Implementing a fullscale quantum computer is a major challenge to modern physics and engineering. Theoretically, this goal should be achievable due to the possibility of faulttolerant quantum computation and has been suggested that faulttolerance can be achieved at the physical level, instead of using quantum errorcorrecting codes. For example, Kitaev chain is a way to construct decoherenceprotected degrees of freedom in onedimensional systems (“quantum wires”). It was first proposed by A. Yu Kitaev in 2001 and consists in a row of noninteracting spinless fermions in the presence of hopping between lattice sites and superconducting pairing, wich link nearest fermions. Although it does not automatically provide faulttolerance 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 zeroenergy 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 manyneighbours 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 manyneighbours problem is lead through the Transfer Matrix method for the calculation of the wave functions of the zeroenergy 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 manyneighbours 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.
Topological phases in disordered longrange Kitaev chains
TIBURZI, EDOARDO MARIA
2021/2022
Abstract
Implementing a fullscale quantum computer is a major challenge to modern physics and engineering. Theoretically, this goal should be achievable due to the possibility of faulttolerant quantum computation and has been suggested that faulttolerance can be achieved at the physical level, instead of using quantum errorcorrecting codes. For example, Kitaev chain is a way to construct decoherenceprotected degrees of freedom in onedimensional systems (“quantum wires”). It was first proposed by A. Yu Kitaev in 2001 and consists in a row of noninteracting spinless fermions in the presence of hopping between lattice sites and superconducting pairing, wich link nearest fermions. Although it does not automatically provide faulttolerance 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 zeroenergy 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 manyneighbours 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 manyneighbours problem is lead through the Transfer Matrix method for the calculation of the wave functions of the zeroenergy 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 manyneighbours 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.File  Dimensione  Formato  

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https://hdl.handle.net/20.500.12608/41615