This thesis concerns two topics where symmetry and strong correlations in time and space are important. The first topic is the theory of dynamical systems, and this theory deals with generic phenomena which are often convenient to study in very simple models. Dynamical systems can be divided in two basic classes: dissipative and conservative systems. Dissipation often effectively reduces the degrees of freedom in a dynamical system, and it is often sufficient to study low-dimensional models. An example is the one- dimensional circle map. The transition from ordered to chaotic motion in a circle map with extra symmetry is studied in Paper I using renormalization group techniques. Special emphasis is put into the possible breakdown of the extra symmetry. An example of a conservative system is a particle moving in a potential field, and typical for the dynamics is the large number of periodic trajectories. A particularly simple class of conservative systems is the integrable systems. In such a system there are continuous manifolds of marginally stable periodic orbits, which are divided into discrete sets of stable and unstable periodic orbits if the potential of the system is nonlinearly perturbed. In Paper II we demonstrate how the stable and unstable orbits in a quartic potential exchange stability several times as the strength of the nonlinear perturbation is increased. The second topic of this thesis is the ceramic high-temperature superconductors. Apart from being superconducting these materials show several other phases dependent on the doping. The 2-dimensional Hubbard model has been suggested as a highly simplified model for the electronic properties of these materials, and we study an extended version of this model at arbitrary temperature and doping using a theory capable of describing a rich variety of states simultaneously. Important is that we use no a priori assumptions of the state of the system. The high degree of freedom in the theory requires a group- theoretical treatment in order to classify the possible states, and this classification relies on the basic symmetries of the Hubbard model. Examples of states that we find are antiferromagnetic, ferromagnetic, superconducting (s- and d-wave), charge density wave and spiral spin-wave states.
This thesis concerns two topics where symmetry and strong correlations in time and space are important. The first topic is the theory of dynamical systems, and this theory deals with generic phenomena which are often convenient to study in very simple models. Dynamical systems can be divided in two basic classes: dissipative and conservative systems. Dissipation often effectively reduces the degrees of freedom in a dynamical system, and it is often sufficient to study low-dimensional models. An example is the one- dimensional circle map. The transition from ordered to chaotic motion in a circle map with extra symmetry is studied in Paper I using renormalization group techniques. Special emphasis is put into the possible breakdown of the extra symmetry. An example of a conservative system is a particle moving in a potential field, and typical for the dynamics is the large number of periodic trajectories. A particularly simple class of conservative systems is the integrable systems. In such a system there are continuous manifolds of marginally stable periodic orbits, which are divided into discrete sets of stable and unstable periodic orbits if the potential of the system is nonlinearly perturbed. In Paper II we demonstrate how the stable and unstable orbits in a quartic potential exchange stability several times as the strength of the nonlinear perturbation is increased. The second topic of this thesis is the ceramic high-temperature superconductors. Apart from being superconducting these materials show several other phases dependent on the doping. The 2-dimensional Hubbard model has been suggested as a highly simplified model for the electronic properties of these materials, and we study an extended version of this model at arbitrary temperature and doping using a theory capable of describing a rich variety of states simultaneously. Important is that we use no a priori assumptions of the state of the system. The high degree of freedom in the theory requires a group- theoretical treatment in order to classify the possible states, and this classification relies on the basic symmetries of the Hubbard model. Examples of states that we find are antiferromagnetic, ferromagnetic, superconducting (s- and d-wave), charge density wave and spiral spin-wave states.
Quasi-one-dimensional quantum models are ideal for theoretically exploring the physical phenomena associated with strong correlations. In this thesis we study three examples where strong correlations play an important role in the static or dynamic properties of the system. Firstly, we examine the behaviour of a doped fermionic two-leg ladder in which umklapp interactions are present. Such interactions arise at special band fillings and can be induced by the formation of charge density wave order in an array of two-leg ladders with long-range (three-dimensional) interactions. For the umklapp which arises from the half-filling of one of the bands, we show that the low-energy theory has a number of phases, including a strong coupling regime in which the dominant fluctuations are superconducting in nature. These superconducting fluctuations carry a finite wave vector – they are the one-dimensional analogue of Fulde-Ferrell-Larkin-Ovchinnikov superconductivity. In a second example, we consider a quantum spin model which captures the essential one-dimensional physics of CoNb2O6, a quasi-one-dimensional Ising ferromagnet. Motivated by high-resolution inelastic neutron scattering experiments, we calculate the dynamical structure in the paramagnetic phase and show that a small misalignment of the transverse field can lead to quasi-particle breakdown – a surprising broadening in the single particle mode observed in experiment. Finally, we study the out-of-equilibrium dynamics of a model with tuneable integrability breaking. When integrability is broken by the presence of weak interactions, we show that the system relaxes to a non-thermal state on intermediate time scales, the so-called “prethermalization plateau”. We describe the approximately stationary behaviour in this regime by constructing a generalised Gibbs ensemble with charges deformed to leading order in perturbation theory. Expectation values of these charges are time-independent, but interestingly the charges do not commute with the Hamiltonian to leading order in perturbation theory. Increasing the strength of the integrability breaking interactions leads to behaviour compatible with thermalisation. In each case we use a combination of perturbative analytical calculations and non-perturbative numerical computations to study the problem at hand.
Since the discovery of the superconducting cuprates and their hitherto unknown electronic properties there has been a continuously growing interest in a theoretical treatment of suitable models of strongly correlated fermions which at least qualitatively reflect the essential aspects of these physical systems. The Hubbard model in two dimensions plays an important role in this context. The present thesis examines instabilities and possible symmetry broken phases of the two-dimensional repulsive Hubbard model with the help of numeric and semi analytic methods. While in the first part it is pointed out how such instabilities can be treated by a mean field theory, the second part of the work supplies the theoretical bases of the numeric procedures developed in part III. For the case of a pure nearest neighbour hopping it is shown that the second order contributions of perturbation theory generate an effective attraction between the electrons. The energy scale of the superconducting instabilities is also examined; it is shown that the vanHove singularity drastically enhances this critical scale at half filling. The investigation of a possible coexistence of two different kinds of spontaneous symmetry breaking forms the emphasis of the thesis: Lately accomplished renormalization group calculations showed (for suitably chosen parameters) the occurrence of a Pomeranchuk instability of the two-dimensional Hubbard model with nonvanishing next nearest neighbour hopping, which should give rise to a ground state that spontaneously breaks the model's lattice symmetry as well as its U(1) symmetry, i.e. the ground state could be characterized by two co-existing order parameters. For treatment of the problem of co-existing instabilities the ansatz of a renormalized perturbation theory is used, which had been suggested in order to avoid the occurrence of unphysical singularities arising within the framework of naive perturbation theory; these singularities are related to interaction-induced shifts of the Fermi surface, which cannot be captured by simply shifting the chemical potential. The attractive Hubbard model is also treated within the framework of renormalized perturbation theory; the well-known reduction of the superconducting order parameter with respect to its BCS value reported in earlier investigations can be confirmed. A spontaneously broken lattice symmetrie does not arise here, in contrast to the repulsive case.
This thesis is about unconventional superconductors out of equilibrium. More precisely, it summarizes our theoretical efforts in addressing a few questions related to the real-time evolution in models displaying unconventional superconductivity. Apart from the purely theoretical interest, the motivation for these studies comes from the recent achievement in the field of ultrafast time-resolved spectroscopy on correlated materials, such as the high-temperature copper-oxide superconductors. The task of describing such systems in nonequilibrium is a very difficult one. In this thesis we attack the problem from two different points of view: • the anisotropic unconventional d-wave symmetry, • the strong electron-electron correlations. The thesis is organized as follows: The first part introduces the basic concepts which lay at the foundation of the work presented in the other two parts. In Chapter 1, the concept of unconventional superconductivity is briefly introduced. This mainly includes the high-temperature copper-oxide superconductors and their phenomenology: d-wave symmetry of the super- conducting gap and strong electron-electron interaction. Chapter 2 is devoted to a review of the recent experimental advances in the field of ultrafast spectroscopy of high temperature superconductors and correlated materials, with a particular focus on time- and angle- resolved photoemission spectroscopy. Finally, Chapter 3 is a short review of some works in the field of cold-atom Fermi gases. In the second part we address the physics of the d-wave superconductor after a sudden excitation. First, in Chapter 4 we introduce in some detail a model of mean-field superconductor with a d-wave symmetry. The principal feature of this model is the anisotropy of the gap and the nodal lines along which the gap vanishes. The results concerning the dynamics of the gap after a “quantum quench” are exposed in Chapter 5 where we compare our model to the s-wave superconductor. This comparison allows to identify the consequences of the unconventional symmetry. In Chapter 6 we discuss the spectral features of the transient nonequilibrium state of the d-wave superconductor. This is particularly important to move a first step towards photoemission experiments. In the third part we concentrate on the attractive Hubbard model as a prototype of strongly correlated superconductor. In Chapter 8 we discuss the superconducting state at equilibrium with the use of dynamical mean-field theory. In Chapter 9 we discuss the extension of this technique to systems out of equilibrium. In particular it is described the implementation in the superconducting phase. At the end of the thesis two appendices give some details of the calculations.
The first part of the thesis deals with so called numerical renormalization group methods, and in particular the density-matrix renormalization group (DMRG). This method performs extremely well for large one-dimensional quantum systems like spin-chains and Hubbard models. We discuss general features of the quantum states obtained by using this method for large systems. One finds that the method works better, i.e. faster convergence and higher accuracy, when applied to systems having an energy gap between the ground state and the first excited state. An investigation of the behavior of the DMRG for a simple gapless system is done, mainly focusing on the properties of correlation functions. The second part of the thesis deals with topics related to superconductivity. It starts with a discussion of magnetic Josephson junctions and the physical processes governing the behavior of such a system. We also introduce and discuss models of high-temperature superconductors, in particular the t-J model and its relation to stripes. Starting from the t-J model on a two dimensional square lattice, we discuss an approximation where the spin-configuration is assumed static. We discuss the properties of the fictitious magnetic field experienced by the electrons due to the static spin-configuration. Configurations being close to ferromagnetic and antiferromagnetic turn out to have very different properties. Furthermore, we derive an effective model for this system and look for different phases of this model. In particular we investigate whether the system takes advantage of a spin-generated fictitious flux. Uniform phases are discussed in a Hartree-Fock approximation and we also consider stripes between antiferromagnetically ordered regions, these being modeled in a self-consistent Hartree approximation. Finally, and inspired by the fictitious fluxes, there is a chapter discussing the statistics of particles living on a circle. It is shown that there are two free parameters in the quantization of such a system and we also sort out the relation between fermions and hard-core bosons in one dimension.
One of the central goals of condensed matter theory is to understand the behavior of electrons under various circumstances. This dissertation focuses on the some relatively unexplored effects brought by lattices, in a general setting.One ingredient that lattices introduce is the orbital degrees of freedom, resulting from the interaction between electrons and background ions. We have non-perturbatively studied a strongly coupled multi-orbital electron liquid, which has a fully polarized ground state. Both Hund’s coupling and electron itinerancy are essential to building up the global spin coherence. At finite temperature, the system exhibits both Curie-Weiss type spin susceptibility and finite compressibility, demonstrating the coexistence of spin coherence and charge incoherence. At low temperature, the single particle excitation extends over the entire Brillouin zone rather than the vicinity of the Fermi surface due to the spin fluctuation.Another dramatic effect of lattices on electrons is the formation of Mott insulators. We consider a special type of Mott insulators occurring at non-integer fillings, which has a finite single-particle gap but strong local charge fluctuation. As an example, we studied 1D Hubbard model with SU(N) symmetry at half-filling. The systems are insulators for all values of N, but the filling is half-integer when N is odd. We demonstrate that the systems with N even and N odd exhibit distinct energy scales of charge and spin gaps. In the case of odd N, the local charge fluctuation leads to spin gap much greater than the super exchange energy scale. The difference between the systems with opposite parity of N is gradually smeared out as N increases.The symmetry of lattices can also enforce band touchings and band crossing, leading to nodal-line and nodal-point semi-metals. We explore the symmetry constraints on electrons imposed by generalized lattices with space-time mixing periodicities. which include the Floquet lattice systems as a special case. Compared to space and magnetic groups, the symmetry group of such systems is augmented by “time-screw” rotations and “time-glide” reflections involving fractional time translations. A complete classification of the 13 space-time groups in 1+1D and 275 space-time groups in 2+1D are performed. Kramers-type degeneracy can arise from space-time symmetries without the half-integer spinor structure, which constrains the winding number patterns of spectral dispersions. In 2+1D, non-symmorphic space-time symmetries enforce spectral degeneracies, leading to protected Floquet semi-metal states.
The subject of condensed matter physics is very rich — there are an infinite number of parameters producing a diversity of exciting phenomena. As a theorist, my goal is to distill general principles out of this complexity — to construct theories that can coherently explain many known examples altogether.This thesis is composed of several attempts to develop such theories in topics related to spontaneously symmetry breaking. A remarkable feature of many-body interacting systems is that although they are described by equations respecting various symmetries, they may spontaneously organize into a state that explicitly breaks symmetries. Examples are numerous: various types of crystalline and magnetic orders, Bose-Einstein condensates of cold atoms, superfluids of liquid helium, chiral symmetry in QCD, neutron stars, and cosmic inflation.These systems with spontaneously broken continuous symmetries have gapless excitations, so called Nambu-Goldstone bosons (NGBs). Although the properties of NGBs are well understood in Lorentz-invariant systems, surprisingly, some basic properties of NGBs such as their number and dispersion in nonrelativistic systems have not been discussed from a general perspective. In the first part of this thesis, we solve this issue by developing and analyzing an effective Lagrangian that coherently captures the low-energy, long-distance physics of many different symmetry-breaking states all at once.Next, we examine whether these NGBs originating from spontaneous symmetry breaking remain to be well-defined excitations inside a metal, where low-energy electrons near Fermi surface can collide with them. Our result is a one equation criterion that specifies whether the interactions between electrons and NGBs can be ignored, or whether it completely changes their character. In the latter case, unusual phases of matter such as non-Fermi liquids may arise; in that case, NGBs are overdamped and cannot form particle-like excitations in spite of the assumed symmetry breaking.In the last part of this thesis, we investigate the possibility of spontaneously breaking of Hamiltonian itself. The homogeneity of time is one of the most fundamental symmetries of nature, underlying the conservation of the energy. The question is whether this symmetry can be spontaneously broken, as suggested recently by Wilczek, in analogy with ordinary crystals. Contrary to his proposal that attracted a significant attention and stimulated many further studies, we prove a no-go theorem that rules out spontaneously breaking of time translation, in the ground state or in the canonical ensemble of a general Hamiltonian.