5 - From Spin-1/2 Cluster c Chains to Majorana c Chains

Summary

Chapter 5 introduces a family of exactly soluble spin-1/2 lattice Hamiltonians: The spin-1/2 cluster “c” chains. Each member of this family is gapped and nondegenerate when periodic boundary conditions hold. The nondegeneracy of the ground state is lifted for all members of this family except for one member, owing to the presence of zero modes bound to the boundaries when open boundary conditions hold. The notion of symmetry fractionalization is thereby introduced. This family of exactly soluble Hamiltonians is mapped of a family of exactly soluble Majorana lattice Hamiltonians, one of which is an example of a Kitaev chain, though the Jordan–Wigner transformation. The stability of the degeneracy of the zero modes to integrability-breaking but symmetry-preserving interactions is derived through the explicit construction of the stacking rules.