Exactly Solvable Spin Tri-Junctions

Abstract

We present a class of exactly solvable tri-junctions of one- and two-dimensional spin systems. Based on the geometric criterion for solvability, we clarify the sufficient condition for the junctions so that the spin Hamiltonian becomes equivalent to Majorana quadratic forms. Then we examine spin tri-junctions using the obtained solvable models. We consider the transverse magnetic field Ising spin chains and reveal how Majorana zero modes appear at the tri-junctions of the chains. Local terms of the tri-junction crucially affect the appearance of Majorna zero modes, and the tri-junction may support Majorana zero mode even if the bulk spin chains do not have Majorana end states. We also examine tri-junctions of two-dimensional SO(5)-spin lattices and discuss Majorana fermions along the junctions.