Topological phases in disordered long-range Kitaev chains

Implementing a full-scale quantum computer is a major challenge to modern physics and engineering. Theoretically, this goal should be achievable due to the possibility of fault-tolerant quantum computation and has been suggested that fault-tolerance can be achieved at the physical level, instead of using quantum error-correcting codes. For example, Kitaev chain is a way to construct decoherence-protected degrees of freedom in one-dimensional systems (“quantum wires”). It was first proposed by A. Yu Kitaev in 2001 and consists in a row of non-interacting spinless fermions in the presence of hopping between lattice sites and superconducting pairing, wich link nearest fermions. Although it does not automatically provide fault-tolerance for quantum gates, it should allow, when implemented, to build a reliable quantum memory. In fact, this model shows, under certain conditions, a topological order phase, having a gapped bulk together with zero-energy unpaired Majorana modes localized at the edges of the system, which are robust against disorder, local impurities, or dynamical perturbations. The aim of the thesis is to investigate the behaviour of the many-neighbours pairing Kitaev chain (and different dependencies of the Hamiltonian parameters on the distance), in the hypothesis of disorder associated with the chemical potential. A proposal to a numerical resolution of the many-neighbours problem is lead through the Transfer Matrix method for the calculation of the wave functions of the zero-energy Majorana edge modes in the basis of the Bogoliubov states, which diagonalize the Kitaev chain’s Hamiltonian. On the other hand, the study of the system in the hypothesis of disorder is carried out with the aim of generalizing the results proposed by N. M. Gergs, L. Fritz, and D. Schuricht to the many-neighbours case, thus looking for the critical values of the chemical potential, typical for each amount of disorder (intended as standard deviation of a normal distribution) at fixed interaction parameters, above which a topological phase cannot exist. In this study we show that, as happens in the nearest neighbour case, also in the general case there is an amount of disorder beyond which the critical chemical potential is zero (i.e., a topological phase is never possible), before which we show the presence of an enhancement of the edge modes.