Title: Characterization of Majorana-Ising phase transition in a helical liquid system,
Abstract: We map an interacting helical liquid system, coupled to an external magnetic
field and s-wave superconductor, to an XYZ spin system, and it undergoes
Majorana-Ising transition by tuning of parameters. In the Majorana state,
lowest excitation gap decays exponentially with system size, and the system has
degenerate ground state in the thermodynamic limit. On the contrary, the gap
opens in the Ising phase even in the thermodynamic limit. We also study other
criteria to characterize the transition, such as edge spin correlation with its
neighbor $C(r=1)$, local susceptibility $\chi_i$, superconducting order
parameter of edge spin $P(r=1)$, and longitudinal structure factor $S(k)$. The
ground state degeneracy and three other criteria lead to the same critical
value of parameters for Majorana-Ising phase transition in the thermodynamic
limit. We study, for the first time, the entanglement spectrum of the reduced
density matrix of the helical liquid system. The system shows finite Schmidt
gap and non-degeneracy of the entanglement spectrum in the Ising limit. The
Schmidt gap closes in the Majorana state, and all the eigenvalues are either
doubly or multiply degenerate.