Correlated electron systems represent a wide class of materials which at equilibrium display fascinating properties. Several recent experimental breakthroughs in the field of femtosecond spectroscopy and cold atomic gases allow nowadays to investigate the real time dynamics of these many-body quantum systems. Since strongly correlated systems usually escape single particle approaches, the theoretical study of their dynamics constitutes a formidable problem which necessitates the development of novel techniques. In this Thesis we investigate the out-of-equilibrium physics of simple paradigmatic models that are believed to capture some essential physics of interacting fermions by means of the time dependent extension of the Gutzwiller Variational Approach. After an introductory Chapter on the recent results in this field, in Chapter 2 we present the Gutzwiller Approach in-and-out of equilibrium. In Chapter 3 we investigate the dynamics for the single band Hubbard model after a linear ramp of the Coulomb interaction. We will show that a dynamical transition appears for any duration of the ramp; this dynamical point is adiabatically connected to the zero temperature Metal-to-Insulator transition. We will then consider the role of quantum fluctuations beyond mean field. In Chapter 4 we consider the dynamics of an initial antiferromagnetic state under a quench of the interaction in the single band fermionic Hubbard model. We will show that non-thermal ordered states survive more than expected and that two different nonequilibrium antiferromagnets can be distinguished. Finally in Chapter 5 we will consider a two-band Hubbard model which we believe captures the main physics of the paradigmatic compound vanadium sesquioxide, V2 O3 . After an investigation of the equilibrium properties for this model, we will provide evidences that non-thermal metallic phases can emerge upon an excitation of a Mott insulator.
Correlated electron systems represent a wide class of materials which at equilibrium display fascinating properties. Several recent experimental breakthroughs in the field of femtosecond spectroscopy and cold atomic gases allow nowadays to investigate the real time dynamics of these many-body quantum systems. Since strongly correlated systems usually escape single particle approaches, the theoretical study of their dynamics constitutes a formidable problem which necessitates the development of novel techniques. In this Thesis we investigate the out-of-equilibrium physics of simple paradigmatic models that are believed to capture some essential physics of interacting fermions by means of the time dependent extension of the Gutzwiller Variational Approach. After an introductory Chapter on the recent results in this field, in Chapter 2 we present the Gutzwiller Approach in-and-out of equilibrium. In Chapter 3 we investigate the dynamics for the single band Hubbard model after a linear ramp of the Coulomb interaction. We will show that a dynamical transition appears for any duration of the ramp; this dynamical point is adiabatically connected to the zero temperature Metal-to-Insulator transition. We will then consider the role of quantum fluctuations beyond mean field. In Chapter 4 we consider the dynamics of an initial antiferromagnetic state under a quench of the interaction in the single band fermionic Hubbard model. We will show that non-thermal ordered states survive more than expected and that two different nonequilibrium antiferromagnets can be distinguished. Finally in Chapter 5 we will consider a two-band Hubbard model which we believe captures the main physics of the paradigmatic compound vanadium sesquioxide, V2 O3 . After an investigation of the equilibrium properties for this model, we will provide evidences that non-thermal metallic phases can emerge upon an excitation of a Mott insulator.
Correlated systems are a wide class of materials in which the strong electron-electron repulsion is the origin of very fascinating and unusual properties, among which metal-to-insulator transitions and high temperature superconductivity are the most striking examples. In the recent years, the fast development of non-adiabatic probing techniques opened new interesting perspectives for the investigation of such materials in non-equilibrium conditions. In this thesis, we discuss the theoretical description of few relevant cases which represent different examples of non-equilibrium phenomena in correlated materials. In particular, we will focus on the dynamics following a sudden excitation and the coupling to an external driving field. As a first example we consider the dynamics across a phase transition, namely we explore the possibility of driving a phase transition as the result of a sudden excitation, as e.g. the coupling with a short light pulse. We consider systems showing different equilibrium phases and study the conditions under which the off-equilibrium dynamics may lead to non-trivial dynamical phase transitions. A different case is represented by the dynamics induced by a driving electric field. This problem is particularly relevant for the possible applications of correlated materials in electronic devices. Here we consider the paradigmatic case of a correlated material coupled to external sources which impose a finite bias across the system. We analyze the formation and the properties of the non-equilibrium stationary states in which a finite current flows through the system. This allows us to study the non-linear response properties of a correlated system. In this context, a particularly relevant aspect is the problem of the dielectric breakdown of a Mott insulator, namely the formation of conducting states in the Mott insulating phase. In this thesis we explore different mechanisms leading to such possibility. First we discuss a quantum tunneling mechanism of carriers driven across the insulating gap by the effect of strong electric-fields. Eventually, we discuss the possibility of a resistive transition from an insulating to a metallic state induced by the application of an external electric-field.
We review the recently proposed extension of the Gutzwiller approximation, M. Schiro' and M. Fabrizio, Phys. Rev. Lett. 105, 076401 (2010), designed to describe the out-of-equilibrium time-evolution of a Gutzwiller-type variational wave function for correlated electrons. The method, which is strictly variational in the limit of infinite lattice-coordination, is quite general and flexible, and it is applicable to generic non-equilibrium conditions, even far beyond the linear response regime. As an application, we discuss the quench dynamics of a single-band Hubbard model at half-filling, where the method predicts a dynamical phase transition above a critical quench that resembles the sharp crossover observed by time-dependent dynamical mean field theory. We next show that one can actually define in some cases a multi-configurational wave function combination of a whole set of mutually orthogonal Gutzwiller wave functions. The Hamiltonian projected in that subspace can be exactly evaluated and is equivalent to a model of auxiliary spins coupled to non-interacting electrons, closely related to the slave-spin theories for correlated electron models. The Gutzwiller approximation turns out to be nothing but the mean-field approximation applied to that spin-fermion model, which displays, for any number of bands and integer fillings, a spontaneous Z2 symmetry breaking that can be identified as the Mott insulator-to-metal transition.
Materials with strongly correlated electrons show a multitude of unconventional phenomena, such as superconductivity or magnetism, that cannot be explained in a single-particle picture and necessitate to take many-body effects into account. A route to explain the origin of these phenomena consists in constructing effective models for these materials that are then solved, often using numerical techniques. Paradigmatic models for two different classes of strongly correlated electron systems are investigated in this thesis: The Hubbard model is a paradigmatic model to describe the Mott metal-insulator transition (MIT) and the Kondo lattice model describes the low-energy physics in heavy fermion compounds. Both models are analyzed numerically using different variations of a quantum cluster technique, the variational cluster approximation (VCA). The first part focuses on the MIT, as it is investigated experimentally in quasi-two-dimensional charge-transfer salts. Its nature and associated universality class are still heavily discussed and both experiments and theoretical approaches come to different conclusions. In this thesis, the Mott transition is investigated on an anisotropic two-dimensional Hubbard model at half-filling. By using a control parameter, which induces this anisotropy, strong evidence for Mott quantum criticality is found in weakly coupled Hubbard chains. The results at zero temperature show that the second-order critical end point Tc of the interaction-driven metal-insulator transition can be tuned down to zero at strong anisotropy. Further results for the antiferromagnetic phase suggest a similar picture and motivate adding the anisotropy as a new axis in the phase diagram to account for a low-temperature critical end point of the Mott transition. The second part of this thesis focuses on unconventional phases that emerge in a paradigmatic model for heavy fermion systems, the Kondo lattice model. Using the VCA, the two-dimensional Kondo lattice model is investigated in the paramagnetic phase as well as in phases with broken symmetry. An antiferromagnetic phase at weak coupling and a phase with Kondo-singlet formation at strong coupling are analyzed. Within the antiferromagnetically ordered region, two phases with different Fermi-surface topology, which are separated by a discontinuous transition, are identified. The model is also tested for s-wave superconductivity as found recently using dynamical mean-field theory, but no indications for robust local pairing are seen within VCA. Instead, nodal d-wave superconductivity is found and analyzed in a large range of couplings and electron fillings, and its interplay with antiferromagnetic order at weak coupling is discussed. This motivates further studies on extended models that might allow for a close comparison with experiments on heavy fermion systems. Understanding the formation of unconventional phases in strongly correlated electron systems is not only of fundamental interest, but it is also essential for designing new functional materials. Although the results of approximate cluster techniques have to be handled with care, the new aspects presented in this thesis pave the way for further investigations in this direction.
For weakly interacting many-body quantum systems one can apply the extensive framework of perturbative quantum field theory. On the other hand, strong interactions can not be treated practically in perturbation theory because of the absence of a small control parameter. In this context variational methods are a powerful tool to obtain reliable informations regarding the properties of the ground state of the strongly correlated electron system. In this thesis we present novel formulations of two widely used methods: the Gutzwiller wave function and the slave-spin theory. These techniques share a similar philosophy and the extensions are based on the common attitude towards a more detailed description of the high-energy incoherent excitations characterizing interacting electron systems together with low-energy quasiparticles excitations, captured by standard approaches. Our task is not only crucial for a faithful description of the insulating phase but also improves the variational characterization of the low-energy quasiparticle excitations. We apply the novel ghost-Gutzwiller wave function technique to tackle an intriguing phenomenon, namely the exciton Mott transition in photoexcited semiconductors. Despite being a quite old topic that goes back to the 1970's, the nature of the exciton Mott transition still defies a complete understanding. By considering an idealised model of photoexcited semiconductors we unveil the important role of the exciton binding energy in determining the nature of the transition. Moreover, our results uncover rather anomalous electron-hole liquid phase next to the transition, which still sustains excitonic excitations although being a degenerate Fermi liquid of heavy mass quasiparticles. By means of the Dirac-Frenkel variational principle, we generalize the ghost-Gutzwiller wave function to study the out of equilibrium dynamics of strongly correlated electron systems. Numerical results on the single-band Hubbard model show a remarkable agreement with those obtained with time-dependent dynamical mean field theory. We believe that the method opens the way to several promising developments and future applications. Concerning the slave-particle approach, we consider Anderson impurity models and we show that, within our formulation, the constraint on the slave-spin variable can be removed, leaving thermal average free by any projection on the "physical" subspace of the enlarged Hilbert space. To the best of our knowledge, this represents an exception and shows that our formulation is more convenient than other slave-particle approaches, where the constraint have to be implemented explicitly. The method, suited to deal with impurity models, finds direct application in studying time-dependent transport across an interacting quantum dot. To this aim we formulate, by means of the Keldysh Green's function, the slave-spin mean-field theory in the out of equilibrium framework and we apply the method to the dynamics of a driven quantum dot. Finally we study the current-voltage characteristic of an interacting quantum dot tunnel-coupled to the edge of a superconductive nanowire. The appearance of the topological Majorana edge mode is signalized by distinctive features in the charge transport properties of the junction. Apart from the well known half-integer zero bias conductance we find that the topological region is characterized by a vanishing Fano factor, that can be measured in experiments to detect the presence of a Majorana zero mode hybridized with the quantum dot.
Strongly correlated quantum systems display a myriad of complex, fascinating behaviour. The competition between charge, spin, orbital, or lattice degrees of freedom and their respective energy scales gives rise to rich and complex emergent phases. More recently, the non-equilibrium properties of correlated quantum many-body systems have come into the focus of active research, with remarkable advances. While the advent of tensor network based simulations on the theoretical side has allowed access to the full time-evolution of the quantum state as it traverses the Hilbert space, the experimental progress in the ultracold atom community has allowed physicists to enter the paradigm of analog quantum simulation. Their tunability and isolation from the environment makes them ideal platforms to emulate and study open problems in condensed matter physics and go beyond the simulational capabilities of state of the art classical simulations. In this thesis, we explore the dynamics of correlated Fermi gases. We develop and present different facets of applicability of radiofrequency modulation techniques to drive the system out of equilibrium. We begin by considering a homogeneous, three-dimensional Fermi gas in the BCS-BEC crossover and develop an off-resonant radiofrequency transfer scheme to excite the Higgs mode of the superfluid. In a subsequent study we investigate the tunability of the coherence between Cooper pairs by tuning the duration of interaction ramps of the internal interaction strength. These works highlight the possibility to externally tune and stabilise complex quantum many-body states away from equilibrium. Using quasi-exact matrix product state simulations, we study the response of an attractively interacting, one-dimensional Fermi-Hubbard model to weak radiofrequency perturbations. We reveal the emergence of two distinct dynamical regimes in the time evolution and are able to relate it back to the underlying excitation spectrum of the system. Finally, we explore the possibility to perform quantum quenches through radiofrequency π-pulses, achieving nearly complete population transfer between different interacting fermionic states. We reveal non-trivial dynamical effects in the pair correlation of the final state, as well as the excitation of a collective trap mode of the system.
We study the out-of-equilibrium dynamics of the fermionic Hubbard model induced by a linear ramp of the repulsive interaction U from the metallic state through the Mott transition. To this extent, we use a time-dependent Gutzwiller variational method and complement this analysis with the inclusion of quantum fluctuations at the leading order, in the framework of a Z2 slave-spin theory. We discuss the dynamics during the ramp and the issue of adiabaticity through the scaling of the excitation energy with the ramp duration $\ensuremath{\tau}$. In addition, we study the dynamics for time scales longer than the ramp time, when the system is again isolated and the total energy conserved. We establish the existence of a dynamical phase transition analogous to the one present in the sudden quench case and discuss its properties as a function of final interaction and ramp duration. Finally, we discuss the role of quantum fluctuations on the mean-field dynamics for both long ramps, where spin-wave theory is sufficient, and for very short ramps, where a self-consistent treatment of quantum fluctuations is required in order to obtain relaxation.