Floquet NEGF: Driven Quantum Systems

• The Floquet NEGF framework is an advanced theoretical approach that combines NEGF and Floquet theory to study periodically driven, interacting quantum systems.
• It transforms time-dependent problems into a Floquet–Sambe matrix format, enabling efficient calculation of spectral, transport, and response functions.
• The method integrates diagrammatic self-energy approximations and innovative computational techniques to address nonequilibrium, dissipative, and many-body effects in nanoscale and correlated materials.

The Floquet Nonequilibrium Green’s Function (Floquet NEGF) framework is an advanced theoretical and computational machinery designed to analyze quantum many-body systems subjected to time-periodic driving. By combining core principles from NEGF, many-body perturbation theory, and Floquet theory, the framework enables nonperturbative treatment of periodically driven, interacting, and open quantum systems, including metals, semiconductors, correlated materials, and nanoscale devices. It provides methods to compute both transient and steady-state properties, including spectral, transport, and response functions, accounting for elastic and inelastic scattering, strong correlations, and nonequilibrium distribution functions.

1. Theoretical Foundations

At its core, the Floquet NEGF framework begins with the nonequilibrium Green’s function formalism, which expresses the dynamics of quantum systems in terms of two-time correlation functions of particle creation and annihilation operators. The essential object is, for example, the "lesser" Green’s function,

$$G^{<}_{ij}(t_1, t_2) = \langle c_j^\dagger(t_2) c_i(t_1) \rangle,$$

which describes single-particle correlations out of equilibrium. In general, NEGF theory generates a hierarchy of equations—the Martin–Schwinger hierarchy—for the evolution of these n-particle correlation functions, which are ultimately reduced to self-consistent equations for single-particle Green’s functions via suitable self-energy approximations (e.g., second Born, GW, T-matrix) (1211.6959, Leeuwen et al., 2013).

When the system's Hamiltonian and its bath couplings are time-periodic, i.e., $H(t) = H(t + T)$ with period $T = 2\pi/\Omega$, the Floquet theorem enables any operator or propagator to be expanded in a discrete Fourier series. This transforms the computationally challenging two-time problem into one in “quasi-energy” space, simplifying the treatment of periodic nonequilibrium steady states and the explicit evaluation of observables such as current, spectral functions, and optical responses (Balabanov, 2018, Mosallanejad et al., 2023).

2. Mathematical Structure and Approximations

A central mathematical innovation in the Floquet NEGF framework is the recasting of time-periodic two-time Green’s functions into “Floquet matrices.” For a function $A(t, t')$, the expansion reads: $$A(t, t') = \sum_{m, n} e^{-i (m \Omega) t} e^{i (n \Omega) t'} A_{mn}(t - t'),$$ or, equivalently, in frequency space as $A(\omega; m, n)$, where each Floquet index $m, n$ represents photon absorption or emission processes, and $\omega$ is the energy variable within the fundamental Floquet “zone.” This reduces convolution integrals and Dyson equations to matrix equations in an extended Hilbert (“Floquet–Sambe”) space (Balabanov, 2018, Mosallanejad et al., 2023).

The Martin–Schwinger hierarchy is closed via a self-energy $\Sigma(G)$ that can be constructed diagrammatically, formally adapted to Floquet space, and systematically improved by employing the GW, fluctuation-exchange (FLEX), or strong-coupling self-energies (Honeychurch et al., 2023, Reeves et al., 14 May 2024). In practice, the “generalized Kadanoff–Baym ansatz” (GKBA) is widely used to reduce the full two-time equations to single-time equations, reconstructing the non-diagonal parts from the density matrix and time-evolution propagators (in the time-periodic case, replaced with Floquet-adapted propagators) (1211.6959).

For open quantum systems, the Floquet NEGF framework is combined with reservoir models and self-energies to describe dissipation and thermalization, with extensions to incorporate inelastic electron–phonon effects via self-consistent Born or more nonperturbative schemes (Honeychurch et al., 2022, Lane et al., 2021).

3. Computational Methodologies

(a) Floquet–Sambe Machinery

The infinite-dimensional Floquet–Sambe formalism “lifts” the problem into a frequency domain: the full Hamiltonian and all self-energies become block matrices indexed by Floquet photon numbers. The retarded Green’s function satisfies: $(\omega^+ I - \mathcal{H}) \mathcal{G}(\omega) = I$ where $\mathcal{H}$ is the block matrix containing all harmonics of the periodic Hamiltonian. Effective truncation to a finite number of Floquet blocks (harmonics) is generally sufficient for modest driving amplitudes, after which recursive Green’s function, matrix inversion, and iterative schemes can be applied (Balabanov, 2018).

(b) Extension to Many-Body Interaction Effects

Interaction self-energies are likewise recast in Floquet space; this involves convolution with photon indices and the use of approximations such as fluctuation-exchange (FLEX) (Honeychurch et al., 2023). The FLEX self-energy,

$$\Sigma_\text{FLEX} = \Sigma^\text{GW} + \Sigma^\text{T-matrix} + \Sigma^\text{particle-hole},$$

sums categories of diagrams capturing screening, particle–hole fluctuations, and vertex corrections, and is expressed as Floquet matrices for self-consistent numerical solution.

(c) Modeling Inelastic and Non-Hermitian Dynamics

Interactions with vibrations are handled via Floquet extensions of the self-consistent Born approximation, where electron and phonon Green’s functions are coupled self-consistently, allowing the paper of photon-assisted and phonon-assisted tunneling and vibrational excitation (Honeychurch et al., 2022).

When including strong system-bath coupling or explicitly tracing out vibrational baths using path integrals, non-Hermitian dynamics can arise in stochastic Hamiltonians and require “three-time” or higher-order NEGF formulations for correct normalization and recovery of physical observables (Lane et al., 2021).

(d) Efficient Large-Scale and Long-Time Algorithms

A significant computational advance is the application of quantics tensor train (QTT) representations for compressing two-time Green’s functions, reducing memory and computational scaling from $\sim t_\text{max}^2$ or higher to nearly linear in $t_\text{max}$, while preserving accuracy (Murray et al., 2023). Additionally, machine learning strategies—such as using recurrent neural networks to learn the nonlinear operator mapping between Green’s functions and collision integrals in Kadanoff–Baym equations—can further reduce computational complexity for long-time evolution (Zhu et al., 13 Jul 2024).

4. Applications and Physical Insights

Floquet NEGF methods have enabled the investigation of a wide range of phenomena:

• Periodically Driven Quantum Transport: The framework accurately describes photon-assisted tunneling, time-resolved currents, and inelastic effects in molecular and quantum dot devices under ac-bias or light irradiation (Balabanov, 2018, Honeychurch et al., 2022, Mosallanejad et al., 2023).
• Spintronics and Magnetism: First-principles Floquet NEGF combined with noncollinear DFT allows computation of spin pumping and spin current response in heterostructures with strong interfacial spin–orbit coupling and antiferromagnetic order. The dependence of pumped spin current on cone angle, chirality, and materials parameters can be systematically analyzed, revealing enhancements when replacing normal with heavy metals and qualitative departures from simple analytic scaling laws (Dolui et al., 2021).
• Correlated Many-Body Systems: Interaction effects under periodic driving in systems such as Hubbard models are captured at the GW, FLEX, or second Born levels, allowing paper of nonequilibrium phase transitions, spectral weight redistribution, and dynamical entropy growth under strong fields (Reeves et al., 14 May 2024, Honeychurch et al., 2023). For example, under sufficiently strong driving, correlation-induced modifications become essential for accurately predicting observables.
• Response and Susceptibility: The framework provides access to nonequilibrium susceptibilities (e.g., spin or charge), capturing features such as dynamical localization, Floquet sidebands, and multiple-peak structures in the presence of strong periodic driving, and identifying the impact of light frequency and polarization on electronic interactions (Ono et al., 2018).
• Full Counting Statistics: Floquet NEGF methods can be used to compute time-resolved statistics of charge and spin transfer, including waiting time distributions and oscillations caused by coherent and incoherent processes, even when strong fields or time-dependent protocols are present (Tang et al., 2014).
• Dissipative Nonequilibrium Phases: In combination with mean-field or renormalization group techniques, NEGF can characterize nonequilibrium phase transitions, such as those modified by current-induced suppression of charge density wave order or the formation of discrete time-crystalline phases in Floquet-driven dissipative quantum gases (Klöckner et al., 2020, Jiao et al., 7 Nov 2024).

5. Challenges, Approximations, and Open Questions

While the Floquet NEGF framework is formally rigorous, several practical challenges and open questions remain:

• Hierarchy Closure and Self-Energy Approximations: Decoupling of the Martin–Schwinger hierarchy via truncation (e.g., neglect of three-particle correlations) is typically justified in weakly or moderately correlated regimes, but its validity in strongly correlated, strongly driven contexts remains under active investigation. Approximations such as GKBA facilitate simulations but may distort energy renormalization and spectral features in the Floquet regime (1211.6959).
• Numerical Scaling and Truncation: While the Floquet–Sambe approach reduces the two-time problem to a matrix problem, the required number of Floquet harmonics can grow rapidly for strong or broadband drives. Compression techniques (QTT, machine learning) are being developed to address these limitations (Murray et al., 2023, Zhu et al., 13 Jul 2024).
• Initial State and Boundary Conditions: Treatment of arbitrary initial conditions—whether generated by a thermal ensemble, quenches, or sudden periodic drives—is handled by advanced contour integration (Konstantinov–Perel’, Keldysh, or Matsubara branches) and by generalized Wick’s theorems for arbitrary initial correlations (Leeuwen et al., 2013).
• Interplay of Periodic Driving and Dissipation: In periodically driven open systems, the balance of drive-induced excitation and reservoir-induced relaxation determines the approach to periodic steady state, the appearance of nonequilibrium distribution functions distinct from equilibrium, and the realization of phenomena such as nonequilibrium phase transitions or time-crystalline order (Jiao et al., 7 Nov 2024).

6. Comparison to Other Many-Body Approaches

Systematic comparisons show that under weak or moderate fields, wavefunction-based methods (e.g., time-dependent coupled cluster) and NEGF-based approaches agree, but in regimes of strong driving and high entanglement, advanced NEGF with self-energy approximations (e.g., GW) better captures the correct density matrix evolution and spectral features (Reeves et al., 14 May 2024). This is linked to the ability of NEGF to redistribute spectral weight under drive, essential for correct physical description outside weak-coupling limits.

7. Outlook and Prospects

The Floquet NEGF framework has established itself as a central tool in the paper of periodically driven, strongly correlated, and open quantum systems. Ongoing developments include:

• Enhanced diagrammatic expansions in Floquet space for higher-order correlation and memory effects,
• Machine-learning-accelerated solvers for efficient, scalable simulations (Zhu et al., 13 Jul 2024),
• Further integration with ab initio electronic structure and density functional theory for realistic modeling of driven materials (Dolui et al., 2021),
• Methodologies for the treatment of non-Hermitian and stochastic dynamics in open environments (Lane et al., 2021),
• Application to quantum devices, quantum materials engineering, and exploration of nonequilibrium phases beyond static analogs.

The framework’s flexibility, rigorous mathematical foundation, and recent computational advances ensure it will remain indispensable in the quantitative prediction and conceptual understanding of Floquet-engineered quantum matter and nonequilibrium dynamics.