Borgnia et al. (2020); Okuma et al. (2020); Lee and Thomale (2019); Zeng et al. (2020); Lee et al. (2019a); Gao et al. (2018); Silveirinha (2019); Longhi (2019); Deng and Yi (2019); Hou et al. (2020); Liu et al. (2019a); Lee et al. (2019b); Zhang et al. (2019a); Ezawa (2019); Zhang and Wu (2019); Hirsbrunner et al. (2019); Chen and Zhai (2018); Wang et al. (2019); Rui et al. (2019); Zhang et al. (2020a); Kunst et al. (2018); Herviou et al. (2019); Zhang et al. (2020b); Wang et al. (2020) and experimentally Zeuner et al. (2015); Ding et al. (2016); Xiao et al. (2017); Zhu et al. (2018); Song et al. (2019a); Zhao et al. (2019); Ghatak et al. (2019); Xiao et al. (2020). Many interesting characters have been found in different non-Hermitian systems Gao et al. (2018); Silveirinha (2019); Longhi (2019); Deng and Yi (2019); Hou et al. (2020); Liu et al. (2019a); Lee et al. (2019b); Zhang et al. (2019a); Ezawa (2019); Zhang and Wu (2019); Hirsbrunner et al. (2019); Chen and Zhai (2018); Wang et al. (2019); Rui et al. (2019); Zhang et al. (2020a). One of their unique features is that not only the edge states but also the nontopologically protected bulk states are localized at the edges, which is called skin effect Kawabata et al. (2019); Borgnia et al. (2020); Lee and Thomale (2019); Zeng et al. (2020); Okuma et al. (2020); Martinez Alvarez et al. (2018); Xiong (2018); Zhang et al. (2019b); Yang et al. (2019). It causes that one cannot characterize the edge states by the topological properties of the bulk spectrum. This is the non-Hermiticity induced breakdown of bulk-boundary correspondence (BBC) Lee (2016); Leykam et al. (2017); Kunst et al. (2018); Yao and Wang (2018); Yao et al. (2018), which lays the foundation for the classification of topological phases in Hermitian systems Hasan and Kane (2010); Qi and Zhang (2011); Bansil et al. (2016); Chiu et al. (2016). To describe the topological features of the edge states, many strategies including biorthogonal eigenstate Kunst et al. (2018), singular-value decomposition Herviou et al. (2019), gauge transformation Ge et al. (2019), and modified periodic boundary condition Imura and Takane (2019) have been proposed. A milestone among these is the non-Bloch band theory established in the generalized Brillouin zone (BZ) for the one-dimensional (1D) chirally symmetric and translation invariant systems Yao and Wang (2018); Yokomizo and Murakami (2019), which is recently generalized to the system without chiral symmetry Zhang et al. (2019b); Yang et al. (2019).

Coherent control via periodic driving dubbed as Floquet engineering has become a versatile tool in artificially synthesizing exotic topological phases in systems of ultracold atoms Eckardt (2017); Meinert et al. (2016), photonics Rechtsman et al. (2013); Cheng et al. (2019), superconductor qubits Roushan et al. (2017), and graphene McIver et al. (2020). Parallel to the topological phases in static systems, the topological phases in periodically driven systems are called Floquet topological phases (FTPs). Many intriguing FTPs absent in static systems Kundu and Seradjeh (2013); Higashikawa et al. (2019); Harper and Roy (2017); Hu et al. (2020); Rudner et al. (2013); Asbóth and Obuse (2013); Račiūnas et al. (2016); Wang et al. (2018); Rodriguez-Vega and Seradjeh (2018); Sajid et al. (2019); Xiao et al. (2018); Zhou and Gong (2018a) have been simulated by periodic driving in Hermitian systems. The key role played by periodic driving is changing symmetry and inducing an effective long-range hopping in lattice systems Tong et al. (2013); Xiong et al. (2016); Liu et al. (2019b). A natural question is what controllable topological characters can periodic driving bring to non-Hermitian systems. Given the fact that the chiral symmetry can be broken by periodic driving, one cannot apply the well developed non-Bloch band theory of 1D chirally symmetric static systems Yao and Wang (2018) to the periodic ones for recovering the BBC and defining topological invariants. Without touching the topological characterization, the transport phenomena of the non-Hermitian Floquet edge states was studied in Refs. Höckendorf et al. (2019); Li et al. (2019). For some special cases in the absence of the skin effect, the topological numbers were defined in the traditional BZ Zhou and Gong (2018b); Zhou (2020). Recent study reveals that the BBC is approximately recoverable only for small intercell coupling Zhang and Gong (2020). Further, the inevitable spatial disorder also invalidates the non-Bloch band theory to restore the BBC. Thus, a general theory to characterize the non-Hermitian FTPs is still lacking.

In this work, we investigate the FTPs in the periodically driven non-Hermitian systems. A general description is established to characterize the FTPs of such nonequilibrium systems both in the momentum and the real spaces. The main idea to characterize the FTPs in both of the spaces is to restore the chiral symmetry of the periodically driven systems by the proposed similarity transformations, which keep the quasienergy spectrum unchanged. Taking the non-Hermitian Su-Schrieffer-Heeger (SSH) model as an example, we find that rich topological phases absent in the static case are created by the periodic driving. The studies on the real-space topological physics in the presence of disorder reveal that the extra phases called non-Hermitian Floquet topological Anderson insulator phases are induced by the disorder. Our results demonstrate that the periodic driving and its constructive interplay with the disorder supply us useful ways to engineer exotic topological phases in the non-Hermitian systems.

A time-periodic system ${H}(t)={H}(t+T)$ with period $T$ has a complete set of basis $|u_{\alpha}(t)\rangle$ determined by Floquet equation $[{H}(t)-i\partial_{t}]|u_{\alpha}(t)\rangle=\varepsilon_{\alpha}|u_{\alpha}(t)\rangle$ such that any state evolves as $|\Psi(t)\rangle=\sum_{\alpha}c_{\alpha}e^{-i\varepsilon_{\alpha}t}|u_{\alpha}(t)\rangle$ Sambe (1973); Chen et al. (2015). Acting as stationary states and eigenenergies of static systems, $|u_{\alpha}(t)\rangle$ and $\varepsilon_{\alpha}$ are called quasistationary states and quasienergies, respectively. Being equivalent to ${U}_{T}|u_{\alpha}(0)\rangle=e^{-i\varepsilon_{\alpha}T}|u_{\alpha}(0)\rangle$ with $U_{T}$ the one-period evolution operator, the Floquet equation defines an effective Hamiltonian ${H}_{\text{eff}}=\frac{i}{T}\ln U_{T}$ whose eigenvalues are the quasienergies. The FTPs are defined in the quasienergy spectrum. Different from the static case, they can occur at both of the quasienergies $0$ and $\pi/T$ Tong et al. (2013).

Chiral symmetry plays an important role in characterizing the non-Hermitian topological phases Leykam et al. (2017); Kunst et al. (2018); Yao and Wang (2018); Jin and Song (2019); Yokomizo and Murakami (2019). However, it cannot be preserved if a periodic driving is applied. Consider a non-Hermitian two-band system $H$ with its parameters periodically driven between two specific $H_{1}$ and $H_{2}$ in the respective time duration $T_{1}$ and $T_{2}$. Applying the Floquet theorem, we obtain $H_{\text{eff}}$ from $U_{T}=e^{-iH_{2}T_{2}}e^{-iH_{1}T_{1}}$. It can be seen that even $H_{j}$ ($j=1,2$) have chiral symmetry $SH_{j}S^{-1}=-H_{j}$ with $S$ being the chiral operator, $H_{\text{eff}}$ breaks the symmetry due to $[H_{1},H_{2}]\neq 0$. The absence of the chiral symmetry in $H_{\text{eff}}$ makes it hard to define the FTPs in a non-Hermitian system by the non-Bloch band theory, which is developed for the chirally symmetric static system Yao and Wang (2018). We propose the following scheme to resolve this problem. Two similarity transformations $G_{j}=e^{i(-1)^{j}H_{j}T_{j}/2}$ covert $U_{T}$ into $\tilde{U}_{T,1}=e^{-iH_{1}T_{1}/2}e^{-iH_{2}T_{2}}e^{-iH_{1}T_{1}/2}$ and $\tilde{U}_{T,2}=e^{-iH_{2}T_{2}/2}e^{-iH_{1}T_{1}}e^{-iH_{2}T_{2}/2}$, from which the defined $\tilde{H}_{\text{eff},j}={i\over T}\ln\tilde{U}_{T,j}$ share the same quasienergies with $H_{\text{eff}}$ while recover the chiral symmetry of $H_{j}$. It can be equivalently understood to define new chiral operators $G_{j}^{-1}S^{-1}G_{j}$ such that $H_{\text{eff}}$ obeys the chiral symmetry. The similar scheme was used in Hermitian systems Asbóth et al. (2014). As we will see later, the recovered chiral symmetry is significant to characterize the FTPs in the non-Hermitian system both for the translation-invariant and variant cases.

If the system is further translation invariant, we can develop a general characterization to the FTPs in the momentum space. The coefficient matrices of $H_{j}$ are written in the momentum space as $\mathcal{H}_{j}(k)=\mathbf{h}_{j}(k)\cdot\boldsymbol{\sigma}$ with $\boldsymbol{\sigma}$ being the Pauli matrices. We readily obtain $\mathcal{H}_{\text{eff}}({k})=\mathbf{h}_{\text{eff}}(k)\cdot{\boldsymbol{\sigma}}=i\ln[e^{-i\mathcal{H}_{2}(k)T_{2}}e^{-i\mathcal{H}_{1}(k)T_{1}}]/T$ with the Bloch vector $\mathbf{h}_{\text{eff}}(k)=-\arccos(\epsilon)\underline{\mathbf{r}}/T$ and

	$\displaystyle\epsilon$	$\displaystyle=$	$\displaystyle\cos(T_{1}E_{1})\cos(T_{2}E_{2})-\underline{\mathbf{h}}_{1}\cdot\underline{\mathbf{h}}_{2}\sin(T_{1}{E}_{1})\sin(T_{2}E_{2}),~{}~{}~{}$		(1)
	$\displaystyle\mathbf{r}$	$\displaystyle=$	$\displaystyle\underline{\mathbf{h}}_{1}\times\underline{\mathbf{h}}_{2}\sin(T_{1}E_{1})\sin(T_{2}E_{2})-\underline{\mathbf{h}}_{2}\cos(T_{1}E_{1})$		(2)
			$\displaystyle\times\sin(T_{2}E_{2})-\underline{\mathbf{h}}_{1}\cos(T_{2}{E}_{2})\sin(T_{1}{E}_{1}),$

where $T=T_{1}+T_{2}$, $\underline{\mathbf{h}}_{j}=\mathbf{h}_{j}/E_{j}$, and $E_{j}=\sqrt{\mathbf{h}_{j}\cdot\mathbf{h}_{j}}$ is the complex eigen energies of $\mathcal{H}_{j}(k)$ SMP . The FTP transition is associated with the closing of the quasienergy bands, which occurs at the exceptional points for the $k$ and driving parameters satisfying

	$\displaystyle T_{j}E_{j}=n_{j}\pi,~{}n_{j}\in\mathbb{Z},$		(3)
	$\displaystyle\text{or}~{}\begin{cases}\underline{\mathbf{h}}_{1}\cdot\underline{\mathbf{h}}_{2}=\pm 1\\ T_{1}{E}_{1}\pm T_{2}{E}_{2}=n\pi,~{}n\in\mathbb{Z}\end{cases}$		(4)

at the quasienergy zero (or $\pi/T$) if $n$ is even (or odd) SMP . As the condition for the phase transition, Eqs. (3) and (4) supply a guideline to manipulate the exceptional points via the driving parameters for engineering various non-Hermitian FTPs at will. They reduce to the results in the Hermitian case Xiong et al. (2016); Liu et al. (2019b) as a special case when the non-Hermitian terms in ${\bf h}_{j}$ vanish.

We see from Eq. (2) that $\mathbf{h}_{\text{eff}}(k)$ generally has three components even though the chirally symmetric $\mathbf{h}_{j}$ have only two. It proves that the chiral symmetry is broken by the periodic driving SMP . Thanks to the similarity transformation $G_{j}$, we obtain $\tilde{\mathcal{H}}_{\text{eff},j}(k)$ preserving the chiral symmetry of $\mathcal{H}_{j}(k)$. Then we can restore the BBC and define proper topological invariants in our periodically driven non-Hermitian system by introducing the generalized BZ in the similar manner as the static system Yao and Wang (2018). The topological properties of the periodic non-Hermitian system are fully characterized by the two winding numbers $\mathcal{W}_{j}$ defined in the generalized BZ associated with $\tilde{\mathcal{H}}_{\text{eff},j}$. The number of $0$- and $\pi/T$-mode edge states relates to $\mathcal{W}_{j}$ as Xiao et al. (2020); Zhou and Gong (2018b)

$$N_{0}=|\mathcal{W}_{1}+\mathcal{W}_{2}|/2,~{}N_{\pi/T}=|\mathcal{W}_{1}-\mathcal{W}_{2}|/2.$$

(5)

Without loss of generality, we demonstrate our method by the 1D non-Hermitian SSH model Su et al. (1979); Zhu et al. (2014); Yin et al. (2018)

$\displaystyle H=\sum_{l=1}^{L}[(t_{1}+\frac{\gamma}{2})a^{{\dagger}}_{l}b_{l}+(t_{1}-\frac{\gamma}{2})b^{{\dagger}}_{l}a_{l}+t_{2}(a^{{\dagger}}_{l}b_{l-1}+\text{h.c.})],~{}~{}~{}$

(6)

where $a_{l}$ ($b_{l}$) are the annihilation operators on the sublattice $A$ ($B$) of the $l$th lattice, and $L$ is lattice length. In momentum space and the operator basis $(\tilde{a}_{k},\tilde{b}_{k})^{T}$ with $\tilde{a}_{k}$ ($\tilde{b}_{k}$) being the Fourier transform of $a_{l}$ ($b_{l}$), it reads

$$\mathcal{H}(k)=d_{x}\sigma_{x}+(d_{y}+i{\gamma}/{2})\sigma_{y},$$

(7)

where $d_{x}=t_{1}+t_{2}\cos k$ and $d_{y}=t_{2}\sin k$. The bands close at $k=\pi$ (or $0$) when $t_{1}=t_{2}\pm\gamma/2$ (or $-t_{2}\pm\gamma/2$). It is in conflict with the result under the open-boundary condition, where the bands close when $t_{1}=\sqrt{t_{2}^{2}+\gamma^{2}/4}$. It is called the non-Hermiticity caused breakdown of BBC Lee (2016); Leykam et al. (2017); Kunst et al. (2018); Yao and Wang (2018); Yao et al. (2018). The problem for the static system with chiral symmetry $\sigma^{-1}_{z}\mathcal{H}(k)\sigma_{z}=-\mathcal{H}(k)$ Chiu et al. (2016) is cured by the skin effect. Via replacing $e^{ik}$ by $\beta=\sqrt{\lvert\frac{t_{1}-\gamma/2}{t_{1}+\gamma/2}\lvert}e^{ik}$, Eq. (7) is converted into $\mathcal{H}(\beta)=\sum_{n=\pm}R_{n}(\beta)\sigma_{n}$ with $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$ and $R_{\pm}(\beta)=t_{1}\pm\frac{\gamma}{2}+\beta^{\mp 1}t_{2}$. Here $\beta$ defines a generalized BZ. Its topological property is described by the winding number $\mathcal{W}=-(\mathcal{W}_{+}-\mathcal{W}_{-})/2$, where $\mathcal{W}_{\pm}=\frac{1}{2\pi}[\arg R_{\pm}(\beta)]_{C_{\beta}}$ with $[\arg R_{\pm}(\beta)]_{C_{\beta}}$ are the phase change of $R_{\pm}$ as $\beta$ counterclockwisely goes along the generalized BZ $C_{\beta}$ Yao and Wang (2018); Yokomizo and Murakami (2019). When $|t_{1}|<\sqrt{t^{2}_{2}+\gamma^{2}/4}$, $\mathcal{W}=1$ and a pair of edge states is formed.

Choosing the periodic driving as

$$t_{2}(t)=\begin{cases}f,&t\in[mT,mT+T_{1})\\ q\,f,&t\in[mT+T_{1},(m+1)T),\end{cases}~{}m\in\mathbb{Z},$$

(8)

we now investigate the FTPs in our periodically driven non-Hermitian SSH model. Figure 1(a) shows the quasienergy spectrum under the open-boundary condition. It indicates that even the static system when $f=0$ is topologically trivial, diverse topological phases at the quasienergies $0$ and $\pi/T$ can be created by the periodic driving. However, this quasienergy spectrum has a dramatic difference from the one under the periodic-boundary condition, which takes $\sqrt{\mathbf{h}_{\text{eff}}(k)\cdot\mathbf{h}_{\text{eff}}(k)}$. It reveals that the non-Hermiticity induced breakdown of BBC occurs in our periodically driven system too. To solve this problem, we introduce the generalized BZ via replacing $e^{ik}$ in $\mathcal{H}_{\text{eff}}(k)$ by $\beta$ SMP . Then the effective Hamiltonian is converted to $\mathcal{H}_{\text{eff}}(\beta)$. First, $\mathcal{H}_{\text{eff}}(\beta)$ correctly explains the exceptional points of the quasienergies under the open-boundary condition. Remembering ${\bf h}(t)=[t_{1}+t_{2}(t)(\beta+\beta^{-1})/2,i[\gamma+t_{2}(t)(\beta^{-1}-\beta)]/2,0]$ and using Eqs. (3) and (4) by setting $t_{1}>\gamma/2>0$, we obtain the phase-transition conditions as follows.

Case I: $\underline{{\bf h}}_{1}\cdot\underline{{\bf h}}_{2}=1$. We can check that Eqs. (4) induce

$\displaystyle T_{1}|\kappa+e^{i\alpha}f|+T_{2}|\kappa+e^{i\alpha}qf|=n_{\alpha}\pi,~{}(n_{\alpha}\in\mathbb{Z})$

(9)

for $k$ in $\beta$ being $\alpha=0$ or $\pi$, where $\kappa=\sqrt{t_{1}^{2}-\gamma^{2}/4}$. Here $\text{sgn}[(\kappa-f)(\kappa-qf)]=1$ is further needed for $\alpha=\pi$.

Case II: $\underline{{\bf h}}_{1}\cdot\underline{{\bf h}}_{2}=-1$ requires $k=\pi$ when $\text{sgn}[(\kappa-f)(\kappa-qf)]=-1$. Then Eqs. (4) give

$\displaystyle T_{1}|\kappa-f|-T_{2}|\kappa-qf|=n_{\pi}\pi.$

(10)

Case III: According to Eq. (3), any $k$ in $\beta$ satisfying

$\displaystyle T_{1}E_{1}=n_{1}\pi,~{}T_{2}E_{2}=n_{2}\pi,~{}(n_{1},~{}n_{2}\in\mathbb{Z})$

(11)

contributes to the band closing.

Taking care of the skin effect via introducing $\beta$, Eqs. (9)-(11) perfectly describe the band closing of the quasienergy spectrum under the open-boundary condition. The $\pi/T$-mode band-closing points at $f\simeq 0.34\gamma$ and $2.27\gamma$ in Fig. 1(a) are obtainable from Eqs. (9) with $n_{0}=n_{\pi}=1$. The $0$-mode ones at $f\simeq 0.97\gamma$ and $1.65\gamma$ are obtainable from Eqs. (10) with $n_{\pi}=0$ and (9) with $n_{0}=2$, respectively. Thus the BBC has been successfully retrieved in our periodically driven system.

Second, the FTPs of the quasienergy spectrum under the open boundary condition are well characterized by the two winding numbers $\mathcal{W}_{j}$ defined in $\tilde{\mathcal{H}}_{\text{eff},j}$. According to Eq. (5), we plot in Figs. 1(b) and 1(c) the numbers of $0$-mode and $\pi/T$-mode edge states calculated from the conventional and generalized BZs. Although qualitatively capturing the exceptional points of the quasienergy under the periodic-boundary condition, the ill-defined topological numbers from the conventional BZ nonphysically take half integers. However, the ones from the generalized BZ correctly count the number of the edge states. It is called the non-Bloch BBC Yao and Wang (2018); Yokomizo and Murakami (2019). Note that, absent in the static system, such correspondence for the $\pi/T$-mode edge states is unique in our periodic system.

Third, the topological change of the quasienergy spectrum is reflected by $\mathcal{H}_{\text{eff}}(\beta)$. We plot in Fig. 2 the trajectories of $R_{\pm}$ in $\mathcal{\tilde{H}}_{\text{eff,1}}(\beta)$ when $f$ increases across the phase borders. Figures 2(a) and 2(b) show that $R_{\pm}$ have no wrapping to the origin and thus $\mathcal{W}_{1}=0$ before the $\pi/T$-mode phase transition. When $f$ increases across the critical point, $R_{\pm}$ at the neighbourhood of $k=0$ changes such that $\varepsilon=\sqrt{R_{+}R_{-}}$ crosses $\pi/T$. Due to its periodicity, $\varepsilon$ abruptly jumps to $-\pi/T$ keeping the direction of $R_{\pm}$ unchanged. Then an anticlockwise and a clockwise wrappings to the origin are formed by $R_{+}$ and $R_{-}$, respectively, and thus $\mathcal{W}_{1}=1$. Figures 2(c) and 2(d) show that $\mathcal{W}_{1}$ changes from 1 to 0, where $R_{\pm}$ at the neighbourhood of $k=\pi$ changes such that $\varepsilon$ crosses the quasienergy 0. This gives a geometric picture to the FTP transition in Fig. 1.

As a useful tool in controlling the exceptional points, the periodic driving enables us to realize not only the topological phases inaccessible in the same static-system condition, but also rich phases completely absent in its original static system. Figure 3 shows the phase diagram in the $T_{1}$-$T_{2}$ plane. A widely tunable number of $\mathcal{W}_{j}$ and edge states are induced by changing the driving parameters. The presence of such rich phases originates from the distinguished role of periodic driving in simulating an effective long-range hopping in different lattices Tong et al. (2013); Xiong et al. (2016); Liu et al. (2019b). The phase boundaries in red solid lines (black dashed lines) are perfectly described by Eq. (9) with $\alpha=0$ [by Eq. (10)]. $T_{2}E_{2}=\pi$ in Eqs. (11) is satisfied by $T_{2}\simeq 2.22/\gamma$. $T_{1}E_{1}=n_{1}\pi$ is satisfied by $T_{1}\simeq n_{1}\pi/(\gamma\sqrt{6+5.66\cos k})$. When $k$ runs from $0$ to $\pi$ for given $n_{1}$, a series line segments with a common $T_{2}\simeq 2.22/\gamma$ (see the blue dot-dashed line in Fig. 3) are formed, which all give the phase boundaries. We see that our analytical method successfully describes the FTPs in the periodically driven non-Hermitian system. The result reveals that, without changing the intrinsic parameters in the static system, the periodic driving supplies us another control dimension to adjust the numbers of the non-Hermitian topological edge states. This is useful in the application of non-Hermitian topological physics.

When the translation invariance of Eq. (6) is broken by the disorder $d\xi_{l}$ in the non-Hermitian term $\gamma$, where $\xi_{l}\in[-0.5,0.5]$ is the disorder in the $l$th cell with strength $d$, we cannot work in the momentum space anymore. The non-Bloch band theory is inapplicable too. However, we still may characterize the FTPs by the chirally symmetric $\tilde{\mathcal{H}}_{\text{eff},j}$ in the real space. Regarding $l\in[1,\ell]$ and $[L-\ell+1,L]$ of the chain as the boundaries, we define the real-space winding numbers Song et al. (2019b)

$$\mathcal{W}_{j}^{\prime}=\frac{1}{2L^{\prime}}\text{Tr}^{\prime}(SQ_{j}[Q_{j},X]).$$

(12)

Here $S_{ls,l^{\prime}s^{\prime}}=\delta_{ll^{\prime}}(\sigma_{z})_{ss^{\prime}}$ is the chiral operator and $X_{ls,l^{\prime}s^{\prime}}=l\delta_{ll^{\prime}}\delta_{ss^{\prime}}$ with $s,s^{\prime}=A,B$ being the sublattices, $Q_{j}=\sum^{\prime}_{n}(|n^{\text{R}}_{j}\rangle\langle n^{\text{L}}_{j}|-S|n^{\text{R}}_{j}\rangle\langle n^{\text{L}}_{j}|S^{\dagger})$ with $\tilde{\mathcal{H}}_{\text{eff},j}|n_{j}^{\text{R}}\rangle=\varepsilon_{j,n}|n_{j}^{\text{R}}\rangle$ and $\tilde{\mathcal{H}}^{\dagger}_{\text{eff},j}|n_{j}^{\text{L}}\rangle=\varepsilon^{*}_{j,n}|n_{j}^{\text{L}}\rangle$, and $\text{Tr}^{\prime}$ denotes the trace over the middle interval with length $L^{\prime}=L-2\ell$ and $\sum_{n}^{\prime}$ denotes the summation to the bulk states. We can check that $\mathcal{W}_{j}^{\prime}$ return to $\mathcal{W}$ when $d=0$. Thus, more general than $\mathcal{W}_{j}$, the real-space $\mathcal{W}_{j}^{\prime}$ can give a unified description to the FTPs of the non-Hermitian system both for the translation-invariant and variant cases.

Figure 4 shows the winding numbers $\mathcal{W}_{j}^{\prime}$ and the quasienergies with the change of the disorder strength. We can see from Figs. 4(a) and 4(c) that the topological trivial character of the disorder-free case is robust when the disorder is weak for $d\lesssim 2t_{1}$. With the increase of $d$, it is remarkable to find that a $0$-mode edge state is triggered in a wide range $d\in(2,10)t_{1}$. The disorder-induced edge state has been found in static Hermitian systems Li et al. (2009); Meier et al. (2018); Guo et al. (2010); Titum et al. (2016). Analogous to that, we call the similar state occurred in our periodically driven non-Hermitian system as Floquet topological Anderson insulator phase. Its presence can be further confirmed by Figs. 4(b) and 4(d), where a $\pi/T$-mode edge state exists in the disorder-free case. Here, it is interesting to observe a coexisted regime of the $\pi/T$-mode edge state and 0-mode Floquet topological Anderson insulator state. Both of the states are absent in the static system. However, in the strong disorder regime, the bands close and all the edge states disappear, which is compatible to the result in the Hermitian case Shtanko and Movassagh (2018).

Note the chiral symmetry is not recoverable in some driving cases Zhang et al. (2017), where the non-Hermitian FTPs can be described by a $Z_{2}$ topological invariant SMP . Our result is realizable in the present experimental state of art of photonics, where the non-Hermitian topological phases of the SSH model Zeuner et al. (2015); Zhu et al. (2020) and the Hermitian FTPs Rechtsman et al. (2013); Cheng et al. (2019) have been observed.

We have investigated the topological phases in periodically driven non-Hermitian systems. A scheme is proposed to retrieve the BBC, based on which a complete description to the FTPs is established for such non-Hermitian systems both with and without the translation invariance. Taking the SSH model as an example, we have found that diverse exotic FTPs can be created from the topologically trivial static system by the periodic driving. Further study reveals that the Floquet topological Anderson topological insulator phases can be triggered by the moderate-strength disorder. Exhibiting a wide perspective of controllability, our results hopefully promote further studies of both fundamental physics and potential applications of rich non-Hermitian FTPs.

The work is supported by the National Natural Science Foundation (Grant Nos. 11875150 and 11834005) and the Fundamental Research Funds for the Central Universities of China.