In the different chapters of this dissertation we investigate multiple non-perturbative approaches that allow the study of solids subjected to time dependent fields. In the second chapter, we develop a very general non-perturbative and flexible approach that allows to study periodically driven quantum systems via an effective time-independent theory. It makes use of renormalization-group-like flow equations to find highly accurate effective Hamiltonians[Phys. Rev. X 9, 021037 (2019)]. The range of validity of the approach is checked numerically for various 1D spin systems and confirmed to be beyond the reaches of ordinary perturbative methods. It also compares favourably to another non-perturbative approach. In the third chapter we examine the wealth of different approximations to the time evolution operator that are known from the literature. Inspired by Hamilton-Jacobi theory we find a useful reformulation of the equation for the time evolution operator [Phys. Rev. A 100, 012132 (2019)]. This allows us to find the various known approximations of the time evolution operator in a unified fashion and new ones. Interestingly the RG-like flow equations from the second chapter are found to be one specific limit of the more general approach developed here. A periodically driven Ising model is used to verify the range of validity of the different approximations. This allows us to put them into a hierarchy. Up until here the methods that were explored are most valid in the high to mid frequency regime. In the fourth chapter we show how for the case of a weakly driven system a simple approach can be used to treat the low frequency regime [Phys. Rev. B 101, 024303 (2020)]. To achieve this we derive a quasi-energy dependent Hamiltonians and very crucially require self-consistency for the quasi-energies. For the simple two band example of single layer graphene it is found to be highly accurate approach that also allows for accurate descriptions of even sensitive quantities like Chern numbers For the last fifth chapter of the thesis we switch gears and consider a specific system of much recent interest - twisted bilayer graphene. We investigate how light that is altered by the boundary conditions of a metallic waveguide can change the strength of interlayer couplings [arXiv:2001.04416 (2020)]. We investigate how this change affects the twist angles at which flat bands appear and find very simple analytic results that are in good agreement with a numerical treatment. In the sixth chapter of the thesis, we investigate the effects circularly polarized light has on twisted bilayer graphene[ arXiv:2002.05124 (2020)]. We derive two effective Hamiltonians valid in different driving regimes. One is valid for weak drives but low frequencies and the other for relatively strong drives. The Hamiltonian for the strongly driven regime is compared to exact numeric results and found to perform better than other common methods. We analyse the newly generated terms in both Hamiltonians for their symmetry.
In the different chapters of this dissertation we investigate multiple non-perturbative approaches that allow the study of solids subjected to time dependent fields. In the second chapter, we develop a very general non-perturbative and flexible approach that allows to study periodically driven quantum systems via an effective time-independent theory. It makes use of renormalization-group-like flow equations to find highly accurate effective Hamiltonians[Phys. Rev. X 9, 021037 (2019)]. The range of validity of the approach is checked numerically for various 1D spin systems and confirmed to be beyond the reaches of ordinary perturbative methods. It also compares favourably to another non-perturbative approach. In the third chapter we examine the wealth of different approximations to the time evolution operator that are known from the literature. Inspired by Hamilton-Jacobi theory we find a useful reformulation of the equation for the time evolution operator [Phys. Rev. A 100, 012132 (2019)]. This allows us to find the various known approximations of the time evolution operator in a unified fashion and new ones. Interestingly the RG-like flow equations from the second chapter are found to be one specific limit of the more general approach developed here. A periodically driven Ising model is used to verify the range of validity of the different approximations. This allows us to put them into a hierarchy. Up until here the methods that were explored are most valid in the high to mid frequency regime. In the fourth chapter we show how for the case of a weakly driven system a simple approach can be used to treat the low frequency regime [Phys. Rev. B 101, 024303 (2020)]. To achieve this we derive a quasi-energy dependent Hamiltonians and very crucially require self-consistency for the quasi-energies. For the simple two band example of single layer graphene it is found to be highly accurate approach that also allows for accurate descriptions of even sensitive quantities like Chern numbers For the last fifth chapter of the thesis we switch gears and consider a specific system of much recent interest - twisted bilayer graphene. We investigate how light that is altered by the boundary conditions of a metallic waveguide can change the strength of interlayer couplings [arXiv:2001.04416 (2020)]. We investigate how this change affects the twist angles at which flat bands appear and find very simple analytic results that are in good agreement with a numerical treatment. In the sixth chapter of the thesis, we investigate the effects circularly polarized light has on twisted bilayer graphene[ arXiv:2002.05124 (2020)]. We derive two effective Hamiltonians valid in different driving regimes. One is valid for weak drives but low frequencies and the other for relatively strong drives. The Hamiltonian for the strongly driven regime is compared to exact numeric results and found to perform better than other common methods. We analyse the newly generated terms in both Hamiltonians for their symmetry.
Driving a quantum system periodically in time can profoundly alter its long-time correlations and give rise to exotic quantum states of matter. The complexity of the combination of many-body correlations and dynamic manipulations has the potential to uncover a whole field of new phenomena, but the theoretical and numerical understanding becomes extremely difficult. We now propose a promising numerical method by generalizing the density matrix renormalization group to a superposition of Fourier components of periodically driven many-body systems using Floquet theory. With this method we can study the full time-dependent quantum solution in a large parameter range for all evolution times, beyond the commonly used high-frequency approximations. Numerical results are presented for the isotropic Heisenberg antiferromagnetic spin-1/2 chain under both local(edge) and global driving for spin-spin correlations and temporal fluctuations. As the frequency is lowered, we demonstrate that more and more Fourier components become relevant and determine strong length- and frequency-dependent changes of the quantum correlations that cannot be described by effective static models.
This thesis covers my work in the field of closed, repeatedly driven, Hamiltonian systems. These systems do not exchange particles with the surrounding environment and their time-evolution is described by Hamilton's equations of motion (in the classical framework) or the Schroedinger equation (in the quantum framework). Their interaction with the environment is encoded into the time-dependence of the system's Hamiltonian. Chapter 1 is an "Overview" in which the status of the field, my contributions and future prospective are outlined. Chapters 2 to 4 provide the theoretical background which is used in Chapters 5 to 7 to derive some original results. These results show that in Hamiltonian systems, after many driving events, universal properties emerge. In particular, using the framework of the linear Boltzmann equation, I have studied the dynamics of a mobile, light impurity in a gas of heavy particles. The impurity's kinetic energy increases and, in the long time limit, approaches a non-thermal asymptotic distribution. The significance of this work is to show explicitly the emergence of a non-thermal distribution in a closed, driven system. Moreover, using the work-fluctuation theorems, I have studied the character of the energy distribution of a generic isolated system driven according a generic protocol. Both thermal and non-thermal distributions can be realized for the same system by changing the characteristics of the driving protocol. These two different regimes are separated by a dynamical phase transition. Finally, I have used the Floquet Theory and the Magnus Expansion to analyze the behavior of a generic interacting system which is driven periodically in time. For fast driving the system is unable to absorb energy and remains localized in the low energy part of the Hilbert space while for slow driving the system absorbs energy and, in the long time limit, it is delocalized in the entire Hilbert space. These two qualitatively different behaviors are separated by a many-body localization transition which is related to the break down of the Magnus expansion at the critical value of the driving frequency.
This dissertation studies interacting electron models for quantum matter from two perspectives with different approaches respectively. The first part of this text focuses on rigorously proving Lieb-Robinson bounds for continuous space integer quantum Hall systems perturbed by short range two-body interactions. A Lieb-Robinson bound is an operator norm inequality that implies a finite rate of information propagation in a quantum system, a low energy analogue to the relativistic limit given by the speed of light. Many results of this nature have been obtained for lattice systems, but over the last several years, progress has been made in identifying classes of continuous space models that admit propagation estimates. In this thesis, we prove Lieb-Robinson bounds for two-dimensional continuum fermions in a constant, perpendicular magnetic field using the strategy of Gebert et al. \cite{gebert:2020}, and use this result to prove strong continuity of the Heisenberg dynamics in the limit where inter-particle interactions occur in infinite volume. We then shift to a different framework, namely that of lattice-localized frames, to represent a class of interacting quantum Hall systems in continuous space, as introduced by Bachmann and De Nittis, \cite{bachmann2025lieb}. With this framework, we define a family of parametrized Hamiltonians and construct a continuum analogue of the Hastings-Wen quasi-adiabatic evolution, i.e. dynamics in parameter space that satisfy Lieb-Robinson bounds themselves. Moreover, we then directly apply this to prove quantization of Hall conductance for these models. The new results of these sections are based on works in preparation, \cite{hingorani2026lieb,hingorani2026quantization}. The second portion of this dissertation focuses on two specific materials and phases: magic angle twisted bilayer graphene (MATBG) and superconducting lanthanum nickel gallium-2 (LaNiGa2). MATBG has garnered a tremendous amount of attention since works such as \cite{MB} and \cite{tarnopolsky2019origin}, which first suggested the existence of flat bands in the electronic band structure of twisted bilayer graphene: bilayer graphene with a relative twist between the layers. Astonishingly, this phenomenon occurs for specific twist angles dubbed magic angles. This quenching of kinetic energy sets the stage for electron-electron interactions to dominate the behavior of the material. The onset of Mott insulating states at several carrier densities as seen through peaks in inverse compressibility \cite{expts-b} drew our interest. In collaboration with Oitmaa and Singh, \cite{hingorani2022onset}, we study the SU(4) Fermi-Hubbard model as a candidate effective model for describing the low energy behavior of MATBG, where we uncover the onset of charge incompressibility and Mott gaps, indicating the existence of the insulating phase.Additionally, we investigate the superconducting pairing mechanism of LaNiGa2, which has been highlighted as a potential time-reversal symmetry breaking (TRSB) material below its critical temperature due to experimental evidence for internal magnetization, \cite{Ghosh2020}, \cite{Hillier2012}, \cite{Sundar2024}. In our recent work, \cite{tsnb-1mxc}, we report the existence of a Hebel-Slichter (HS) peak in the nuclear-spin relaxation rate, an effect attributed to the divergence in the electronic density of states as the superconducting gap opens. Furthermore, we study models with triplet and singlet spin pairing respectively to ascertain a mechanism that reproduces a HS peak. In doing so, we find that two distinct superconducting gaps and TRSB may not coexist, which raises questions about whether LaNiGa2 really does break time reversal symmetry or whether a different model is required to fully characterize this material in its superconducting phase.
The Chapter 2 is devoted to the study of the transport properties of a quantum wire, described by the Tomonaga-Luttinger model, in the presence of a backscattering potential provided by several point-like or extended time-dependent impurities (barriers) and find regimes of conductance enhancement. In Chapte 3 we study a set of crossed 1D systems, which are coupled with each other via tunnelling at the crossings. Concentrating on the case of a single crossing, we were able to explain recent experiments on crossed metallic and semiconducting nanotubes, which showed the presence of localized states in the region of crossing. The Chapter 4 treats the effects of disorder and interactions in a quantum Hall ferromagnet. We study the problem by projecting the original fermionic Hamiltonian into magnon states, which behave as bosons in the vicinity of the ferromagnetic ground state. The approach permits the reformulation of a strongly interacting model into a non-interacting one. The latter is a non-perturbative scheme that consists in treating the two-particle neutral excitations of the electron system as a bosonic single-particle. Indeed, the employment of bosonization facilitates the inclusion of disorder in the study of the system. It has been shown previously that disorder may drive a quantum phase transition in the Hall ferromagnet. However, such studies have been either carried out in the framework of the nonlinear sigma model, as an effective low-energy theory, or included the long-range Coulomb interaction in a quantum description only up to the Hartree-Fock level. Here, we establish the occurrence of a disorder-driven quantum phase transition from a ferromagnetic 2DEG to a spin glass phase by taking into account interactions between electrons up to the random phase approximation level in a fully quantum description. In the Chapter 5 we derive and discuss an experimentally realistic model describing ultracold atoms in an optical lattice including a commensurate, but staggered, Zeeman field. The resulting band structure is quite exotic; fermions in the third band have an unusual rounded picture-frame Fermi surface (essentially two concentric squircles), leading to imperfect nesting. We develop a generalized theory describing the spin and charge degrees of freedom simultaneously, and show that the system can develop a coupled spin-charge-density wave order. This ordering is absent in studies of the Hubbard model that treat spin and charge density separately.
The work presented in this dissertation is centered around the phenomena of correlations between members of a system, and how these correlations lead to unique effects which the system is in a non-equilibrium state. The systems described in this work are divided into two sections: dynamical systems and networks. The work on dynamical systems is focused on how correlated systems can gradually evolve when strongly driven. The work on networks is focused on how percolation is affected by correlations.Dynamical systems focuses on the motion of particles in the presence of an external driving force, specifically focusing on nonlinear effects observable in three different examples of correlated systems which are strongly driven by an external field which is rapidly oscillating in time. The first system is a classical system of coupled oscillators, the second is a quantum system with avoided level crossings and the third is the conduction level of a semiconductor. All three systems are strongly driven and specifically driven far from an initial equilibrium. It is posited that these strongly driven systems will reach quasi-stationary states, where the effects of the external driving can be classified on two time scales, slow evolution and rapid oscillation. The classical system is studied as a classical field theory, the effects of the strong driving force is observed in the qualitative properties of the system's effective potential. In the presence of a strong driving force this effective potential will exhibit points whose stability will change depending on the parameters of the driving force. In the quantum systems the effects of the external driving are seen in the Floquet quasienergy spectrum. In the third system, the strong driving force results in a renormalization of the bandwidth. If the driving force is strong enough the band can be completely inverted. Again working in the Floquet representation, the inverted band will exhibit multiplication of the driving frequency, specifically the third harmonic will become the dominant frequency and the fundamental frequency will be suppressed. Correlations in networks have many different aspects, correlations can affect how a network is formed (via attachment models) or how a random process is able to percolate across a random graph. The effects of correlated percolation is discussed in the second part of this dissertation, where a process can affect single sites on a graph or a cluster of sites simultaneously. It is shown that a lower bound on the percolation threshold can be established, and this bound is dependent on the size of clusters present.
Dynamique des systèmes quantiques en basses dimensions guidée hors équilibre Cette thèse analyse certains aspects de la dynamique hors équilibre de systèmes quantiques unidimensionnels lorsqu’ils sont soumis à des champs externes dépendant du temps. Nous considérons plus particulièrement le cas des forçages périodiques, et le cas d’une variation temporelle lente d’un paramètre de l’Hamiltonien qui permet de traverser une transition de phase quantique. La première partie contient une présentation des notions, des modèles et des outils nécessaires pour comprendre la suite de la thèse, avec notamment des rappels sur les modèles quantiques critiques (en particulier sur les chaines de spin et sur le modèle de Bose-Hubbard), le mécanisme de Kibble-Zurek, et la théorie de Floquet. Ensuite, nous étudions la dynamique hors équilibre des gaz de Tonks-Girardeau dans un potentiel harmonique dépendant du temps par différentes techniques : développements perturbatifs, diagonalisation numérique exacte et solutions analytiques exactes basées sur la théorie des invariants dynamiques d’Ermakov-Lewis. Enfin, nous analysons la dynamique hors équilibre des systèmes quantiques ouverts markoviens soumis à des variations périodiques des paramètres du système et de l’environnement. Nous formulons une théorie de Floquet afin d’obtenir des solutions exactes des équations de Lindblad périodiques. Ce formalisme de Lindblad-Floquet est utilisé pour obtenir une caractérisation exacte du fonctionnement en temps fini des machines thermiques quantiques.