Phases and phase transitions in Kitaev ladders: symmetries and finite size scaling

We discuss phases and phase transitions in Kitaev ladders with and without interactions. In the noninteracting system, the phase diagram of a two-leg Kitaev ladder with time-reversal (TR) symmetry contains three distinct phases possessing 0, 2, and 4 Majorana edge modes and one tricritical line separating these phases. We discuss the finite-size corrections to energy along the tricritical line and finite-size scaling away from it which exhibits a non-trivial universal shape. Upon breaking the protecting TR symmetry, we find that the phase described by four Majorana edge modes in TR protected state crosses over to a trivial phase with no edge modes via symmetry broken path G in parameter space. The energy gap remains finite along G indicating that Z-classification (class BDI )reduces to Z_2 (class D) classification upon breaking of TR. Remarkably, the finite size scaling function in the vicinity of a critical line separating trivial phase from the nontrivial one in TR broken phase, exhibits a new nontrivial behavior. We also show that there is a phase transition of Ising universality in an interacting Kitaev chain and discuss its implications in ladder models.