The Hilbert-space structure of free fermions in disguise

TL;DR

This work characterizes the Hilbert-space structure of free fermions in disguise (FFD) spin chains, proving an exact factorization $\mathcal{H}=\mathcal{H}_F\otimes\mathcal{H}_D$ into a fermionic sector and a degenerate sector. It then refines the degenerate part as $\mathcal{H}_D=\mathcal{H}_{\widetilde{D}}\otimes\mathcal{H}_{F'}$, identifying an representation-dependent symmetry algebra on $\mathcal{H}_{\widetilde{D}}$ generated by commuting Pauli strings, and an ancillary free-fermion algebra $\mathcal{A}_{F'}$ on $\mathcal{H}_{F'}$. The authors construct explicit operator algebras and ancillary fermions that resolve all Hamiltonian eigenspace degeneracies, providing a framework potentially enabling efficient computation of spin correlations both in and out of equilibrium. The results illuminate how to map complex spin-degenerate structures to tractable fermionic and auxiliary degrees of freedom, with implications for correlation functions and generalizations to other FFD models. Practically, this enhances understanding of non-JW-free-fermion solvable systems and offers tools for analyzing dynamics in frustrated or degenerate quantum chains.

Abstract

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 $\mathcal{H}=\mathcal{H}_F\otimes \mathcal{H}_D$, where $\mathcal{H}_F$ hosts the fermionic operators, while $\mathcal{H}_D$ accounts for the exponential degeneracy of the energy eigenspaces. By constructing a family of spin operators generating the operator algebra supported on $\mathcal{H}_D$, we further show that $\mathcal{H}_D=\mathcal{H}_{F'}\otimes \mathcal{H}_{\widetilde{D}}$, where $\mathcal{H}_{F'}$ hosts ancillary free fermions in disguise, while $\mathcal{H}_{\widetilde{D}}$ is generated by the common eigenstates of an extensive set of commuting Pauli strings. Our construction allows us to fully resolve the exponential degeneracy of all Hamiltonian eigenspaces and is expected to have implications for the computation of spin correlation functions, both in and out of equilibrium.