Solvable Nonunitary Fermionic Long-Range Model with Extended Symmetry

We define and study a long-range version of the xx model, arising as the free-fermion point of the xxztype Haldane-Shastry (HS) chain. It has a description via nonunitary fermions, based on the free-fermion Temperley-Lieb algebra, and may also be viewed as an alternating gl(1 1) spin chain. Even and odd lengths behave very differently; we focus on odd length. The model is integrable, and we explicitly identify two commuting Hamiltonians. While nonunitary, their spectrum is real by PT symmetry. One Hamiltonian is chiral and quadratic in fermions, while the other is parity invariant and quartic. Their one-particle spectra have two linear branches, realizing a massless relativistic dispersion on the lattice. The appropriate fermionic modes arise from "quasi-translation" symmetry, which replaces ordinary translation symmetry. The model exhibits exclusion statistics, like the isotropic HS chain, with even more "extended symmetry" and larger degeneracies.