Topological Phases in a One-dimensional Majorana-Bose-Hubbard Model*

Majorana zero modes (MZM-s) occurring at the edges of a 1D, p-wave, spinless superconductor, in absence of fluctuations of the phase of the superconducting order parameter, are quintessential examples of topologically-protected zero-energy modes occurring at the edges of 1D symmetry-protected topological phases. In this work, we numerically investigate the properties of the topological phase in the presence of phase-fluctuations using the density matrix renormalization group (DMRG) technique. To that end, we consider a one-dimensional array of MZM-s on superconducting islands at zero temperature. We show that the system can be in either a Mott-insulating phase, a Luttinger liquid (LL) phase of Cooper-pairs or another gapless phase. The latter phase can be viewed as a LL of charge-e bosons where nonlocal string correlation functions decay algebraically. These nonlocal correlation functions, together with correlation functions of the Cooper-pair creation operators, distinguish the different phases of the model. We map the system to an interacting model of spins coupled to rotors and use DMRG to characterize the different phases and the phase-transitions.