Characterization of Majorana-Ising phase transition in a helical liquid system

TL;DR

This paper investigates the Majorana-Ising phase transition in an interacting helical liquid system coupled to magnetic and superconducting elements, identifying key signatures and the entanglement spectrum behavior across phases.

Contribution

It maps the helical liquid to an XYZ spin model and characterizes the phase transition using multiple criteria, including entanglement spectrum analysis, for the first time.

Findings

Majorana state exhibits exponentially decaying excitation gap with system size.

Degenerate ground state appears in the Majorana phase in the thermodynamic limit.

Entanglement spectrum shows finite Schmidt gap in Ising phase and degeneracy in Majorana phase.

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.