Uncovering Phases of the One-Dimensional Majorana Ladder

Majorana, particles that are their own anti-particles, have been an interesting yet elusive particle class studied across the different disciplines of physics. With the novel prediction by Fu-Kane that Majorana fermion zero-modes are predicted to form in vortices on the surface of topological insulators, there has been a surge of interest to study the many-body physics of Majorana fermion systems. Recently, the full phase diagram was found for the interacting Majorana chain. The chain contains a transition between two different extended critical gapless phases, a free Majorana Ising phase and an Ising + Luttinger liquid phase. In this research, we evolve the single chain into the simplest two-dimensional extension: the Ladder. The Majorana Ladder simply consists of two legs of interacting chains coupled by an inter-leg hopping. We use density matrix renormalization group (DMRG) calculation methods to find low lying energy state measurements and show how added inter-leg hopping effects the phases of the Ladder. We show that both critical phases are broken by the introduction of inter-leg hopping, and as this hopping is increased, there is a single smooth crossover into a paramagnetic like spin phase.