Free fermions beyond Jordan and Wigner

Understand

The Jordan-Wigner transformation is frequently utilised to rewrite quantum spin chains in terms of fermionic operators.

• When the resulting Hamiltonian is bilinear in these fermions, i.e.
• the fermions are free, the exact spectrum follows from the eigenvalues of a matrix whose size grows only linearly with the volume of the system.
• However, several Hamiltonians that do not admit a Jordan-Wigner transformation to fermion bilinears still have the same type of free-fermion spectra.