The main goal of the present thesis is to set up a functional renormalization group formalism in Floquet space to treat interacting, time periodic quantum dots and to investigate the consequential renormalization of the parameters and the transport through the dot. Building upon the time independent, steady state description in frequency space (Karrasch 2010, Jakobs 2009) and a time dependent FRG formulation (Kennes 2014), we tackle the steady state of periodically driven quantum dots. We focus on the long time behavior where all transients have died out, and therefore the entire system is characterized by the same periodicity as given by the driven external fields. As a consequence, we can transform the according time dependent flow equation to Floquet space using Floquet-Green’s functions. It allows us to study quantum dot systems in the whole range of driving frequency and amplitude in the presence of a small Coulomb interaction. We exemplify the potential of our approach by applying it to the interacting resonant level model (IRLM), describing an idealized single level quantum dot dominated only by charge fluctuations.We investigate the role of the driving frequency Ω in the RG flows of the time periodic parameters of the IRLM. The small driving amplitude limit allows to complement our numerical solution by analytic expressions of the renormalization of all dot parameters to the leading order of driving amplitude over mean value. Four different configurations are studied, where distinct combinations of the hopping and/or onsite energy of the dot are chosen to be time periodic. The transparent structure of the renormalization of the parameters in this limit, allows for an analytic treatment of all higher harmonics, where the various protocols reveal very different RG flows. Even beyond the small driving amplitude limit the RG flow is discussed with the help of an effective reservoir distribution function which can be defined in this setup and its form is defined by the ratio of driving amplitude and frequency (Suzuki 2015). The second focus is directed to the transport in the time periodic quantum dot systems and how it is affected by the Coulomb interaction. Based on the well studied parameter pump in the adiabatic limit (Brouwer 1998), where two parameters are varied periodically and phase shifted, we like to investigate the pumped charge in the whole regime of driving frequency and amplitude including interaction in such a setup. Further a single parameter pump is realized and the charge susceptibility as well as the mean current are studied. The latter reveals power law behavior on the driving frequency. A quantum master equation calculation in Floquet-Liouville space complements the FRG results to study the requirements of such a single parameter quantum pump. Finally, the conductance and the current are considered for several time periodic hoppings of non-sinusoidal form.
The main goal of the present thesis is to set up a functional renormalization group formalism in Floquet space to treat interacting, time periodic quantum dots and to investigate the consequential renormalization of the parameters and the transport through the dot. Building upon the time independent, steady state description in frequency space (Karrasch 2010, Jakobs 2009) and a time dependent FRG formulation (Kennes 2014), we tackle the steady state of periodically driven quantum dots. We focus on the long time behavior where all transients have died out, and therefore the entire system is characterized by the same periodicity as given by the driven external fields. As a consequence, we can transform the according time dependent flow equation to Floquet space using Floquet-Green’s functions. It allows us to study quantum dot systems in the whole range of driving frequency and amplitude in the presence of a small Coulomb interaction. We exemplify the potential of our approach by applying it to the interacting resonant level model (IRLM), describing an idealized single level quantum dot dominated only by charge fluctuations.We investigate the role of the driving frequency Ω in the RG flows of the time periodic parameters of the IRLM. The small driving amplitude limit allows to complement our numerical solution by analytic expressions of the renormalization of all dot parameters to the leading order of driving amplitude over mean value. Four different configurations are studied, where distinct combinations of the hopping and/or onsite energy of the dot are chosen to be time periodic. The transparent structure of the renormalization of the parameters in this limit, allows for an analytic treatment of all higher harmonics, where the various protocols reveal very different RG flows. Even beyond the small driving amplitude limit the RG flow is discussed with the help of an effective reservoir distribution function which can be defined in this setup and its form is defined by the ratio of driving amplitude and frequency (Suzuki 2015). The second focus is directed to the transport in the time periodic quantum dot systems and how it is affected by the Coulomb interaction. Based on the well studied parameter pump in the adiabatic limit (Brouwer 1998), where two parameters are varied periodically and phase shifted, we like to investigate the pumped charge in the whole regime of driving frequency and amplitude including interaction in such a setup. Further a single parameter pump is realized and the charge susceptibility as well as the mean current are studied. The latter reveals power law behavior on the driving frequency. A quantum master equation calculation in Floquet-Liouville space complements the FRG results to study the requirements of such a single parameter quantum pump. Finally, the conductance and the current are considered for several time periodic hoppings of non-sinusoidal form.
We present an extension of the functional renormalization group to Floquet space, which enables us to treat the long time behavior of interacting time periodically driven quantum dots. It is one of its strength that the method is neither bound to small driving amplitudes nor to small driving frequencies, i.e., very general time periodic signals can be considered. It is applied to the interacting resonant level model, a prototype model of a spinless, fermionic quantum dot. The renormalization in several setups with different combinations of time periodic parameters is studied, where the numerical results are complemented by analytic expressions for the renormalization in the limit of small driving amplitude. We show how the driving frequency acts as an infrared cutoff of the underlying renormalization group flow which manifests in novel power laws. We utilize the tunability of the effective reservoir distribution function in a periodically driven onsite energy setup to show how its shape is directly reflected in the renormalization group flow. This allows us to flexibly tune the power-law renormalization generically encountered in quantum dot structures. Finally, an in-phase quantum pump as well as a single parameter pump are investigated in the whole regime of driving frequency, demonstrating that the new power law in the driving frequency is reflected in the mean current of the latter.
We introduce a functional renormalization group framework formulated directly in the Floquet steady state that systematically incorporates frequency-dependent interaction effects. By retaining the frequency structure of the two-particle vertex up to second order in interaction strength, our approach provides controlled access to dynamical response functions and nonequilibrium transport in driven, interacting systems. Using the periodically driven single-impurity Anderson model as a paradigmatic example, we benchmark our results against state-of-the-art Floquet Green's function methods and find quantitative agreement for finite-frequency observables up to intermediate interaction strengths. Remarkably, we also show that static properties are often captured reliably by much simpler approximations, suggesting practical pathways for modeling driven quantum materials. Finally, we demonstrate that although periodic driving of the dot strongly broadens the Kondo resonance through inelastic scattering, it leaves the many-body Kondo cloud largely intact. This robustness suppresses Floquet replicas of the Kondo peak and leads to a partial persistence of Kondo pinning, highlighting the resilience of emergent many-body correlations under local periodic driving.
Time-periodic quantum systems are present all over science and technology. They have been successfully studied using Floquet theory, and a broad diversity of interesting phenomena such as coherent destruction of tunnelling, dynamical localization and creation of new phases of matter have been discovered. Furthermore, periodic driving has also been introduced as a tool for engineering novel quantum systems that have no static analogue. Besides these remarkable achievements for isolated systems, in reality, systems always interact with their surroundings. Taking the effect of the environment into account, time-periodic quantum systems have been mainly described using the Floquet-Markov approach, which consists in deriving a weak coupling master equation in the Floquet basis. In this thesis, we focus on how to overcome these standard approximations in the description of driven open quantum systems. We start by providing a general review on the topics of isolated driven systems, open quantum systems and the Floquet-Markov approach, before presenting the main results of this work. Under the effect of high-frequency driving, we provide an analytic solution to a large class of open quantum systems. Our method relies on discrete symmetries of the system-bath Hamiltonian and grants access to the time evolution operator of the full system, including bath degrees of freedom. The solution can be interpreted as the stroboscopic evolution of a family of observables under the influence of an effective static Hamiltonian. This provides a flexible simulation procedure of nontrivial Hamiltonians. The results are exemplified with the driven spin-boson model, where a coherent control of dissipation is observed. By a novel combination of Floquet theory, full counting statistics and a Markovian embedding strategy (reaction coordinate mapping), more intricate scenarios with additional driven system-environment interactions are addressed. Hereby, we can access the dynamics and thermodynamics of a periodically driven thermal machine. For the particular case where the working medium is a two-level system, we identify four different operating regimes, including a heat engine and a refrigerator. As the coupling strength is increased, their efficiency and coefficient of performance decrease and the refrigerator regime disappears. Notably, our model reproduces the setup of laser cooling of trapped ions in a specific parameter limit. Following the same combination of methods, we investigate electron pumping through a single quantum dot with periodically modulated energy and tunnelling amplitudes. In the high-frequency regime, the direction of the current can be controlled with the driving phase and a rectification effect that is exclusive to the non-Markovian character of the model is observed. In the low-frequency regime, we apply an additional transformation to view the model as three independent transport channels and observe that the coupling strength also controls the current direction. Studying charge fluctuations we validate that our model behaves as a single-electron source.
We study transport properties of quantum impurity systems using the functional renormalization group. The latter is an RG-based diagrammatic tool to treat Coulomb interactions in a fast and flexible way. Prior applications, which employed a simple first-order (Hartree-Fock-like) scheme to truncate the FRG flow equations within the Matsubara formalism, succeeded in accurately describing linear transport of various quantum dot geometries at zero temperature T=0. In a nutshell, advance in this Thesis is three-fold. First, we introduce a frequency-dependent second-order approximation in order to eventually compute finite-energy properties such as the conductance at T>0 (mainly focusing on the single impurity Anderson model). Second, a generalisation of the Hartree-Fock-like approach to Keldysh space allows for addressing the non-equilibrium steady-state dynamics of the interacting resonant level model. Third, we investigate the physics of a quantum dot Josephson junction as well as the charging of a single narrow level using the first-order scheme.
We introduce a real time version of the functional renormalization group, which allows us to study correlation effects on nonequilibrium transport through quantum dots. Our method is equally capable to address (i) the relaxation out of a nonequilibrium initial state into a (potentially) steady state driven by a bias voltage and (ii) the dynamics governed by an explicitly time-dependent Hamiltonian. All time regimes from transient to asymptotic can be tackled; the only approximation is the consistent truncation of the flow equations at a given order. As an application we investigate the relaxation dynamics of the interacting resonant level model, which describes a fermionic quantum dot dominated by charge fluctuations. Moreover, we study incoherence and relaxation phenomena within the ohmic spin-boson model by mapping the latter to the interacting resonant level model.
We study driven and open quantum systems described by a time-periodic Lindblad master equation. In closed systems, the stroboscopic dynamics can always be described by an effective time-independent Floquet Hamiltonian; this idea is the basis of Floquet engineering. However, in the presence of dissipation, the existence of an effective time-independent Floquet Lindbladian is not guaranteed due to the non-unitary nature of the evolution. Using Floquet theory, we construct a well-defined time-independent Floquet Lindbladian in an extended Sambe-Liouville space, transforming the initial time-dependent problem to a static and non-Hermitian eigenvalue problem. For harmonic driving, we introduce a matrix continued fraction method to nonperturbatively resum multiphoton processes and construct an effective Floquet Lindbladian acting only on the physical Liouville space. Compared to other high-frequency expansions, this method has the advantage of providing the whole infinite series expansion at once. Using a resolvent formalism, we show how to obtain a spectral Floquet representation of correlation functions of an open quantum system. As an application, we consider a dissipating two-level system in a linearly polarized field and calculate its resonance fluorescence spectrum. Furthermore, we consider a parametrically driven quantum dot with pump and loss for which we calculate its spectral function and current-voltage characteristics.