The physics of Majorana fermion occurs at the edge and interface of many one-dimensional quantum many body system with gapped excitation spectrum. We present the renormalization group equations for strongly interacting helical liquid. We present the results of both Majorana- Ising topological quantum phase transition and also Berezinskii-Kosterlitz-Thouless topological phase transition. We show that the Majorana-Ising topological quantum phase transition for non-interacting case is an exactly solvable line but in presence of interaction the system has no exactly solvable line. We study the effect of umklapp scattering on the renormalization group flow diagrams and also the condition for the appearence of Majorana fermion modes.
We consider an interface separating the Moore-Read state and Halperin 331 state in a half-filled Landau level, which can be realized in a double quantum well system with varying interwell tunneling and/or interaction strengths. In the presence of electron tunneling and strong Coulomb interactions across the interface, we find that all charge modes localize and the only propagating mode left is a chiral Majorana fermion mode. Methods to probe this neutral mode are proposed. A quantum phase transition between the Moore-Read and Halperin 331 states is described by a network of such Majorana fermion modes. In addition to a direct transition, they may also be separated by a phase in which the Majorana fermions are delocalized, realizing an incompressible state which exhibits quantum Hall charge transport and bulk heat conduction.
We analyze a series of interacting Majorana Fermion chains with finite range pair interactions with coupling strength
g that all exhibit a tri-critical point that separates an Ising critical phase from a supersymmetric gapped phase. We first notice that the interacting models exhibit an even-odd asymmetry depending on the number of sites,
δ, over which the interaction ranges. The even case exhibits competing order, thereby making it numerically untractable while the odd case exhibits an exactly solvable point at
g=−0.5 where the entanglement entropy vanishes. By introducing a swirling geometrical twist, we map our 1D
δ-range chains to a series of 2D
δ/2-width models. Our new 2D models possess a unique helical boundary condition, constructed from 1D chains with the end of one connected to the start of another. We propose that the phase transition in the 1D system can be understood as a finite-system size transition in 2D. That is, the
gc−δ behavior is controlled by a 2D tri-critical universality class at
δ→∞ limit and is predicted by finite-size scaling theory.