The Hilbert-space structure of free fermions in disguise

Free fermions in disguise (FFD) Hamiltonians describe spin chains which can be mapped to free fermions, but not via a Jordan-Wigner transformation. Although the mapping gives access to the full Hamiltonian spectrum, the computation of spin correlation functions is generally hard. Indeed, the dictionary between states in the spin and free-fermion Hilbert spaces is highly non-trivial, due to the non-linear and non-local nature of the mapping, as well as the exponential degeneracy of the Hamiltonian eigenspaces. In this work, we provide a series of results characterizing the Hilbert space associated to FFD Hamiltonians. We focus on the original model introduced by Paul Fendley and show that the corresponding Hilbert space admits the exact factorization H=HF⊗HD, where HF hosts the fermionic operators, while HD accounts for the exponential degeneracy of the energy eigenspaces. By constructing a family of spin operators generating the operator algebra supported on HD, we further show that HD =HF′ ⊗HD, where HF′ hosts ancillary free fermions in disguise, while HD is generated by the common eigenstates of an extensive set of commuting Pauli strings. Our construction allows us to fully resolve the exponential degeneracyofallHamiltonianeigenspacesandisexpectedtohaveimplications forthecomputationofspincorrelationfunctions,bothinandoutofequilibrium.