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

• Published in: Journal of magnetism and magnetic materials Vol. 475; pp. 257 - 263
• Main Authors: Saha, Sudip Kumar, Dey, Dayasindhu, Roy, Monalisa Singh, Sarkar, Sujit, Kumar, Manoranjan
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.04.2019
• Subjects: DMRG; Helical liquid; Majorana fermion; Spin chain; Spin chain; Helical liquid; Majorana fermion; DMRG
• ISSN: 0304-8853
• DOI: 10.1016/j.jmmm.2018.11.088

•Majorana-Ising phase transition in a helical liquid system is studied.•In the Majorana state lowest excitation gap decays exponentially with system size.•Different criteria is used to characterize the Majorana-Ising phase transition.•In thermodynamic limit all criteria lead to the same critical value of parameters.•The Schmidt gap closes in the Majorana state.•The entanglement spectrum of the system is doubly or multiply degenerate in Majorana state. 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 study various criteria to characterize the transition, such as edge spin correlation with its neighbor C(r=1), local susceptibility χi, superconducting order parameter of edge spin P(r=1), and longitudinal structure factor S(k). All these criteria lead to the same critical value of parameters for Majorana-Ising phase transition in the thermodynamic limit. We study 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.