Non-Hermitian Floquet Theory

• Non-Hermitian Floquet theory is a framework for analyzing time-periodic systems with non-Hermitian Hamiltonians, characterized by complex quasienergies and exceptional points.
• The theory uncovers novel topological phases with distinctive invariants, such as fractional winding numbers and the non-Hermitian skin effect, enhancing edge state predictions.
• Applications in driven photonic, quantum, and condensed-matter systems demonstrate how engineered gain/loss and periodic driving yield robust transport and emergent dynamical behaviors.

Non-Hermitian Floquet theory addresses the dynamics, spectral properties, and topological classification of quantum and classical systems governed by time-periodic, non-Hermitian (dissipative or gain/loss) Hamiltonians. Unlike conventional Hermitian Floquet systems—characterized by real quasienergy spectra and unitary evolution—non-Hermitian Floquet systems admit complex spectra, nonorthogonal eigenstates, the non-Hermitian skin effect, exceptional points, and a host of novel topological phases. This framework has deeply influenced contemporary understanding of driven photonic, quantum, and condensed-matter systems, enabling the engineering of robust topological transport, greatly enriched edge state physics, and even the possibility of entirely new dynamical phases unique to the interplay between non-Hermiticity and periodic driving.

1. Fundamental Formalism of Non-Hermitian Floquet Theory

A time-periodic non-Hermitian system evolves according to the Schrödinger equation $i\partial_t \psi(t) = H(t)\psi(t)$ with $H(t+T)=H(t)$ and $H(t) \neq H^\dagger(t)$. The formal Floquet solution assumes the form

$\psi(t) = e^{-i E t}u(t), \qquad u(t+T) = u(t),$

but now $E$ (the quasienergy) is generally complex and the evolution operator $U(T)$ is not unitary. As a consequence:

• The eigenvalues of $U(T)$, $e^{-iE T}$, occupy the complex plane (rather than the unit circle).
• Eigenvectors may be nonorthogonal or even coalesce at exceptional points.
• The biorthogonal formalism is required: for a right Floquet eigenvector $|\psi_R\rangle$, one also defines a left eigenvector $\langle\psi_L|$ such that $U(T)|\psi_R\rangle = e^{-iE T}|\psi_R\rangle$, $\langle\psi_L|U(T) = e^{-iE T}\langle\psi_L|$.

Scattering in non-Hermitian Floquet systems exhibits haLLMarks absent in Hermitian counterparts. For example, an oscillating non-Hermitian potential well may be engineered (via the Darboux transformation) to completely suppress Floquet sidebands for all incident particle energies—so-called scattering-free or even “fully invisible” potentials (Longhi et al., 2013).

2. Topological Phases and Invariants in Driven Non-Hermitian Systems

The interplay between periodic driving and non-Hermiticity produces a rich zoo of topological phases:

• Winding Numbers and Symmetric Time Frames:

Most 1D non-Hermitian Floquet models are classified by a pair of winding numbers, $W_1$ and $W_2$, defined in two symmetric time frames (i.e., Floquet operators split as $U_1 = FG$ and $U_2 = GF$, with $F$ and $G$ propagators over subintervals of the period). The physical invariants

$$W_0 = \frac{W_1 + W_2}{2},\quad W_\pi = \frac{W_1 - W_2}{2}$$

count edge state pairs pinned at quasienergies $0$ and $\pi$ (Zhou et al., 2018, Zhou et al., 2019).

• Exceptional Points and Skin Effect:

When non-Hermitian parameters are tuned, band touchings in the quasienergy spectrum become “exceptional points” (EPs) where both eigenvalues and eigenvectors coalesce. Each EP generically carries a half-integer defect winding number (Zhang et al., 2019). Simultaneously, bulk eigenstates can accumulate at system boundaries—a phenomenon known as the non-Hermitian skin effect (NHSE)—invalidating conventional bulk–boundary correspondence (BBC).

• Bulk–Edge Correspondence and Generalized Brillouin Zone (GBZ):

In presence of NHSE, correct prediction of edge states requires replacing momentum $k$ by a complex quasi-momentum $\beta = e^{ik}$ and performing calculations over the GBZ contour satisfying $|\beta_N|=|\beta_{N+1}|$. This restores BBC even for driven systems with long-range couplings induced by the Floquet drive (Roy et al., 7 Oct 2024, Roy et al., 31 Jul 2025).

• Unbounded and Fractional Topological Invariants:

Certain classes of non-Hermitian Floquet systems feature phases with arbitrarily large or even unbounded winding numbers—implying the possibility of many robust, real-quasienergy edge states—driven by non-Hermitian effects (Zhou et al., 2018, Zhou, 2019). Fractional (e.g., half-integer) values of winding numbers are also possible, revealing unique exceptional topology (Zhou et al., 2020).

• Second-Order and Higher-Order Topological Phases:

Two-dimensional periodically driven non-Hermitian lattices can realize Floquet second order topological insulators (SOTIs), supporting corner-localized modes with quantized invariants $(\nu_0,\nu_\pi)$ governing the number of corner states at zero and $\pi$ quasienergy (Pan et al., 2020).

3. Spectral Features: Exceptional Points, Band Structures, and Dynamical Responses

Non-Hermitian Floquet systems exhibit a complex spectral landscape:

• Complex Quasienergy Surfaces:

Eigenvalues of non-Hermitian Floquet operators $U(T)$ trace out surfaces in the complex energy–momentum plane, supporting momentum gaps and exceptional points. These have been mapped experimentally in photonic lattices subject to time-periodic loss/gain and nonreciprocity (Park et al., 2021).

• Decaying Dressed States and Absorption:

In driven atomic or condensed-matter systems where decay is present, the steady-state absorption spectrum is determined by the decaying dressed states—eigenstates of a non-Hermitian Floquet Hamiltonian—each contributing Lorentzian-like terms with widths given by imaginary parts of their quasienergies (Potvliege, 24 Jun 2024).

• Wannier–Stark Localization and Floquet Band Gaps:

In periodic non-Hermitian systems, eigenstates can exhibit Wannier–Stark localization in frequency space (non-Floquet engineering), with Floquet band gaps visible in the lattice representation in frequency space (Wang et al., 2021). Modulation parameters or the initial time of the drive can tune localization behaviors.

• Resonance and Spreading Dynamics:

In non-Hermitian Floquet Wannier–Stark ladders, resonant driving leads to power-law spreading of energy levels with exponents $z=1$ or $z=1/2$, distinguishing real versus imaginary hopping modulation (Zhang et al., 24 Jan 2024).

4. Bulk–Boundary Correspondence and Topological Classification

Retrieving the bulk–boundary correspondence (BBC) in non-Hermitian Floquet systems demands careful consideration of symmetry constraints, finite-size effects, and the generalized Brillouin zone:

• Symmetry Restoration and Similarity Transformations:

Periodic driving can break chiral symmetry, obstructing the definition of topological invariants. By applying suitable similarity transformations or defining time-frame shifted evolution operators, effective chiral symmetry can be restored, enabling standard topological classifications even without translation invariance (Wu et al., 2020, Roy et al., 7 Oct 2024).

• Comprehensive Topological Classification (Periodic Tables):

Full classification of Floquet non-Hermitian systems follows the 54-fold generalized Bernard–LeClair (GBL) symmetry classes, each associated with a Clifford algebra extension and classifying space—allowing construction of periodic tables enumerating all possible topological invariants in arbitrary spatial dimension and gap class (angle-gapped vs angle-gapless Floquet operators) (Liu et al., 2021).

• Dual Topological Characterization:

Simultaneous momentum-space (winding number) and real-space (open-boundary winding number or Q-matrix) analysis elucidates phases unaffected by NHSE, providing experimentally accessible signatures and clarity even in models with intractable GBZ construction (Zhou et al., 2020, Zhou et al., 2020).

• Floquet Time Crystals and Coexistence of Multiple Edge Modes:

Non-Hermitian ladder models under certain driving support simultaneous zero and $\pi$ modes and bi-directional NHSE—distinctive of “Floquet time crystal” phenomenology (Roy et al., 7 Oct 2024).

5. Physical Realizations and Applications

Experimental and proposed platforms demonstrate wide-ranging phenomena originating from non-Hermitian Floquet theory:

• Photonic and Acoustic Lattices:

Gain/loss modulated lattices, especially in the mid-IR to THz domain (e.g. graphene, quantum wells with ultrafast population inversion), enable robust, nonreciprocal topological transport and edge laser operation (Li et al., 2018, Park et al., 2021).

• Topolectrical Circuits:

Electrical network analogs (TEC) of tight-binding Hamiltonians allow mapping of non-Hermitian Floquet phenomena onto impedances and voltage profiles, providing direct access to skin modes and impedance signatures of Floquet edge states. The proposed design includes interlayer couplings (simulate Floquet replicas), staggered resistors (gain/loss), and negative resistance for phase inversion (Roy et al., 31 Jul 2025).

• Cold Atom and Superconducting Quantum Simulators:

Periodically driven non-Hermitian Kitaev and SSH chains may be realized using ultracold atoms subject to controlled dissipation or superconducting qubits with engineered loss channels, granting access to Majorana $\pi$-modes, Floquet Anderson insulators, and bulk–edge correspondence recovery (Zhou, 2019, Wu et al., 2020).

• Quantum State Stabilization via Non-Hermitian Driving:

Protocols using ancillary qubits for non-Hermitian driving—where qubits are coupled, information extracted, and then traced out—remove entropy and correct errors, enabling indefinite stabilization of Floquet topological states. Such schemes have been proposed for robust topological charge pumping and dynamical localization akin to Anderson localization (Timms, 2023).

6. Outlook and Open Questions

Ongoing research and open challenges in non-Hermitian Floquet theory include:

• Extending BBC and Topological Invariants to Disordered or Many-Body Systems:

Though bulk–boundary correspondence has been formally retrieved for models with translation invariance and chiral symmetry, the interplay of inhomogeneities, strong driving-induced long-range hoppings, and NHSE raises nontrivial open questions for both noninteracting and interacting systems (Wu et al., 2020, Zhou et al., 2020).

• Experimental Challenges in Precise Potential Engineering:

Realizing the designed oscillating, spatially inhomogeneous, and temporally modulated non-Hermitian potentials poses significant experimental complexity, though advances in photonics, cold atoms, and electronic circuits offer promising testbeds (Longhi et al., 2013, Roy et al., 31 Jul 2025).

• Non-Bloch Topological Phenomena in Higher Dimensions:

Extension of Floquet non-Bloch theory to higher-dimensional (and higher-order) phases, with attention to measurement of topological invariants and corner or hinge modes, remains a prime target (Pan et al., 2020).

• Time-Crystalline and Anomalous Edge Physics:

The potential for discrete time crystal phases, subharmonic Floquet responses, and the coexistence of zero and $\pi$ modes, particularly in the strong non-Hermitian limit, presents opportunities for accessing dynamical phases beyond those achievable in Hermitian or static counterparts (Roy et al., 7 Oct 2024).

• Bulk–Edge Correspondence in Presence of NHSE and Strong Driving:

As driving enhances the effective range of couplings, constructing the generalized Brillouin zone becomes intractable in generic cases, necessitating either alternative numerical or analytic approaches or a real-space invariant formulation.

Non-Hermitian Floquet theory thus provides a framework with wide applicability across quantum, classical, and photonic systems, linking symmetry, topology, periodic driving, and dissipation/gain in a unified manner. Its methodologies and classifications—ranging from explicit winding numbers in symmetric time frames to K-theoretic periodic tables—have led to the prediction and realization of phenomena without static or Hermitian analogues, reshaping the landscape of topological phases in out-of-equilibrium systems.