Interplay of Superconductivity and Magnetism in the Two Dimensional Kondo Lattice Model

Since the original discovery of heavy fermion behavior in the late seventies by Andres, heavy fermions keep attracting scientific interest due to their exotic and unusual properties. These are inter-metallic compounds that contain rare earth elements, like cerium, praseodymium, and ytterbium, and actinides like uranium. The term ``heavy'' refers to their large effective electronic mass, as large as 1000 times the normal metal ones. The active physics in these materials results from the magnetic moments, associated to the partially filled $f$-shells of rare earth or actinide ions, being immersed into a quantum sea of mobile conduction electrons. In most rare earth metals and insulators, local moments tend to order magnetically, but in heavy electron metals the quantum mechanical jiggling of the local moments induced by delocalized electrons is fierce enough to melt magnetic order. The mechanism by which this takes place involves a remarkable piece of quantum physics known as the ``Kondo effect'' that describes the process by which a magnetic impurity get screened by conduction electrons, forming the so-called Kondo singlet below a characteristic temperature/energy scale named the Kondo temperature, $T_K$. Even though the Kondo effect refers strictly speaking to a very dilute concentration of magnetic ions, typically few part per million, the same physics is believed to play a role in heavy fermions. Heavy fermion materials have become recently popular also in the study of the quantum critical behavior of matter in the vicinity of a zero temperature second-order phase transition. Indeed, heavy fermions realize prototypical examples of quantum critical points that separate at zero temperature magnetic and paramagnetic phases. Experimentally, quantum critical points are attained by tuning non-thermal control parameters, such as pressure, chemical doping or applied magnetic field, so as to drive continuously to zero the magnetic ordering temperature. One of presently lively discussions up to date is about the appearance of two types of magnetic quantum critical points, depending on the behavior of the Kondo singlet as the transition is approached from paramagnetic side. If the Kondo singlet remains across the magnetic transition, the latter is of a spin-density-wave type in which the only critical degrees of freedom are the fluctuations of the magnetic order parameter. In this scenario, the Fermi volume does not change and contains both $f$ and conduction electrons. The alternative scenario invokes instead a local quantum criticality, where the Kondo singlet breaks down right at the magnetic transition, bringing about novel critical modes. Across such a quantum critical point, one expects a sudden collapse of the large Fermi surface of the paramagnetic side to a small magnetic one that contains only conduction electrons. Around a quantum critical point interesting phenomena such as non-Fermi liquid behavior or the appearance of exotic phases may appear. Indeed, many heavy fermions show superconductivity right after the magnetic transition. There are also evidences of coexisting magnetism and superconductivity. Emergence of superconductivity in heavy fermions is at first glance quite surprising, since in the conventional wisdom magnetic impurity scattering is pair-breaking. The evidence of non-$s$ wave symmetry of the order parameter brings these materials in the class of unconventional superconductors, where pairing is not phonon-mediated but likely caused by magnetic fluctuations. This issue has attracted a lot of experimental and theoretical interest. From the theoretical point of view, already building up a microscopic Hamiltonian that could capture the main physics and reproduce the phase diagram of heavy fermions is a challenge that is still ongoing. One of the first attempts to attack this issue was done by Anderson, who proposed in 1961 the model that is nowadays universally known as the Anderson impurity model. Later on, Doniach introduced a lattice version believed to describe heavy fermions, the so-called Kondo lattice model. The latter one has been studied extensively and there is a strong belief that it indeed captures the basic physics of heavy fermions. In this Thesis we study the ground-state phase diagram of various versions of the Kondo lattice model in two dimensions, starting from the simplest Doniach's one, with special focus on the possible appearance of superconductivity in the phase diagram. To attack this problem, we adopt a variational Monte Carlo scheme that allows treating quite large lattices, thus going beyond the one-dimensional and, at the opposite, the infinite-dimensional cases where most of the numerical studies have been restricted so far. Using Gutzwiller projected wave functions we are able to satisfy the local constraint of one electron per $f$ orbital locally not in average: this is the main advantage of the variational Monte Carlo against the mean-field approach. The flexibility of this variational method makes it possible to account for different types of correlations (specially pairing correlations) in the trial wave function, which are not present at the mean-field level. A full optimization of the variational wave function allows us to finally depict the phase diagram.