Level degeneracy of the transverse ising model

[wdlang]

the hamiltonian is

H=-J \sum_{l=1}^N \sigma_l^z \sigma_{l+1}^z+ g \sigma_l^x

here we assume periodic boundary condition

my problem is, what is the highest possible degeneracy of the levels?

initially i expected 2

but numerically i find that it is 4

i cannot understand it

 

[wdlang]

it is not even 4

it can be higher

i just noticed that since the single particle excitation spectrum is symmetric, the multiple excitation spectrum can be highly degenerate

but i cannot see the symmetry behind