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In this paper we first demonstrate explicitly that the new models ofintegrable nearest-neighbour Hamiltonians recently introduced in PRL 125 (2020)031604 satisfy the so-called free fermion condition. This both implies that allthese models are amenable to reformulations as free fermion theories, andestablishes the universality of this condition. We explicitly recast thetransfer matrix in free fermion form for arbitrary number of sites in the6-vertex sector, and on two sites in the 8-vertex sector, using a Bogoliubovtransformation. We then put this observation to use in lower-dimensionalinstances of AdS/CFT integrable R-matrices, specifically pure Ramond-Ramondmassless and massive AdS_3, mixed-flux relativistic AdS_3 and massless AdS_2.We also attack the class of models akin to AdS_5 with our free fermionmachinery. In all cases we use the free fermion realisation to greatly simplifyand reinterpret a wealth of known results, and to provide a very suggestivereformulation of the spectral problem in all these situations.

Free Fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT