Non-Hermitian Floquet Topological Matter—A Review

The past few years have witnessed a surge of interest in non-Hermitian Floquet topological matter due to its exotic properties resulting from the interplay between driving fields and non-Hermiticity. The present review sums up our studies on non-Hermitian Floquet topological matter in one and two spatial dimensions. We first give a bird’s-eye view of the literature for clarifying the physical significance of non-Hermitian Floquet systems. We then introduce, in a pedagogical manner, a number of useful tools tailored for the study of non-Hermitian Floquet systems and their topological properties. With the aid of these tools, we present typical examples of non-Hermitian Floquet topological insulators, superconductors, and quasicrystals, with a focus on their topological invariants, bulk-edge correspondences, non-Hermitian skin effects, dynamical properties, and localization transitions. We conclude this review by summarizing our main findings and presenting our vision of future directions.

With all these developments, a natural follow-up is to consider the system in a more general situation, in which it is subject to both time-periodic drivings and non-Hermitian effects. Such non-Hermitian Floquet systems may possess exotic dynamical phenomena and topological phases with no static or Hermitian analogies. On the theoretical side, the investigation of driven non-Hermitian systems may lead to the discovery of new topological states and bring about the improvement of classification schemes for nonequilibrium phases of matter in general. On the practical side, the exploration of non-Hermitian Floquet matter is helpful to the design of new approaches for preparing or stabilizing topologically nontrivial states and controlling material properties. It also stimulates new ideas for realizing quantum devices and quantum computing protocols that are robust to perturbations caused by the environment. Though still at the early stage, many progresses have been made in the realization and characterization of non-Hermitian Floquet phases . In this review, we limit our scope to the discussion of a number of typical topological phases we discovered in non-Hermitian Floquet systems [254][255][256][257][258][259][260][261][262][263][264]. In Sec. II, we give a pedagogical introduction to some key aspects of Floquet systems, including their dynamical and topological characterizations. In Sec. III, we present typical examples of non-Hermitian Floquet topological insulators, superconductors, quasicrystals and summarize their main physical properties, with a focus on the features that are unique to driven non-Hermitian systems. In Sec. IV, we conclude this review, briefly mention some relevant studies and discuss potential future work.

II. BACKGROUNDS
We start with a recap of the basic description of a non-Hermitian Floquet system. The Hamiltonian of such a system satisfiesĤ(t) =Ĥ(t + T ), and there exists t ∈ [0, T ] such that H(t) ̸ = [Ĥ(t)] † . Here t denotes time and T is the driving period. The state of the system evolves according to the Schrödinger equation where we have set ℏ = 1. We first show that this equation can be solved by Floquet states even thoughĤ(t) is non-Hermitian. This is followed by different ways of obtaining the Floquet states in general situations, in the high-frequency regime, and in the adiabatic regime.
We next discuss the symmetry, topological invariants, and dynamical characterizations of non-Hermitian Floquet states, with a focus on the types of physical systems explored in our previous studies. We conclude this section by presenting some tools for characterizing the spectrum properties and localization transitions in non-Hermitian Floquet disordered systems.
It also implies that The latter equation indicates that the only change of the state |Ψ ε (t)⟩ after undergoing a oneperiod evolution is to pick up an exponential factor e −iεT . We refer to the set {|Ψ ε (t)⟩|Re(ε) ∈ [−π/T, π/T )} as Floquet eigenstates of the system. They form an orthonormal and complete basis at each instant of time t.
Since the evolution from |Ψ ε (t)⟩ to |Ψ ε (t + T )⟩ is governed by the Schrödinger equation, we can also express Eq. (8) asÛ (t + T, t)|Ψ ε (t)⟩ = e −iεT |Ψ ε (t)⟩, whereÛ (t + T, t) =Te −i´t +T tĤ (t ′ )dt ′ is nothing but the Floquet operator (evolution operator over one driving period) of the system. When we are only concerned with the stroboscopic dynamics, the initial time dependence of Eq. (9) is not important. In this case, we can set t = 0 in Eq. (9) and express the Floquet eigenvalue equation aŝ whereÛ =Te −i´T 0Ĥ (t)dt and we have introduced E = εT as the dimensionless quasienergy, whose first BZ is given by [−π, π). To sum up, we find that the solution of The resulting state after an evolution over n driving periods is then given by Note that for a non-HermitianĤ(t),Û is generally non-unitary and the quasienergy E may have a nonvanishing imaginary part. In this case, the real part of E still belongs to the range of [−π, π) and our arguments leading to the general solution (11) can be satisfied.

B. Floquet eigenvalue equation
In the most general situations, we can solve Eq. (10) numerically by the split-operator method [3]. Dividing the evolution periodic T into N segments with a large enough N , we can express the Floquet operatorÛ approximately aŝ where ∆t = T /N . Over each small time interval ∆t,Ĥ(t) is approximately time-independent and we can diagonalize it numerically at t = ℓ∆t aŝ Each column of V ℓ represents a right eigenvector ofĤ(ℓ∆t). The evolution operator over the one-time interval ∆t then takes the form The multiplication of all the V ℓ e −iD ℓ ∆t V −1 ℓ from right to left for ℓ = 0 to N −1 yields Eq. (12), which further converges to the exact Floquet operator in the limit N → ∞. We can thus numerically solve the Floquet eigenvalue equation by diagonalizing the approximatedÛ in Eq. (12). This approach works in principle for systems with any individual or multiple driving frequencies. But it may become time-consuming in practice for certain continuously or slowly driven systems.
When the driving field takes the form of periodic kicking or quenching, the series in Eq. (12) can be greatly simplified and even obtained exactly. Here we give several examples.
Consider a time-periodic Hamiltonian of the form where δ(t/T − ℓ) is the delta function peaked at t = ℓT , i.e., each integer multiple of the driving period. The dynamics over each driving period thus constitutes a free evolution part controlled byĤ 0 and a delta kicking force controlled byĤ 1 . The quantum kicked rotor is one representative example of such a system [28]. Focusing on the one-period evolution from Similarly, if there are two kicks separated by a time interval τ within each driving period, the Hamiltonian could take the form of For the evolution from time t = ℓT − 0 + to t = (ℓ + 1)T − 0 + , the Floquet operator now takes the form ofÛ One typical example of such a system is the double-kicked quantum rotor [24]. For a periodically quenched Hamiltonian in the form of where T = T 1 + T 2 , we can also directly obtain the corresponding Floquet operator from The discrete-time quantum walk can be viewed as one example of such a periodically quenched system [31]. Time-periodic quenches are also frequently implemented in the study of discrete time crystals [19][20][21]. When [Ĥ 1 ,Ĥ 2 ] ̸ = 0, the quenches (or kicks) may effectively generate long-range coupling in the system according to the Baker-Campbell-Hausdorff formula, leading to Floquet phases with large topological invariants and many boundary states.
This point will be explicitly demonstrated by the examples discussed in Sec. III.
For a continuously driven system, the solution of the Floquet eigenvalue equation may also be obtained approximately in terms of the frequency (Sambe) space formalism [267][268][269].
Inserting the Floquet state in Eq. (7) into the Schrödinger equation (1) and reorganizing the terms, we find Using the Fourier expansion ofĤ(t) and |u ε (t)⟩ = n e −iωnt |u ε (ω n )⟩, we further obtain Multiplying 1 T e iω ℓ t from the left on both sides of Eq. (22) and performing the integral over a driving period T , we arrive at whereĤ and Note that eachĤ m−n has the same Hilbert space dimension d as the original Hamiltonian H(t). As an example, for a harmonically driven system described by the Hamiltonian In practical calculations, one should truncate the infinite-dimensional matrix in Eq. (25) at a sufficiently high harmonics N ω, leading to an N d × N d dimensional Floquet effective Hamiltonian, whose eigenvalue problem can be numerically solved. Assuming the characteristic energy scale ofĤ m−n for all m − n to be Ω, we can take a smaller N to do the truncation for a larger ratio of ω/Ω, i.e., for a high-frequency driving field. Instead, for a resonantly or slowly driven system, more harmonics should be kept during the truncation.
All the discussions presented in this subsection hold for both Hermitian and non-Hermitian HamiltoniansĤ(t).

C. Floquet effective Hamiltonian and high-frequency expansion
From the Floquet operatorÛ of a periodically driven system, one can formally define its Floquet effective Hamiltonian aŝ Taking into account the fact that the quasienergies are defined modulus 2π/T , theĤ eff contains the same physical information asÛ . Yet, it provides us with more room to treat the properties of Floquet systems in analogy with static Hamiltonian models. For a continuously driven system, the explicit form ofĤ eff is usually involved. This can be inspected from Eq. (12), as theĤ(ℓ∆t) andĤ(m∆t) do not commute for ℓ ̸ = m in general. When the frequency of the driving field is high enough, an approximate series expression forĤ eff can be obtained via high-frequency expansion methods [61][62][63][64]. Here we recap one such method in its full generality, which is applicable to both Hermitian and non-Hermitian systems.
We first assume that the time-periodic HamiltonianĤ(t) can be decomposed into a static partĤ 0 and a periodically modulated partV (t), i.e., Here T = 2π/ω is the driving period with ω denoting the driving frequency. Next, we apply a similarity transformation to the evolved state |Ψ(t)⟩ in the Schrödinger equation (1), yielding a rotated state HereK(t) is sometimes called the kick operator. It encodes the information regarding the micromotion dynamics of the system. Plugging |Ψ(t)⟩ = e −iK(t) |Φ(t)⟩ into Eq. (1) leads to the transformed Schrödinger equation The aim of the high-frequency expansion method is to find a time-independentĤ eff by transferring all time-dependent terms into the kick operatorK(t). When such a purpose is formally achieved, we can express the Floquet evolution of a system aŝ That is, the system is subject to an initial kick associated with the operatorK(t 0 ), then evolved under the time-independentĤ eff , and finally subject to a second kick carried out by the operatorK(t 1 ). There is no need to perform any time-ordered integral in the calculation ofÛ (t 1 , t 0 ).
Assuming the driving frequency ω to be large, we may expandĤ eff andK(t) into power In the meantime, we can apply Taylor expansions to the two terms in Eq. (32) and expresŝ Note that we have concealed the explicit time-dependence ofK(t) andK (n) (t) in the above two equations for brevity. To proceed, we impose two further requirements forK(t), Inserting Eq. (34) and the Fourier expansion into Eq. (35), we arrive at The high-frequency expansions ofĤ eff andK are obtained by equating both sides of Eq. (38) at each order of 1/ω.
We can now carry out the calculations order-by-order to find the first few terms in the series ofĤ eff andK. At the zeroth order, we could obtain from Eq. (38). To ensure thatĤ (0) eff is time-independent, we must choose such thatK Therefore, up to the zeroth order of 1/ω, we have the following approximation for the effective HamiltonianĤ Up to the first order of 1/ω, we have the following approximation for the kick operator At the first order, we could find from Eq. (38) that By definition, any non-vanishing elements of i K (1) ,Ĥ 0 and − 1 ω ∂ tK (2) are time-dependent.
The only time-independent term that could contribute toĤ (1) eff comes from the term , which can be written explicitly as The time-independent terms are those with m ′ = −m. Collecting these terms together, we Therefore, up to the first order of 1/ω, we have the following approximation for the effective The form ofK (2) is determined by the remaining time-dependent terms. Performing the integration over time directly, we obtain Up to the second order of 1/ω, we can thus approximate the kick operator bŷ We could continue these self-consistent calculations to obtain higher-order terms in the high-frequency expansion ofĤ eff andK(t). For example, for the second-order component Dropping the time-dependent terms, we are left witĥ Therefore, up to the second-order correction in 1/ω, the effective HamiltonianĤ eff turns out to beĤ This approximation holds for both Hermitian and non-Hermitian Floquet systems under high-frequency driving fields. Note that for stroboscopic evolution, we have t 1 − t 0 = N T with N ∈ Z andK(t 1 ) =K(t 0 ) in Eq. (33). In this case, the termK(t 0 ) in Eq. (33) describes an initial phase, which can be set to zero as the expansion ofK(t) at every order of 1/ω is only determined up to a constant [see Eqs. (41) and (48)]. Therefore, for stroboscopic dynamics, we can simply use theĤ eff to capture most of the essential physics. In Sec. III, we will showcase the application of the high-frequency methods presented here to the Floquet engineering of non-Hermitian quasicrystals.
To be concrete, we illustrate the usage ofĤ eff here with a simple example. Consider a harmonically driven two-level Hamiltonian of the form where h x , h z , γ ∈ R, and σ α for α = x, y, z are Pauli matrices. It is not hard to verify that by choosing the kick operator as the Floquet effective Hamiltonian in the rotating frame can be exactly obtained from where σ 0 denotes the 2 × 2 identity matrix. We see that the two levels of H(t) have the instantaneous eigenenergies which are time-independent and complex in general. They could meet with each other at a second-order EP with E = 0 when h x = 0 and h z = ±γ/2. Meanwhile,Ĥ eff in Eq. (55) has two quasienergy levels defined by They could meet with each other at a second-order Floquet EP with ε = ω/2 when h x = 0 and h z = ω/2 ± γ/2. It is clear that the EP is shifted by the driving in both its energy and location in the parameter space. If we define εT as the dimensionless quasienergy E, the Floquet EP of our two-level system appears at the quasienergy E = π. This could lead to Floquet exceptional topology and anomalous π edge modes, which are unique to driven non-Hermitian systems. We will give an explicit example to demonstrate this point in Sec. III.

D. Adiabatic perturbation theory
We now consider a non-Hermitian system that is subject to slow-in-time and cyclic modulations. It can be viewed as the opposite limit of a high-frequency driven system. Introducing a re-scaled and dimensionless time variable s = t/T , we can express the Schrödinger equation Our purpose is to find an expansion for |Ψ(s)⟩ in the series of 1/T . It constitutes an adiabatic perturbation theory (APT) of the system [270], providing that different energy levels ofĤ ⟨ m(s)|n(s)⟩ = δ mn , n |n(s)⟩⟨ n(s)| = 1.
Here E n (s) is the instantaneous eigenenergy associated with |n(s)⟩, which could be complex ifĤ(s) ̸ =Ĥ † (s). To proceed, we introduce an ansatz solution for the time-evolved state where Ω n (s) ≡ Tˆs where Ω mn (s) ≡ Ω m (s) − Ω n (s). We have made the parallel transport gauge choice so that ⟨ n(s)|ṅ(s)⟩ = 0. Performing the integration over s and keeping the terms on the right-handside up to the first order of 1/T (i.e., up to the first order non-adiabatic correction), we arrive at where ∆ mn (s) = E m (s)−E n (s).
Since we have employed the formalism of biorthonormal eigenvectors, we may consider another ansatz solution for the left eigenvector which obeys a conjugate Schrödinger equation Following the same reasoning in deriving Eq. (67), we find, up to the first order of 1/T , that Assuming that initially only the state with m = ℓ is occupied such that c m (0) = δ mℓ , we can further simplify Eqs. (67) and (70) to For any observableÔ, up to the correction of order 1/T , we can now express its biorthogonal We notice that this average does not contain any time-oscillating phase factors.
We now illustrate this APT with an application in the study of dynamical topological phenomena. Let us consider noninteracting particles in a one-dimensional (1D) periodic lattice, whose onsite potential is also varied slowly and periodically in time. This is the typical situation encountered in the topological Thouless pump [16][17][18]. The group velocity of the particle can be expressed asv = ∂ kĤ , where k is the quasimomentum. At the initial time s = 0, we assume that the band ℓ is uniformly filled, and it is separated from the other bands at all k and s. The pumped number of particles over one adiabatic cycle due to this initially filled band is then given by With the aid of Eq. (59), it is not hard to identify that where E ℓ (k, s) denotes the energy dispersion of the ℓth adiabatic Bloch band. Plugging Eqs. (75), (76), and (73) into Eq. (74), we obtain Noting that E ℓ (k = −π, s) = E ℓ (k = π, s) and m |m(k, t)⟩⟨ m(k, t)| = 1, we can simplify the above equation and arrive at the pumped number of particles over an adiabatic cycle Equation (78) describes nothing but the Chern number of the adiabatic Bloch band ℓ, whose The conditions for Eq. (78) to hold are as follows. First, the Hamiltonian of the system should be quasi-Hermitian with a real spectrum (e.g., PT-invariant) in our considered parameter regime. Second, the band E ℓ should be well-gapped from the other bands throughout the 2D torus (k, t) ∈ [−π, π) × [0, T ), and ℏ/T should be much smaller than |∆ ℓm | for all m ̸ = ℓ in order to guarantee the adiabatic condition. Third, the evolution of left and right vectors of the system should follow Eqs. (58) and (69). Note here that the expression of Chern number is not sensitive to the choice of biorthonormal basis. We will get the same N ℓ after changing ℓ → ℓ or ℓ → ℓ in Eq. (78), as proved before for non-Hermitian Chern bands [139].

E. Symmetry and topological characterization
Over the past few years, rich symmetry classifications and topological invariants have been identified for non-Hermitian topological matter [139][140][141][142][143][144][145][146][147]. In this subsection, we mainly recap two symmetries together with their associated topological numbers. They are the most relevant ones for the characterization of non-Hermitian Floquet topological phases reviewed in this work.
We first discuss the PT-symmetry, which is associated with the operator PT . Here The PT-symmetry ofĤ then implies that Therefore, PT |ψ⟩ is also an eigenstate ofĤ with the energy E * . If |ψ⟩ andĤ share the same PT-symmetry, |ψ⟩ should be the common eigenstate ofĤ and PT . PT |ψ⟩ can thus only differ from |ψ⟩ up to a global phase, which means that E = E * ∈ R. However, with the change of system parameters (e.g., the strengths of gain and loss), the PT-symmetry of |ψ⟩ could be spontaneously broken and the spectrum ofĤ could switch from real to complex after undergoing a PT-symmetry breaking transition. A topological invariant, defined as [188] w =ˆ2 might be employed to characterize such a real-to-complex spectral transition. Here E 0 is a base energy chosen appropriately on the complex plane. The parametrized Hamiltonian Ĥ (θ) =Ĥ(θ + 2π), where θ can be viewed as the quasimomentum along an artificial dimension. We have also taken the periodic boundary condition (PBC) forĤ before implementing its θ-parametrization. The w in Eq. (81) thus depicts a spectral winding number with respect to the base energy E 0 , i.e., it counts the number of times that the spectrum ofĤ(θ) winds around E 0 on the complex plane when the synthetic quasimomentum θ is varied over a cycle. When the spectrum ofĤ(θ) is real, we must have w = 0, as a spectral loop cannot be formed on the complex plane in this case. When the spectrum ofĤ(θ) is complex, w may take an integer-quantized value ifĤ(θ) possesses spectral loops around E 0 . A suitably chosen E 0 could then yield a nonzero w when the first spectral loop appears on the complex-E plane, thereby detecting the topological changes in the spectrum ofĤ across the PT-breaking transition. We next consider the chiral (or sublattice) symmetry, whose associated operator will be denoted by S. When the HamiltonianĤ of a system respects the chiral symmetry S, where S is both Hermitian and unitary [272]. The implication of this symmetry on the spectrum ofĤ is as follows. Suppose that |ψ⟩ is an eigenstate of a chiral-symmetricĤ with the energy E, i.e.,Ĥ|ψ⟩ = E|ψ⟩. We havê Therefore, S|ψ⟩ is also an eigenstate ofĤ with the energy −E. The eigenstates of a chiral-symmetricĤ should then come in pairs of {|ψ⟩, S|ψ⟩} with the energies {E, −E} that are symmetric with respect to E = 0. This further leads to a chiral-symmetry protected degeneracy for any eigenstate with E = 0. When the spectrum ofĤ is gapped at E = 0, we can group its energy levels into two clusters with ReE < 0 and ReE > 0. If these two clusters meet with each other at E = 0 and then separate with the change of certain system parameters, the system may undergo a phase transition. In one dimension, the change of band topology of the system before and after such a transition could be characterized by a winding number w 0 , defined as [272][273][274] Here the PBC has been assumed and k ∈ [−π, π) denotes the quasimomentum. The signresolved projector Q(k) is obtained from the spectral decomposition ofĤ = k |k⟩H(k)⟨k| with H(k) = n E n (k)|n(k)⟩⟨ n(k)| at each k by attributing +1 (−1) to every energy band n with ReE n > 0 (ReE n < 0). Q(k) can thus be expressed as [261] Here Here S is the chiral symmetry operator ofĤ,N is the position operator in real space, and Q is the flat band projector defined as where the summation is now taken over all the bulk eigenstates {|ψ j ⟩} ofĤ under the OBC.
The whole lattice of length L = L B + 2L E is decomposed into three segments, with a bulk region of length L B in the middle and two edge regions of the same length L E at the left and right boundaries of the open chain. The trace Tr B (·) is only taken over the bulk region, which excludes all possible interruptions caused by the NHSE in the edge regions. The resulting W 0 was found to be able to faithfully capture the topological phase transitions and bulkedge correspondence in 1D, chiral symmetric non-Hermitian systems even in the presence of NHSE [168]. It was also suggested to be equivalent to the topological winding number defined through the generalized Brillouin zone of non-Hermitian systems. By definition, the winding number W 0 in Eq. (85) is also robust to perturbations induced by symmetrypreserved disorders and impurities, making it applicable to more general situations. In the clean, Hermitian, and thermodynamic limit, the W 0 in Eq. (85) can be further reduced to the w 0 in Eq. (83) [272][273][274].
For a non-Hermitian Floquet system, we can state its chiral symmetry as follows [254].
From Eq. (28), we can express the Floquet operator asÛ = e −iĤ eff , where we have set the driving period T = 1 for brevity. Viewing theĤ eff as a static Hamiltonian, we say that it respects the chiral symmetry if there exists a unitary and Hermitian operator S such that SĤ eff S = −Ĥ eff . At the level ofÛ , the chiral symmetry then implies that As in the case of static systems, the chiral symmetry ofÛ has a direct implication for the symmetry of its spectrum. If |Ψ E ⟩ is an eigenstate ofÛ with the quasienergy E, i.e., which means that S|Ψ E ⟩ is also an eigenstate ofÛ with the quasienergy −E. The Floquet spectrum ofÛ is then symmetric with respect to both the quasienergies E = 0 and E = π.
The latter is because −E and E are identified at the quasienergy π. When the spectrum is gapped at E = 0 and π, the quasienergy levels ofÛ could be grouped into two clusters.
One of them has the quasienergy ReE ∈ (−π, 0) and the other one has ReE ∈ (0, π). They could meet with each other at either the quasienergy zero or π, leading to two possible phase transitions. This implies that a complete topological characterization of a chiral symmetric Floquet system should require at least two winding numbers, which is rather different from the case of static systems where a single winding number is sufficient. To identify these winding numbers, let us consider the example of a periodically kicked 1D system, whose Hamiltonian and Floquet operator take the forms of Eqs. (15) and (16). We also assume that the two parts of HamiltoniansĤ 0 andĤ 1 in Eq. (15) have the same chiral symmetry S. At the level ofÛ = e −iĤ 0 e −iĤ 1 (assuming T = 1), it is not straightforward to identify the form of a chiral symmetry. However, we can apply similarity transformations toÛ and express it in two symmetric time frames [69] aŝ It is then clear that SÛ α S =Û −1 α for α = 1, 2, that is, the Floquet operatorsÛ 1 andÛ 2 in the two symmetric time frames respect the same chiral symmetry S. We can thus introduce a winding number for each of them under the PBC as [254] where k ∈ [−π, π), and The E n (k) in Eq. (92) now denotes the quasienergy of the Floquet eigenstate |n α (k)⟩ ofÛ α at the quasimomentum k under the PBC. Using the w 1 and w 2 , we can construct another pair of winding numbers w 0 and w π , given by [254] In Subsecs. III B and III C, we will demonstrate with explicit examples that the w 0 (w π ) could correctly capture the bulk topological transitions of non-Hermitian Floquet bands through the gap closing/reopening at the quasienergy E = 0 (E = π) in various chiral symmetric, non-Hermitian Floquet insulating and superconducting models. Furthermore, in the absence of NHSE, the w 0 and w π could also capture the numbers of Floquet edge modes at zero and π quasienergies under the OBC, and thus are capable of describing the bulk-edge correspondence of the related models. In the presence of NHSE, we can retrieve the characterization of topological transitions and bulk-edge correspondence in chiral symmetric, Here, the meanings of L B , Tr B , S andN are the same as those in Eq. (85). The Floquet band projector in the time frame α is given by where |ψ α j ⟩ is the jth bulk eigenstate ofÛ α (α = 1, 2) with the quasienergy E j under the OBC. The linear combinations of W 1 and W 2 lead to another pair of winding numbers [261] In Subsec. III B, we will illustrate that with the help of the winding numbers (w 0 , w π ) and (W 0 , W π ), a dual topological characterization of the phase transitions, edge states, and bulk-edge correspondence can be established for 1D, chiral symmetric non-Hermitian Floquet systems under different boundary conditions, regardless of whether the NHSE is present or not [261]. Interestingly, the winding numbers (w 0 , w π ) may both become halfinteger quantized due to the presence of Floquet EP in the bulk, thus revealing the presence of Floquet exceptional topology. Meanwhile, the open-bulk winding numbers (W 0 , W π ) are always integer quantized.

F. Dynamical indicators
In this subsetion, we review two complementary dynamical probes in position and momentum spaces. Both of them can be used to characterize the topological properties of 1D non-Hermitian Floquet systems with chiral symmetry. These indicators allow us to extract the topological winding numbers of the system from its long-time stroboscopic dynamics [254][255][256]. The measurement of these indicators could thus provide evidences for the existence of non-Hermitian Floquet topological matter.

Dynamic winding number (DWN)
The DWN [275,276], obtained from the long-time stroboscopic average of spin textures, could provide us with information about the bulk topological properties of non-Hermitian Floquet systems [256]. Let us consider a 1D, chiral symmetric non-Hermitian Floquet system with two quasienergy bands. Under the PBC, we can express its Floquet operator in the Here n = ± are the indices of the two Floquet bands with the quasienergies E ± (k) ≡ ±E(k).
The biorthonormal relationship requires We consider the case in which the system is prepared in a general initial state |ψ α (k, 0)⟩ = n=± c n (k)|n α (k)⟩. The corresponding initial state in the left Hilbert space reads | ψ α (k, 0)⟩ = n± c n (k)| n α (k)⟩, such that initially n=± |c n (k)| 2 = 1 at each k. After the stroboscopic evolution over a number of ℓ driving periods, the right initial state becomes For the left initial state, we assume it to be evolved by a different effective Hamiltonian so that after the evolution over ℓ driving periods it reaches the state Note that the dynamical equation of | ψ α (k, 0)⟩ we used here is different from that employed in our study of the APT in Subsec. II D.
The stroboscopic average of an observableÔ over | ψ α (k, t)⟩ at the time t = ℓT is then given by Without the loss of generality, we can consider the chiral symmetric H α (k) to be in the form Note that any two out of the three Pauli matrices (σ x , σ y , σ z ) can be chosen to enter the H α (k), and the rest Pauli matrix [e.g., σ z for the H α (k) in Eq. (103)] plays the role of the chiral symmetry operator S. By diagonalizing H α (k), we obtain the biorthonormal eigenvectors as where E n (k) = n h 2 αx (k) + h 2 αy (k) for n = ±. We can now compute the stroboscopicaveraged spin textures in the long-time limit. For the H α (k) in Eq. (103), this means that we need to find the multi-cycle averages of σ x and σ y . According to Eq. (102), they are given by where j = x, y and α = 1, 2. N counts the total number of driving periods. From the averaged spin textures [r α x (k), r α y (k)], we can define the dynamic winding angle as The net winding number of θ α yx (k) over a cycle in k-space defines the dynamic winding number (DWN) in the time frame α, i.e., In Ref. [256], it was proved with straightforward calculations that in the limit N → ∞, the ν α converges to the w α in Eq. (91) if the initial condition satisfies c ± (k) ̸ = 0 at each k. Therefore, by preparing the initial state at different k under this condition and measuring the averaged spin textures over a long stroboscopic time, we can obtain the winding numbers (w 0 , w π ) of a chiral symmetric 1D Floquet system (either Hermitian or non-Hermitian) through the following combinations of (ν 1 , ν 2 ) in two symmetric time frames [256], i.e., In Sec. III, we will illustrate the application of this dynamical-topological correspondence to non-Hermitian Floquet topological insulators in one dimension. We will see that both integer and half-integer quantized topological winding numbers can be extracted from the DWN.

Mean chiral displacement (MCD)
The MCD allows us to detect the winding numbers of a Floquet system from the longtime averaged chiral displacement of an initially localized wavepacket. It was first proposed as a means to probe the topological invariants of chiral symmetric topological insulators in one dimension [277][278][279]. Later, the MCD was used to obtain the winding numbers of Floquet systems [78,81] and also generalized to 2D higher-order topological insulators [80].
Its applicability was demonstrated both theoretically and experimentally [277,279].
For a chiral symmetric non-Hermitian system, the chiral displacement in the symmetric time frame α (= 1, 2) can be defined as Here, t = ℓT denotes the stroboscopic time, with T being the driving period.N is the unit cell position operator and S is the chiral symmetry operator. The initial state ρ 0 can be chosen as a state localized in the middle of the lattice. For example, ρ 0 may take the form of (|0⟩⟨0| ⊗ σ 0 )/2 for a 1D bipartite lattice, where |0⟩ is the eigenbasis of the central unit cell and σ 0 is the identity operator acting on the internal space of the two sublattices.
Both the Floquet operatorÛ α and its dualˆ U α respect the chiral symmetry S. In the lattice representation, they can be expressed aŝ Here L denotes the total number of degrees of freedom of the lattice. |ψ α j ⟩ and ⟨ ψ α j | denote the right and left eigenvectors ofÛ α with the quasienergy E j . They form a biorthonormal basis such that Note thatˆ U α is not the Hermitian conjugate ofÛ α in general.
We now consider the stroboscopic long-time average of C α (t) in the time frames α = 1, 2 for a 1D non-Hermitian Floquet system with chiral symmetry. Under the PBC, taking the initial state to be ρ 0 = (|0⟩⟨0| ⊗ σ 0 )/2 and performing the Fourier transformation from position to momentum representations, we find the C α (t) in Eq. (109) to be [255] C Here k ∈ [−π, π) is the quasimomentum. U α (k) = ⟨k|Û α |k⟩ and U α (k) = ⟨k|ˆ U α |k⟩ act on the internal degrees of freedom (spins and/or sublattices) of the system at a fixed k. Taking the long-time stroboscopic average and incorporating the normalization factor we find the expression of MCD as [255] C α = lim For a given H α (k) of the form in Eq. (103), we have S = σ z , U α (k) = e −iHα(k) , and U α (k) = One can then work out Eq. (113) explicitly and obtain [255] Here the w α is nothing but the winding number ofÛ α as defined in Eq. (91). Therefore, by measuring the MCDs (C 1 , C 2 ) in two symmetric time frames, we can determine the topological winding numbers (w 0 , w π ) of a 1D chiral symmetric Floquet system through the relations [255] w The MCD provides a real-space complementary to the DWN. In Sec. III, we will demonstrate the usage of MCD to dynamically probing the winding numbers of first-and second-order non-Hermitian Floquet topological insulators in one [255] and two [259] spatial dimensions.

G. Localization transition and mobility edge
In the last part of this subsection, we recap some tools that can be used to characterize the real-to-complex spectral transitions, localization transitions, and mobility edges in disordered non-Hermitian Floquet systems [262,264]. Considering the relevance to this review, we focus on a 1D quadratic lattice model of the form Here ⟨n, n ′ ⟩ includes the lattice site indices n and n ′ with n ′ > n.ĉ † n (ĉ n ) creates (annihilates) a particle on the site n and the parameter γ ∈ R. The HamiltonianĤ(t) is non-Hermitian if γ n ̸ = 0 for some n (nonreciprocal hopping) or V n (t) ̸ = V * n (t) (onsite gain and loss).Ĥ(t) is further time-periodic if J n (t) = J n (t + T ) and V n (t) = V n (t + T ) for all n, with T being the driving period. The disorder terms may be included within V n (t) (diagonal disorder) or J n (t) (off-diagonal disorder). As an example, for a 1D non-Hermitian quasicrystal (NHQC) with correlated onsite disorder, the V n (t) may take the form of V n (t) = V (t) cos(2παn + iβ). Here α is irrational and β ∈ R, with iβ describing an imaginary phase shift in the superlattice potential V n .
The Floquet operator of the system described by theĤ(t) in Eq. (116) Here E j is the jth quasienergy eigenvalue ofÛ . L counts the Hilbert space dimension of U , i.e., the total number of degrees of freedom of the lattice. N (ImE ̸ = 0) denotes the number of quasienergy eigenvalues whose imaginary parts are nonzero. ρ thus describes the density of states with complex quasienergies in the system. In the PT-invariant phase, we would have max |ImE| = ρ = 0, implying that all the quasienergies ofÛ are real. In the PT-broken phase, we would have max |ImE| > 0 and ρ ∈ (0, 1], meaning that there is a finite number of eigenstates ofÛ whose quasienergies are complex. Specially, we would have ρ ≃ 1 if almost all the Floquet eigenstates ofÛ possess complex quasienergies. Therefore, by locating the positions where both the max |ImE| and ρ start to deviate from zero in the parameter space, we can identify the PT transition points for the Floquet spectrum ofÛ . We will see that due to the interplay between periodic drivings and non-Hermitian effects, Along the real axis, the spacing between the jth and the (j − 1)th quasienergies ofÛ is given by ϵ j = ReE j − ReE j−1 , from which we obtain the ratio between two adjacent spacings of quasienergy levels as Here the max(ϵ j , ϵ j+1 ) and min(ϵ j , ϵ j+1 ) are the maximum and minimum between the ϵ j and ϵ j+1 , respectively. The statistical property of adjacent gap ratios can then be obtained by averaging over all g j in the thermodynamic limit, i.e., We have g → 0 if all the bulk Floquet eigenstates ofÛ are extended. Comparatively, we expect g to approach a constant g max > 0 if all the bulk eigenstates ofÛ are localized.
If we find g ∈ (0, g max ), extended and localized eigenstates ofÛ should coexist and be separated by some mobility edges in the quasienergy spectrum. The g can thus be utilized to distinguish between phases with different localization nature in 1D non-Hermitian Floquet systems from the perspective of level statistics. In the lattice representation, we can expand the right eigenvector |ψ j ⟩ as |ψ j ⟩ = L n=1 ψ j n |n⟩, where ψ j n = ⟨n|ψ j ⟩ and n |ψ j n | 2 = 1. The inverse and normalized participation ratios of |ψ j ⟩ in the real space can then be defined as For a localized Floquet eigenstate |ψ j ⟩, we have IPR j → λ j and NPR j → 0 in the thermodynamic limit, where the Lyapunov exponent λ j of |ψ j ⟩ could be a function of its quasienergy where the IPR ave = 1 L L j=1 IPR j and NPR ave = 1 L L j=1 NPR j are the averages of IPR j and NPR j over all Floquet eigenstates. It is not hard to see that in the metallic phase where all Floquet eigenstates are extended, we have IPR max → 0, IPR min → 0 and ζ ∼ − log 10 (L) → −∞ in the thermodynamic limit L → ∞. Instead, in the insulating phase with all Floquet eigenstates being localized, we have IPR max > 0, IPR min > 0 and ζ → −∞. In the critical phase where extended and localized Floquet eigenstates are coexistent, we have IPR max > 0, IPR min → 0 together with a finite ζ. Assembling the information obtained from IPR max , IPR min , and ζ thus allows us to distinguish the extended, localized, and critical mobility edge phases of a non-Hermitian Floquet system with disorder. These tools will be applied to our study of the Floquet NHQC in Subsec. III D.
We can also probe the transport nature of distinct non-Hermitian Floquet phases from the wave packet dynamics. Let us consider a generic and normalized initial state |Ψ(0)⟩ in the lattice representation. After the stroboscopic evolution over a number of ℓ driving periods byÛ , the final state turns out to be |Ψ ′ (t = ℓT )⟩ =Û ℓ |Ψ(0)⟩. Since the Floquet operatorÛ is not unitary for a non-HermitianĤ(t) in general, the norm of |Ψ(0)⟩ cannot be preserved during the evolution. We can express the normalized state at t = ℓT as |Ψ(t)⟩ = |Ψ ′ (t)⟩/ ⟨Ψ ′ (t)|Ψ ′ (t)⟩. The real space expansion |Ψ(t)⟩ = L n=1 ψ n (t)|n⟩ then provides us with the probability amplitude ψ n (t) = ⟨n|Ψ(t)⟩ of the normalized final state |Ψ(t)⟩ on the lattice site n at time t. Using the collection of amplitudes {ψ n (t)|n = 1, ..., L}, we can define the following dynamical quantities where the stroboscopic time t = ℓT and ℓ ∈ Z. It is clear that the ⟨x(t)⟩, ∆x(t), and v(t) describe the center, standard deviation, and spreading speed of the wavepacket |Ψ(t)⟩ in the lattice representation, respectively. For simplicity, we usually choose the initial state |Ψ(0)⟩ to be exponentially localized at a single site that is deep inside the bulk of the lattice. If the Floquet system described byÛ resides in a localized phase, we expect the ⟨x(t)⟩ and ∆x(t) to stay around their initial values, which means that the wavepacket almost does not move and spread. In this case, the speed v(t) should also tend to zero for a long-time evolution (ℓ ≫ 1).
If the system stays in an extended phase, we expect the increasing of ∆x(t) with time due to the spreading of the initial wavepacket. Meanwhile, the ⟨x(t)⟩ may or may not change with time, depending on whether the hopping amplitudes are symmetric. For a nearestneighbor, nonreciprocal hopping as in Eq. (116), we may have ⟨x(t)⟩ ∝ t and ∆x(t) ∝ √ t for a metallic phase of the system. The v(t) should take a maximal possible value v max (t) in this case. If the system is prepared in a critical mobility edge phase, we expect both the ⟨x(t)⟩ and ∆x(t) to show intervening behaviors, while the averaged spreading speed v(t) should satisfy 0 < v(t) < v max (t). Therefore, we can exploit the dynamical quantities ⟨x(t)⟩, ∆x(t), and v(t) to discriminate phases with different localization properties in disordered non-Hermitian Floquet systems. We will illustrate such an application for Floquet NHQC in Subsec. III D.

A. Non-Hermitian Floquet exceptional topology
Let us start with a simple model, whose Floquet effective Hamiltonian and quasienergy dispersion are given by Eqs. (55) and (57). To be explicit, we choose where k ∈ (−π, π] denotes the quasimomentum in the first Brillouin zone, and the system parameter µ ∈ R. The H eff in Eq. (55) thus describes the Bloch Hamiltonian of a 1D two-band lattice model under the PBC in the rotating frame, where µ corresponds to the amplitude of onsite potential and the nearest-neighbor hopping amplitude has been chosen to be the unit of energy. From Eqs. (57) and (127), we find that the two quasienergy bands of H eff could meet with each other at the quasienergy ω/2 when If µ satisfies one of the above two equalities, there will be a second-order Floquet EP at k = 0 or k = π in the conventional Brillouin zone. Note that both the quasienergy of this Floquet EP and its location in the parameter space depend on the frequency ω, which highlights the impact of the harmonic driving field on phase transitions in the system. Moreover, the appearance of an EP in k-space usually implies the breakdown of bulk-edge correspondence in conventional topological phases. This issue might be overcome by incorporating the formalism of generalized Brillouin zone (GBZ) and non-Bloch band theory [163,165].
Following the standard recipe, we first make the substitutions e ik → β and e −ik → β −1 , where the complex number β ∈ GBZ. The effective Hamiltonian in Eq. (55) now takes the form of with the quasienergy bands We next focus on the non-constant part of ε ± (β), whose square takes the form of where The ε 2 ± (β) is a Laurent polynomial of β. According to Ref. [165] (see also Ref. [280]), β ∈ GBZ for the H eff (β) if ϵ 2 ± (β) = ϵ 2 ± (βe iθ ), with θ being a phase factor. For Eq. (131), this means that β 2 = ce −iθ /a. The GBZ is thus a circle of the radius It is clear that the GBZ radius is controlled by both the driving field (through ω) and the non-Hermitian effect (through γ) . In the Hermitian limit (γ → 0), we have r → 1 and the GBZ is reduced to the conventional BZ, as expected. The GBZ becomes ill defined if µ − ω/2 = ±γ/2, where the radius r is zero or infinity. Finally, we can use the GBZ to determine the gap closing (phase transition) points of the system under the OBC. Setting the ϵ 2 ± (β) = 0 in Eq. (131), we find These bulk gap-closing conditions are clearly different from those found in conventional BZ under the PBC in Eq. (128). Using the transfer matrix method [167], we can further obtain the parameter regions in which degenerate Floquet edge states appear at the quasienergy ω/2 (anomalous Floquet π edge modes), i.e., Finally, we can characterize the different topological phases of the system by a non-Bloch winding number [168] W =ˆG where σ y is the chiral symmetry operator of the shifted effective Hamiltonian H eff (β) − ω 2 σ 0 . The Q matrix is defined as Q(β) = |ψ + (β)⟩⟨ ψ + (β)| − |ψ − (β)⟩⟨ ψ − (β)|. |ψ ± (β)⟩ are the right eigenvectors of H eff (β) with the quasienergies ε ± (β). ⟨ ψ ± (β)| are the corresponding left eigenvectors. For our model, their explicit expressions are given by In Fig. 2

B. Non-Hermitian Floquet topological insulators
In this subsection, we review three types of non-Hermitian Floquet topological insulators in one and two spatial dimensions. For all the cases, we uncover that the collaboration between driving and gain/loss or nonreciprocal effects could induce topological insulating states unique to non-Hermitian Floquet systems. They are featured by large topological invariants, many topological edge or corner modes, and separated by rich topological phase transitions. The bulk-edge (or bulk-corner) correspondence will also be established for each class of systems considered in this subsection.

First-order topological phase
We start with the characterization of 1D non-Hermitian Floquet topological insulators.
One typical model that incorporates their rich topological properties is the following periodically quenched dimerized tight-binding lattice, whose time-dependent Hamiltonian reads Here T is the driving period and we will set ℏ = T = 1 in our calculations.ĉ † n (ĉ n ) creates (annihilates) a particle in the unit cell n of the lattice. σ α , α = x, y, z, are Pauli matrices acting on the two sublattice degrees of freedom A and B within each unit cell. The system parameters r x , r y , µ, γ are all real. The non-Hermitian effect is introduced by the nonreciprocal intracell coupling term 2iγσ y applied over the first half of each driving period [see Fig. 3(a) for an illustration of the model]. Experimentally, such a term might be realized by coupled-resonator optical waveguide with asymmetric internal scattering [281].
It is clear that both U 1 (k) and U 2 (k) respect the chiral (sublattice) symmetry S = σ z . We can thus characterize the non-Hermitian Floquet topological phases of U (k) by the winding numbers (w 0 , w π ) according to Eq. (93). In Figs. 3(f) and 3(g), we present the w 0 and w π of U (k) versus (r x , r y ) respectively, yielding the topological phase diagram of the system.
We find various non-Hermitian Floquet insulating phases. They are characterized by large integer winding numbers and separated by a series of topological phase transitions with quasienergy level crossings at E = 0 [white solid lines in Fig. 3(f)] and E = π [white dashed lines in Fig. 3(g)]. The winding numbers (w 0 , w π ) could become arbitrarily large with the increase of system parameters (e.g., the intercell hopping amplitude r x ), which implies that and E = π under the OBC. That is, we have the bulk-edge correspondence for our 1D chiral symmetric non-Hermitian Floquet topological insulator as [254] n 0 = 2|w 0 |, n π = 2|w π |. The winding numbers of non-Hermitian Floquet topological insulators can be experimentally probed by measuring the stroboscopic time-averaged spin textures [254] or the MCDs [255], as introduced in Subsecs. II F 1 and II F 2. Following Ref. [256], we consider a periodically quenched non-Hermitian lattice model, whose Floquet operator in the momentum space reads U (k) = e −iJ 2 sin kσy e −iJ 1 cos kσx . Here J 1 = u 1 + iv 1 , J 2 = u 2 + iv 2 , and u 1 , The winding numbers (w 0 , w π ) of U (k) can be obtained using the approach of symmetric time frames [see Eqs. Floquet matter in different types of experimental setups [213,222].
The models we considered above in this subsection do not possess the NHSE. It remains unclear whether the rich non-Hermitian Floquet topology could coexist with NHSEs, and how to characterize the topological bulk-edge correspondence in the presence of Floquet NHSE. To address this issue, we consider a periodically quenched non-Hermitian SSH model [261], whose k-space Hamiltonian under the PBC takes the form of Here µ and J 1 /2 + J 2 /2 are the intracell and intercell hopping amplitudes. The non-Hermiticity is introduced by the asymmetric parts of hopping amplitudes ±iλ/2 between the two sublattices. A sketch of the model is shown in Fig. 5. In symmetric time frames, it respects the chiral symmetry S = σ z [261]. We may thus use the winding numbers (w 0 , w π ) to characterize the Floquet topological phases of U (k). The calculation of such winding numbers following Eqs. (89)- (93) leads to the topological phase diagrams in Fig. 5(a)-(b). Interestingly, we find that despite Floquet topological insulators with large integer winding numbers, there are also various phases with half-integer quantized topological invariants. These unique phases are absent in the Hermitian limit of the model. Floquet edge modes (n 0 , n π ) through the bulk-edge correspondence (n 0 , n π ) = 2(|ν 0 |, |ν π |).
This relation holds even with NHSEs in our system [261]. Therefore, our work provides 5(k)] in our system, which are also consistent with the theoretical predictions [261].
Putting together, we found different types of 1D non-Hermitian Floquet topological insulators in a series of studies [254-256, 258, 261]. All the models considered in these studies exhibit rich non-Hermitian Floquet phases with large topological invariants, many topo-logical edge states, and unique topological transitions induced by the interplay between non-Hermitian effects and time-periodic driving fields. The bulk-edge correspondences and dynamic topological characterizations of these intriguing new phases are also established, leading to an explicit and all-round physical picture of 1D non-Hermitian Floquet topological matter. In the following subsections, we uncover the essential role of Floquet engineering in other types of non-Hermitian topological matter.

Second-order topological phase
We next consider the example of a non-Hermitian Floquet second-order topological insulator (SOTI) in two dimensions. An nth-order topological insulator in d (d ≥ n) spatial dimensions possesses localized eigenmodes along its (d − n)-dimensional boundaries, which are topologically nontrivial and protected by the symmetries of the d-dimensional bulk.
For example, an SOTI in two dimensions usually owns localized topological states around its zero-dimensional geometric corners. Assisted by time-periodic drivings, degenerate corner modes can further appear at different quasienergies in Floquet second-order topological phases [80,282].
In this subsection, we focus on one typical model of non-Hermitian Floquet SOTI, which is obtained following the recipe of coupled-wire construction [80]. A schematic diagram of the static lattice model is shown in Fig. 6(a). The system is formed by a stack of SSH chains along the y-direction, with dimerized couplings (J 1 , J 2 ) between adjacent chains. We where (k x , k y ) ∈ [−π, π) × [−π, π) are the quasimomenta. σ 0 and τ 0 are two by two identity matrices. The Hamiltonians of the 1D subsystems H x (k x ) and H y (k y , t) are explicitly given by Here T is the driving period and ℓ ∈ Z. σ x,y,z and τ x,y,z are Pauli matrices acting on the sublattice degrees of freedom in x and y directions, respectively. Setting ℏ = T = 1, J = δ = ∆/2, and choosing µ = u + iv (u, v ∈ R), the Floquet operator of the system in k-space is found to be [259] U (k x , k y ) = e −i∆(cos kxσx+sin kxσy) ⊗ e −ihz(ky)τz e −ihx(ky)τx , where h x (k y ) = J 1 cos k y , h z (k y ) = u + iv + J 2 sin k y .
The non-Hermitian effect is brought about by the onsite gain and loss encoded in the term ivτ z .
In symmetric time frames, the system has the chiral (sublattice) symmetry S = σ z ⊗ τ y [259]. One can thus characterize its Floquet second-order topological phases by integer winding numbers [259] Here ν α = ww α for α = 1, 2. w is the winding number of the static Hamiltonian ∆(cos k x σ x + sin k x σ y ), which is always equal to one for the topological flat-band limit of the SSH model.
(w 1 , w 2 ) are the winding numbers of the subsystem Floquet operator e −ihz(ky)τz e −ihx(ky)τx , which can be defined via the recipe outlined in Subsec. II E. They both take integer-quantized values even with the non-Hermitian effects considered in our system [259].
We thus established the bulk-corner correspondence for a class of non-Hermitian Floquet SOTI with chiral symmetry. With the growth of the non-Hermitian parameter v, topological phase transitions accompanied by the increase of bulk winding numbers (ν 0 , ν π ) and corner mode numbers (n 0 , n π ) are observed from the gap functions (F 0 , F π ) ≡ can be found analytically [259]. Finally, even with non-Hermitian effects, the bulk winding numbers (ν 0 , ν π ) can be dynamically probed by 2D generalizations of the MCD as introduced in Subsec. II F 2 (see Ref. [259] for more details).
Overall, we discovered rich non-Hermitian Floquet SOTI phases in a periodically quenched 2D lattice with balanced gain and loss. Different from the static case, each SOTI phase is now depicted by two topological invariants (ν 0 , ν π ). Many non-Hermitian Floquet SOTIs with large topological invariants and gain-or loss-induced topological phase transitions were identified. Under the OBC, the winding numbers (ν 0 , ν π ) determine the numbers of symmetry-protected Floquet corner modes at zero and π quasienergies. The interplay between driving and dissipation thus results in a series of non-Hermitian Floquet SOTI phases with multiple zero and π corner modes, which may find applications in topological state preparations, information scrambling and quantum computation. The work of Ref. [259] offers one of the earliest findings of rich non-Hermitian Floquet topological phases beyond one spatial dimension. In the meantime, it introduces a generic scheme to construct non-Hermitian Floquet higher-order topological phases across different physical dimensions, which is expected to be applicable across insulating, superconducting and semi-metallic systems. Note in passing that some latter studies also considered the NHSEs and anomalous π modes in Floquet higher-order topological phases with somewhat different focuses [235,238,246].

qth-root topological phase
We now consider an exotic class of non-Hermitian Floquet topological phase, which could have symmetry-protected boundary states at the quasienergies pπ/q, with p, q ∈ Z ± and p/q ̸ = 0, 1. A systematic recipe of obtaining these new phases is to take the nontrivial qthroot of a propagatorÛ that describes Floquet topological matter. The general procedure is developed in Ref. [263], which generalizes the previous schemes of taking 2 n th and 3 n th roots for static Hamiltonians [283][284][285][286][287][288][289][290][291][292] to Floquet systems. The basic idea is first outlined in Ref. [293], and then expanded by utilizing Z q generalizations of Pauli matrices as ancillary degrees of freedom before taking the qth-root of a Floquet operator in an enlarged Hilbert space. It is in parallel with Dirac's original idea of adding internal degrees of freedom for electrons before taking the square-root to get their relativistic wave equation [294].
To be explicit, we consider the construction of a cubic-root non-Hermitian Floquet topological insulator. An illustration of the scheme is given in Fig. 7. Let us consider a three-step periodically quenched parent system, whose time-periodic Hamiltonian takes the form of where ℓ ∈ Z and we have assumed ℏ = T = 1. The Floquet operator of the system is then Following the methodology of Ref. [263], the nontrivial cubic-rootÛ 1/3 ofÛ is found to bê The cube ofÛ 1/3 then giveŝ It is clear thatÛ 3 1/3 contains three identical copies ofÛ , in the sense that its three diagonal blocks are all related by similarity transformations and thus sharing the same spectrum and topological properties withÛ [263].
As an example, we choose the Floquet model introduced in Ref. [261] as the parent system, whose Hamiltonian can now be expressed aŝ The cubit-root Floquet operatorÛ 1/3 of the system then takes the form of Eq.
After these transitions, more and more degenerate Floquet edge modes emerge at the frac- More precisely, we have the following bulk-edge correspondence for our cubic-root non-Hermitian Floquet topological insulatorÛ 1/3 , i.e., [263] n 0 = n 2π/3 = 2|ν 0 |, n π/3 = n π = 2|ν π |, where n 0 and n π are the numbers of zero and π degenerate Floquet edge modes. Finally, we notice that the system also possesses NHSE under the OBC, as reflected by the profiles of its Floquet bulk states in Fig. 7 The formalism developed in Ref. [263] is equally applicable to the construction of qth-root Floquet topological insulators, superconductors, and semimetals in other (non-)Hermitian systems and across different physical dimensions. The fractional quasienergy edge modes at E = pπ/q might also be employed to generate boundary time crystals with different temporal periodicity and topological properties [98]. They may also find applications in the Floquet quantum computing.

C. Non-Hermitian Floquet topological superconductors
Similar to the static case, non-Hermitian Floquet topological phases could also appear in superconducting systems. We review one such example in this subsection, which admits many Floquet Majorana zero and π edge modes even with non-Hermitian effects.
The model we are going to consider describes a Floquet Kitaev chain, whose superconducting pairing terms are subject to time-periodic kicks. The Hamiltonian of the model takes the form of where J is the nearest-neighbor hopping amplitude, ∆ is the p-wave superconducting pairing amplitude, and µ is the chemical potential. δ T (t) ≡ ℓ∈Z δ(t/T − ℓ) implements delta kickings periodically with the period T . The system is made non-Hermitian by setting J = J r + iJ i or µ = µ r + iµ i with J i ̸ = 0 or µ i ̸ = 0, respectively. Possible experimental realizations of the delta kickings and non-Hermitian terms in this model are discussed in Ref. [257].
Under the PBC and in symmetric time frames, the Floquet operator of H(t) respects the chiral (sublattice) symmetry S = σ x . Its bulk topological phases can then be characterized by the winding numbers (w 0 , w π ) [see Eqs. (89)- (93)], for which we choosê and assume ℏ = T = 1. Typical topological phase diagrams of the system are shown boundaries between different phases can be analytically determined from the gap-closing conditions of the system (see Ref. [257]). Two notable features are brought about by the interplay between periodic drivings and non-Hermitian effects. First, the system could undergo rich topological phase transitions and enter Floquet superconducting phases with arbitrarily large topological winding numbers in principle. These phases could support arbitrarily many zero and π Majorana edge modes in the thermodynamic limit, which might be employed to engineer boundary time crystals and realize Floquet quantum computing protocols. Second, Floquet superconducting phases with larger winding numbers and therefore stronger topological signatures may appear with the increase of the non-Hermitian parameter. Therefore, new topological states could be induced and stabilized solely by non-Hermitian effects in Floquet superconducting systems. This is not expected in the non-driven counterparts of Fig. 8(e)] (w 0 , w π ) (n 0 , n π ) J r [in Fig. 8  Under the OBC, the quasienergy spectrum and edge states of the system can be obtained by diagonalizing the Floquet operatorÛ = e −iĤ 0 e −iĤ 1 in Majorana or BdG bases [257].
As examples, the real parts of Floquet spectra and gap functions (F 0 , F π ) versus µ i [with  Table I). The gap functions are defined as where ε includes all the quasienergies ofÛ under the OBC. The eigenmodes with F 0 = 0 and F π = 0 thus have the quasienergies zero and π, respectively. In Table I, we list the numbers of Floquet Majorana edge modes (n 0 , n π ) with the quasienergies (0, π) and the bulk topological winding numbers (w 0 , w π ). A simple bulk-edge correspondence can be further found between these numbers, i.e., n 0 = 2|w 0 |, n π = 2|w π |.
These relations hold in other parameter regions of our model as well. We conclude that our non-Hermitian Floquet Kitaev chain could indeed possess many Majorana zero and π edge modes due to its large winding numbers. The localization nature of these Majorana modes was also demonstrated in Ref. [257].
Therefore, our study in Ref. [257] established the topological characterization and bulkedge correspondence of 1D non-Hermitian Floquet topological superconductors that belong to an extended BDI symmetry class. The collaboration between time-periodic driving fields and non-Hermitian effects was found to produce rich Floquet superconducting phases with large topological winding numbers and many Majorana edge modes at two distinct quasienergies zero and π. These non-Hermitian Floquet Majorana modes might allow Floquet quantum computing schemes to be more robust to environmental-induced nonreciprocity, dissipation, and quasiparticle poisoning effects. The existence of many pairs of Floquet Majorana modes may also create stronger transport signals at the ends of the chain, making it easier to experimentally detect their topological properties in open-system settings [226].
In future studies, topics like non-Hermitian Floquet topological superconductors under different driving protocols, in other symmetry classes, with more complicated lattice effects (e.g., sublattice structures, long-range hoppings or disorder), in higher spatial dimensions and with NHSEs, deserve to be explored. Possible changes in topological classifications due to many-body effects in non-Hermitian Floquet superconductors are also awaited to be revealed.

D. Non-Hermitian Floquet quasicrystals
In this subsection, we go beyond the clean limit of driven non-Hermitian lattices and showcase that the interplay among correlated disorder, temporal driving, and non-Hermitian effects could yield Floquet quasicrystals with rich PT-symmetry breaking transitions, localization transitions, and topological phase transitions. We will consider systems under both high-frequency and near-resonant driving fields.
We start with the example of a 1D non-Hermitian quasicrystal under high-frequency harmonic driving forces. The time-periodic Hamiltonian of the model takes the form of  An illustration of the model is given in Fig. 9(a). Here γ ∈ R and Je −γ (Je γ ) describes the right-to-left (left-to-right) nearest-neighbor hopping amplitude. The hopping is nonreciprocal and thusĤ(t) ̸ =Ĥ † (t) if γ ̸ = 0.ĉ † n (ĉ n ) creates (annihilates) a particle on the lattice site n. V ∈ R is the amplitude of an onsite potential, and α is chosen to be irrational in order for the potential to be spatially quasiperiodic. K ∈ R is the driving amplitude and ω is the driving frequency. This model can be viewed as a Floquet and quasicrystal variant of the Hatano-Nelson model [183]. In the absence of the driving force and under the PBC, all the eigenstates of the system are extended (localized) with complex (real) eigenvalues if |V | < |2J|e |γ| (|V | > |2J|e |γ| ) [189]. The system could thus undergo a PT transition, a localization-delocalization transition, and also a topological transition accompanied by the quantized change of its spectral winding number [Eq. (81)] at |V | = |2J|e |γ| [189].
Transforming the HamiltonianĤ(t) in Eq. (169) to the rotating frame and applying the method discussed in Subsec. II C, we can obtain the Floquet effective Hamiltonian of our system in the high-frequency limit [262], i.e., Here J 0 (K/ω) is the Bessel function of first kind, which is a non-monotonous function of K/ω. Under the PBC, the Floquet NHQC described byĤ F could thus undergo multiple and reentrant PT, localization and topological transitions with the change of the ratio K/ω between the driving amplitude and driving frequency whenever [262] |V | = |2JJ 0 (K/ω)|e |γ| .
This is indeed the case, as demonstrated by the maximum of the imaginary parts of quasienergies, the minimum of IPRs, and the winding numbers in Figs The Lyapunov exponent λ = ln |(V e −|γ| )/[2JJ 0 (K/ω)]| is quasienergy-independent for all the Floquet eigenstates, which is positive (negative and thus ill-defined) in the localized (extended) phases [262]. Therefore, it is clear that the phases and transitions in the quasicrystal Hatano-Nelson model can be significantly modified by Floquet driving fields even in the high-frequency limit. The periodic driving also provides us with a flexible knob to control and engineer different types of phase transitions in NHQCs, with further examples discussed in Ref. [262].
We move on to the example of an NHQC under near-resonant driving fields. In this case, the interplay between driving and non-Hermitian effects not only induces multiple and reentrant localization transitions but also generates critical mobility edge phases that are absent in non-driven limits. A schematic diagram of our model is shown in Fig. 10(a). Its Hamiltonian takes the form ofĤ(t) =K for t ∈ [ℓT, ℓT + T 1 ) andĤ(t) =V for t ∈ [ℓT + T 1 , ℓT + T 1 + T 2 ), where ℓ ∈ Z and the driving period T = T 1 + T 2 . The system is piecewisely quenched betweenK = J (e γĉ † nĉ n+1 + e −γĉ † n+1ĉ n ) andV = V n cos(2παn)ĉ † nĉ n over each driving period. For an irrational α, we thus arrive at a periodically quenched, spatially quasiperiodic variant of the Hatano-Nelson model, whose Floquet operator is given by [264] U = e −iV n cos(2παn)ĉ † nĉn e −iJ (e γĉ † nĉn+1 +e −γĉ † n+1ĉ n) , where V = V T 2 /ℏ and J = JT 1 /ℏ. Solving the eigenvalue equationÛ |ψ⟩ = e −iE |ψ⟩ and using the tools introduced in Subsec. II G, we could obtain the maximal imaginary parts of quasienergies max |ImE|, the density of states with complex eigenvalues ρ, the minimum of IPRs and the spreading velocity of an initially localized wavepacket v. A collection of these quantities versus the strength of quasiperiodic potential V is shown in Fig. 10(b) for a typical case [with (J, γ, α) = (π/6, 0.8, √ 5−1 2 ) and the length of lattice L = 4181]. We observe that with the increase of V from V = 0, the system first undergoes a complex-toreal PT transition in its quasienergy spectrum, which is also accompanied by a localization transition of all its Floquet eigenstates from spatially extended to localized. Interestingly, with the further enhancement of the quasiperiodic potential V (thus with stronger correlated disorder), some localized Floquet states can again become extended with real eigenvalues.
The system then enters a critical phase, in which extended and localized eigenstates coexist and are separated by mobility edges on the complex quasienergy plane (see also the Fig. 3 of Ref. [264]). The further increase of V leads to reentrant transitions between localized and critical mobility edge phases in the system. This is also reflected by the two-parameter phase diagrams in Figs. 10(c)-(f) [see Subsec. II C for the definitions of ρ, g, IPR min , and ζ].
Note that both the critical phases and the reentrant localization transitions are absent in the non-driven Hatano-Nelson quasicrystal. They are brought about by the nearest-resonant Floquet driving field. It induces long-range spatial couplings and quasienergy windings in the system, thus yielding the observed phenomena (see Ref. [264] for more detailed discussions).
In experiments, Floquet NHQCs might be realized by ultracold atoms in driven and quasiperiodic optical superlattices with particle losses [180,223]. Static NHQCs have also been realized by non-unitary photonic quantum walks [191,192]. Signatures of PT breaking transitions, localization transitions, mobility edges, and topological properties related to NHSEs can be extracted from dynamical observables of initially localized wavepackets [191,192]. Since quantum walk models are intrinsically dynamical, they provide natural platforms to realize and detect the Floquet NHQCs reviewed in this subsection. Meanwhile, as the discrete-time quantum walk carries a synthetic spin-half degree of freedom, it may also be utilized to simulate other types of NHQCs, such as those with lattice dimerizations or non-Abelian potentials [295][296][297]. Topological Anderson insulators induced by uncorrelated disorder could also be explored in similar settings [187].
Overall, we find that both high-frequency and near-resonant driving fields could be used to trigger, control, and enrich the phases and transitions in NHQCs, and even create unique Floquet NHQC phases that are absent in the static limit [262,264]. Non-Hermitian disordered systems thus provide further playgrounds for the exploration of new physics that are enabled by Floquet engineering and time-periodic driving fields.

IV. CONCLUSION AND OUTLOOK
In this review, we recapitulated some progress we made in the study of intriguing topological phases in non-Hermitian Floquet systems. After a general introduction about the backgrounds and motivation, we first formulated the theoretical frameworks underpinning our study. These include the Floquet theory applicable in general situations and some approximation schemes suitable for exploring fast and slowly driven systems. Efficient means of characterizing the symmetry, topology, dynamical, and localization nature of non-Hermitian Floquet systems were then introduced. Equipped with these tools, we discussed prototypical examples of non-Hermitian Floquet topological phases in insulating, superconducting, and quasicrystalline systems, where the PT transitions, topological transitions, bulk-edge/corner correspondences and localization transitions within these non-Hermitian Floquet phases of matter were systematically characterized. Therefore, the collection of our works [254][255][256][257][258][259][260][261][262][263][264] lays a solid foundation for the study of topological phenomena in non-Hermitian Floquet systems and further uncovers the richness of topological phases that could emerge due to the interplay between periodic drivings and non-Hermitian effects.
We note that some other schemes concerning the topological classification and bulk-edge correspondence of non-Hermitian Floquet matter were discussed in Refs. [224,228,230,236,245]. The EPs and gapless topological phases in non-Hermitian Floquet systems were also considered in Refs. [215,231,234,237,239,244,247]. From the perspective of quantum control, the periodic driving fields may provide efficient means to stabilize non-Hermitian systems and control the PT transitions therein [210]. Some interesting aspects regarding the quantum dynamics and anomalous diffusion were revealed in several chaotic non-Hermitian Floquet systems [227,233,242,248]. Aspects of PT-symmetry in non-Hermitian Floquet phases were also investigated in Refs. [209,216,217,221,232]. On the experimental side, the quantum walks in cold atoms, photonic setups, and circuit QED provide useful frameworks to realize and detect non-Hermitian Floquet topological matter [191, 192, 208, 211-214, 222, 223, 241].
Even with all the progress mentioned above, our knowledge about non-Hermitian Floquet topological matter is still quite limited. This is especially the case for systems beyond one spatial dimension, with uncorrelated disorders, and with many-body interactions. Dynamical quantum phase transitions may exhibit unique critical signatures triggered by the non-Hermitian Floquet exceptional topology [298][299][300][301][302]. The entanglement and transport properties of non-Hermitian Floquet systems [303] may also deviate significantly from their Hermitian or static counterparts. The exact connection between the topological phases of non-Hermitian Floquet Hamiltonians and the Floquet master equation (or Floquet Liouvillian) of driven open systems is unclear. More experimental efforts deserve to be made in order to realize and observe non-Hermitian Floquet phases with large topological invariants, many topological edge states, and multiple topological phase transitions. All these facts indicate that the area of non-Hermitian Floquet matter is still in its infancy, and further substantial developments are eagerly needed in this area.