Majorana zero mode and 1D Ising model

Question

It is known that the one-dimensional (1D) Ising model can be mapped to a free Majorana model using a Jordan-Wigner transformation and its two degenerated ground states are well interpreted by the two Majorana zero modes at the two ends of the chain in "Majorana language".

It is evident that the ground states of an infinite (without ends) 1D Ising chain are degenerate. How can I understand the degeneracy in its Majorana counterparts which is also infinite (without ends)?

Another question which confuses me is whether we can treat this two-fold degeneracy as two Majorana zero modes in 1D Ising model. If not, what is the relationship between the two-fold degeneracy in the Ising model and the two Majorana zero modes in the free Majorana model?

This post imported from StackExchange Physics at 2015-02-01 01:46 (UTC), posted by SE-user Dong

Answer 1

We should be very careful when we do a non-local transformation such as Jordan-Wigner transformation. If we start from 1D Ising chain (assuming open boundary condition), the ground state degeneracy is coming from two symmetry breaking ground states. Then, by doing Jordan-Wigner transformation, the degeneracy is moved to the ends of the Majorana chain. Two Majorana zero modes at the ends are just a signal of degenerate ground states. The reverse logic is also applied. Suppose we have a 1D symmetry protect topological (SPT) order where the degeneracy is due to the end states. We can do a non-local transformation such that the new hamiltonian has a corresponding ground state degeneracy caused by symmetry breaking. A typical example is the so-called hidden symmetry breaking in the Haldane chain. People at that time didn't know much about the mathematical structure of SPT phases while studying symmetry breaking was familiar. Such non-local transformation then became a way to study SPT phases in 90s.