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

TL;DR

This paper demonstrates that certain integrable models satisfy the free fermion condition, allowing their reformulation as free fermion theories, and applies this insight to simplify spectral problems in various AdS/CFT integrable systems.

Contribution

It explicitly shows that recent integrable Hamiltonians satisfy the free fermion condition and applies this to lower-dimensional AdS/CFT models, providing new reformulations and simplifications.

Findings

Models satisfy free fermion condition

Transfer matrix recast in free fermion form

Simplified spectral analysis in AdS/CFT models

Abstract

In this paper we first demonstrate explicitly that the new models of integrable nearest-neighbour Hamiltonians recently introduced in PRL 125 (2020) 031604 satisfy the so-called free fermion condition. This both implies that all these models are amenable to reformulations as free fermion theories, and establishes the universality of this condition. We explicitly recast the transfer matrix in free fermion form for arbitrary number of sites in the 6-vertex sector, and on two sites in the 8-vertex sector, using a Bogoliubov transformation. We then put this observation to use in lower-dimensional instances of AdS/CFT integrable R-matrices, specifically pure Ramond-Ramond massless 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 fermion machinery. In all cases we use the free fermion realisation to greatly simplify and reinterpret a wealth of known results, and to provide a very suggestive reformulation of the spectral problem in all these situations.