Kitaev chain

Example

Let $N \ge 2$ and let $c_j, c_j^\dagger$, $j = 1, \dots, N$, be fermion annihilation and creation operators, satisfying $\set{c_i, c_j^\dagger} = \delta_{ij}$ and $\set{c_i, c_j} = 0$. For real parameters $\mu$, $t$ and $\Delta$, the open Kitaev chain has Hamiltonian $$H = \sum_{j=1}^{N} -\mu\, c_j^\dagger c_j + \sum_{j=1}^{N-1}\Big( -t\big(c_j^\dagger c_{j+1} + c_{j+1}^\dagger c_j\big) + \Delta\big(c_j c_{j+1} + c_{j+1}^\dagger c_j^\dagger\big) \Big).$$ Write each fermion through two Majorana operators, $c_j = \tfrac12(\gamma_{j,1} + i\gamma_{j,2})$, with $\gamma_{j,a}$ self-adjoint, $\gamma_{j,a}^2 = 1$, and distinct ones anticommuting. Check at the special point $\mu = 0$, $t = \Delta > 0$: a direct computation gives $$H = i t \sum_{j=1}^{N-1} \gamma_{j,2}\, \gamma_{j+1,1}.$$ The two operators $\gamma_{1,1}$ and $\gamma_{N,2}$ do not appear in $H$. Each of them anticommutes with both factors of every term, so it commutes with every term and hence with $H$.