Phase Diagram of the Interacting Majorana Chain Model

The Hubbard chain and spinless fermion chain are paradigms of strongly correlated systems, very well understood using Bethe ansatz, Density Matrix Renormalization Group (DMRG) and field theory/renormalization group (RG) methods. They have been applied to one-dimensional materials and have provided important insights for understanding higher dimensional cases. Recently, a new interacting fermion model has been introduced, with possible applications to topological materials. It has a single Majorana fermion operator on each lattice site and interactions with the shortest possible range that involve 4 sites. We present a thorough analysis of the phase diagram of this model in one dimension using field theory/RG and DMRG methods. It includes a gapped supersymmetric region and a novel gapless phase with coexisting Luttinger liquid and Ising degrees of freedom. In addition to a first order transition, three critical points occur: tricritical Ising, Lifshitz and a novel generalization of the commensurate-incommensurate transition. We also survey various gapped phases of the system that arise when the translation symmetry is broken by dimerization and find both trivial and topological phases with 0, 1 and 2 Majorana zero modes bound to the edges of the chain with open boundary conditions.