Study Of One-dimensional Quantum Systems With P-wave Superconducting Pairing

This thesis investigates the topological and Anderson localization phenomenon in the one-dimensional(1D)non-interacting system with p-wave superconducting pairing.We consider both the commensurate and incommensurate case,and also explore the influences of the physical gain and loss on the system's properties,thus give a relatively complete study on such systems and pave the way to further understandings of 1D quantum systems.First,modulation functions,which are used to modulate the on-site potentials and the hopping amplitudes between the nearest lattice sites,are introduced in the 1D topological Kitaev chain.By calculating the energy spectra and the topological numbers of the system,we find that the system shows three different phases: topologically trivial phase,topological superconducting phase and Su-Schrieffer-Heeger(SSH)topological phase.The phase diagram and the phase boundaries are also determined.A phenomenological explanation is given based on the Majorana fermion representation.As to the incommensurate 1D lattices,we mainly focus on the Anderson localization phenomenon.We use the mean inverse participation ratio(MIPR)to characterize the localization process and find that due to the introduction of p-wave pairing,a critical phase shows up before the system goes into the localized phase from the extended phase as the incommensurate on-site potential is varied.We further explore these different phases by exploiting the quantum quench method.The system is prepared in the extended phase initially,then the on-site potential is suddenly changed so that the system will be in the extended,critical or localized phase after the quench.Then the time evolutions of the entanglement entropy,Loschmidt echo and width of wave packets of this system are inspected.These quantities behave quite distinctively in different phases and thus can be used to characterize the Anderson localization phenomena and the phase transitions in this system.Finally,considering the influences of the interaction processes between the system and the environment,we introduce physical gain and loss into the system and make the Hamiltonian non-Hermitian.For the topological phases in the commensurate case,the topological state are robust as long as the physical gain and loss are not stronger than certain critical values.While for the Anderson localization in the incommensurate system,different patterns of physical gain and loss have very different effects on the localization process.