Periodically driven nonequilibrium many-body systems are interesting because they have quasi-energy spectra, which can be tailored by controlling the external driving fields. We derive the general spectral representation of retarded Green functions in the Floquet regime, thereby generalizing the well-known Lehmann representation from equilibrium many-body physics. The derived spectral Floquet representation allows us to prove the non-negativity of spectral densities and to determine exact spectral sum rules, which can be employed to benchmark the accuracy of approximations to the exact Floquet many-body Green functions.
Periodically driven nonequilibrium many-body systems are interesting because they have quasi-energy spectra, which can be tailored by controlling the external driving fields. We derive the general spectral representation of retarded Green functions in the Floquet regime, thereby generalizing the well-known Lehmann representation from equilibrium many-body physics. The derived spectral Floquet representation allows us to prove the non-negativity of spectral densities and to determine exact spectral sum rules, which can be employed to benchmark the accuracy of approximations to the exact Floquet many-body Green functions.
We derive an analytical expression for the steady-state quantum statistics of periodically driven quantum systems coupled to a bath using the nonequilibrium Green's function (NEGF) formalism. By embedding Floquet theory into NEGF, we obtain closed expressions for the retarded, advanced, and lesser Green's functions in the Floquet representation, yielding the Floquet Fermi distribution in which the steady-state occupation is expressed as a weighted sum of Fermi functions shifted by integer multiples of the driving frequency. The weights are determined solely by the Fourier components of the micromotion operator, providing a transparent interpretation of Floquet sideband occupations. Our analysis extends beyond the diagonal commuting Hamiltonians treated in earlier work, and further shows that the robust Floquet distribution remains valid for a broad class of weakly coupled bath spectral functions beyond the ideal featureless-bath approximation. Finally, we establish a Floquet version of the Landauer formula for the DC part of the current, in which the equilibrium Fermi functions are replaced by their Floquet-modified counterparts. Together, these results provide a coherent description of Floquet quantum statistics and transport in periodically driven open quantum systems.
Periodically driven quantum systems provide a versatile platform for engineering and exploring non-equilibrium states of matter that have no equilibrium counterparts– a field commonly re- ferred to as Floquet engineering. The research presented in this thesis addresses the microscopic interpretation of experimental spectroscopic signatures, the stability of these driven states in the presence of strong many-body interactions, and the emergence of novel forms of non-equilibrium criticality encoded in quantum entanglement. The first part of the thesis focuses on the interpretation of time- and angle-resolved photo- emission spectroscopy (tr-ARPES), the primary experimental tool for detecting Floquet–Bloch sidebands (FBs). The measured signal reflects interference between processes occurring both inside the material and in the vacuum. In particular, Floquet-dressed initial states coexist and interfere with Volkov-dressed final states arising from the laser-assisted photoelectric effect (LAPE). Their experimental identification requires careful comparison with theoretical predic- tions. Using graphene as a model Dirac material, we perform a systematic comparison between two theoretical frameworks: the first-order perturbative Born approximation (PB1) and time- dependent nonequilibrium Green’s functions (tdNEGF), and compare them with experimental results. We disentangle intrinsic Floquet states, LAPE, and their interference, and quantify how polarization, incidence angle, and near-surface screening shape the observed spectra. Our findings show that, once matrix elements are included, both methods capture qualitative trends, but tdNEGF is necessary to describe the full energy–momentum-resolved features, including hybridization gaps. While non-interacting systems exhibit well-defined FBs, interacting systems absorb energy and can evolve toward an infinite-temperature state. To study this competition between coherent driving and many-body heating, in the second part we consider a one-dimensional chain of interacting spinless fermions using exact diagonalization (ED) and matrix product state (MPS) methods. We compute time-dependent spectral functions and correlation measures, and compare the spectral width of the many-body excitation continuum with the driving frequency to define low- and high-frequency regimes. In our parameter regime, low-frequency driving leads to strong overlap between Floquet sectors, resulting in rapid heating and loss of coherence. For weakly populated continua, we find a transient timescale for the suppression of FBs. When the driving frequency exceeds the continuum width, FBs are well separated and long-lived. Thefinalpartofthethesisinvestigatesnon-equilibriumphenomenafromaninformation-theoretic perspective. Wereportanewclassofdynamicalphasetransitions,termedtemporalentanglement transitions (TET) characterized by non-analytic reorganizations of the entanglement spectrum, in a periodically driven transverse-field Ising chain. These transitions occur in the spectrum of the entanglement Hamiltonian and are signaled by dynamical spontaneous symmetry breaking, marked by the closure of the Schmidt gap and parity flips. These transitions remain invisible to conventional local observables. Finite-size scaling reveals that TET belong to a universality class with a correlation-length exponent ν = 1, mirroring equilibrium Ising universality despite their purely dynamical origin.
Abstract Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.
Abstract The spectral properties of the nonequilibrium Green’s functions are explored. Causality and sum rules are shown to be completed by the extended quasiparticle picture. The off-shell motion is seen to become visible in satellite structures of the spectral function. Different forms of ansatz to reduce the two-time Green’s function to a one-time reduced density matrix are discussed with respect to the consistency to other approximations. We have seen from the information contained in the correlation function that the statistical weight of excitations with which the distributions are populated are given by the spectral function. This momentum-resolved density of state can be found by the retarded and advance functions.
Abstract We derive a general procedure for evaluating the nth derivative of a time-dependent operator in the Heisenberg representation and employ this approach to calculate the zeroth to third spectral moment sum rules of the retarded electronic Green’s function and self-energy for a system described by the Holstein–Hubbard model allowing for arbitrary spatial and time variation of all parameters (including spatially homogeneous electric fields and parameter quenches). For a translationally invariant (but time-dependent) Hamiltonian, we also provide sum rules in momentum space. The sum rules can be applied to various different phenomena like time-resolved angle-resolved photoemission spectroscopy and benchmarking the accuracy of numerical many-body calculations. This work also corrects some errors found in earlier work on simpler models.
We derive exact expressions for the zeroth and the first three spectral moment sum rules for the retarded Green's function and for the zeroth and the first spectral moment sum rules for the retarded self-energy of the inhomogeneous Bose-Hubbard model in nonequilibrium, when the local on-site repulsion and the chemical potential are time-dependent, and in the presence of an external time-dependent electromagnetic field. We also evaluate these expressions for the homogeneous case in equilibrium, where all time dependence and external fields vanish. Unlike similar sum rules for the Fermi-Hubbard model, in the Bose-Hubbard model case, the sum rules often depend on expectation values that cannot be determined simply from parameters in the Hamiltonian like the interaction strength and chemical potential but require knowledge of equal-time many-body expectation values from some other source. We show how one can approximately evaluate these expectation values for the Mott-insulating phase in a systematic strong-coupling expansion in powers of the hopping divided by the interaction. We compare the exact moment relations to the calculated moments of spectral functions determined from a variety of different numerical approximations and use them to benchmark their accuracy.