Edge modes in a frustrated quantum Ising chain

TL;DR

This paper investigates the edge modes in a frustrated quantum Ising chain with competing interactions, revealing the presence of multiple edge modes in different ordered phases through numerical methods.

Contribution

It demonstrates the existence and behavior of Majorana-like edge modes in a frustrated Ising chain with complex interactions, using DMRG and mean-field techniques.

Findings

Four edge modes in the double-staggered phase

Two edge modes in ferromagnetic and Ne9el phases

Edge mode count varies near J_1=0

Abstract

The ground state properties of an Ising chain with nearest ($J_{1}$) and next-nearest neighbor ($J_{2}$) interactions in a transverse field are investigated using the density matrix renormalization group and cluster mean-field theory methods. Its quantum phase diagram has two ordered regions (separated by a quantum disordered phase), one of which has simple ferromagnetic or N\'eel order depending upon the sign of $J_{1}$, and the other is the double-staggered antiferromagnetic phase. The presence of the Majorana-like edge-modes in these ordered phases is inferred by calculating the end-to-end spin-spin correlations on open chains. It is found that there occur four edge modes in the double-staggered phase, while the ferromagnetic and N\'eel phases support two edge modes, except very near $J_{1}=0$ where it seems they have four edge modes like the exact case at $J_{1}=0$ itself.