The Hilbert-space structure of free fermions in disguise

TL;DR

This paper characterizes the Hilbert space structure of free fermions in disguise (FFD) Hamiltonians, revealing a factorization that clarifies the exponential degeneracy and aids in computing spin correlations.

Contribution

It provides an exact Hilbert space factorization for FFD Hamiltonians, including a decomposition into fermionic and degeneracy spaces, and constructs operators to resolve eigenspace degeneracies.

Findings

Hilbert space factorizes as H=H_F⊗H_D

H_D further decomposes into H_{F'}⊗H_{~D}

Constructed operators resolve eigenspace degeneracies

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.