The one-dimensional Kondo Lattice in the Majorana Fermion representation

Abstract: This thesis analyses the possibilities offered by the representation of quantum mechanical operators in terms of Majorana fermions. These objects can be imagined as algebraic constituents of the fermionic degrees of freedom, allowing for a different representation of many Hamiltonians. Given a strongly correlated electron system, the Majorana fermion representation can be used to define new fundamental modes that grant a more convenient perspective on the problem and that can be used in the framework of standard analytical and numerical techniques. The main reason behind the usefulness of this approach is the advantageous form taken by the group of the unitary transformations of the Hilbert space, when represented in terms of Majorana fermions. To test such a new approach to strongly correlated quantum systems, the one dimensional Kondo lattice at zero temperature has been studied. Using the Majorana fermion representation a good description of the ferromagnetic phases of the model is obtained, already at mean-field level. This is possible because even very involved many-body processes, such as the emergence of the spin-selective Kondo insulator, or the deconfinement process of a fermion-spinon bound state, are described in a simple way in terms of Majorana fermions. These results prove that thanks to the redefinition of the degrees of freedom used in the analysis of the system, it becomes possible to obtain quite non-trivial results already at mean field level, or to consider very involved (but meaningful) correlated quantum states via simple variational trial states. This will potentially permit a more judicious and profitable choice of the fundamental degrees of freedom, allowing for an improvement of the efficiency of the analytical and numerical techniques used in the analysis of many strongly correlated quantum systems.