A solvable non-unitary fermionic long-range model with extended symmetry

TL;DR

This work introduces a solvable non-unitary, long-range fermionic model that emerges as the q = i point of the xxz-type Haldane–Shastry family. It is formulated via a free-fermion Temperley–Lieb algebra and can be viewed as a long-range alternating GL(1|1) spin chain, possessing an extended symmetry and a highly structured spectrum organized by motifs. Two commuting charges, a chiral quadratic Hamiltonian H^l and a quartic, parity-even Hamiltonian H, are shown to have real spectra due to PT symmetry, with H^full vanishing at q = i and G replacing ordinary lattice translation as a quasi-translation. A fermionic realisation with quasi-translated modes reveals a spectrum that is sums of linear dispersions on two branches, with exclusion statistics and strong degeneracies from the extended symmetry GL(1|1); the work also outlines open conjectures and paths to a continuum CFT interpretation. The framework provides a lattice regularization for non-unitary systems with current-algebra symmetry and suggests futures directions toward a full eigenbasis construction and a CFT limit with an extended symmetry algebra.

Abstract

We define and study a long-range version of the XX model, arising as the free-fermion point of the XXZ-type Haldane--Shastry (HS) chain. It has a description via non-unitary fermions, based on the free-fermion Temperley--Lieb algebra, and may also be viewed as an alternating $\mathfrak{gl}(1|1)$ spin chain. Even and odd length behave very differently; we focus on odd length. The model is integrable, and we explicitly identify two commuting hamiltonians. While non-unitary, 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, realising 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.

Figures (4)

• Figure 1: The dispersion relations \ref{['eLodd']} alternate between two linear branches, realising chiral and 'full' (up to a shift) massless relativistic dispersions on the lattice.
• Figure 2: Left. The structure of the Hilbert space for $N=3$. Each label represents an eigenstate, which in this case are just Fock states. The vertical axis records the fermion number. The eigenspaces are labelled by motifs, with the parity-conjugate pair linked by a ' '. $\mathrm{C}\,\mathrm{T}$ acts by reflection in the 'equator' $\mathrm{N} = N/2$ up to a possible sign. The lines ' ' and ' ' indicate the action of the global-symmetry generators $\mathrm{F}_1^\pm$ and $\mathrm{F}_2^\pm$, respectively. Middle. The corresponding spectrum: quasi-momenta (note that $\frac{4\pi}{3} = -\frac{2\pi}{3} \, \mathrm{mod}\,2\pi$), energies, and degeneracies. Right. The action of the discrete symmetries on the Fock basis. The bar denotes the complement, e.g. $\lvert\bar{0}\rangle = \lvert12\rangle$, $\lvert\bar{\varnothing}\rangle=\lvert012\rangle$. For parity note that reordering may give a sign, e.g. $\mathrm{P}\,\lvert\bar{\varnothing}\rangle = \lvert210\rangle = -\lvert\bar{\varnothing}\rangle$.
• Figure 3: Left. The structure of the Hilbert space for $N=5$. The states with $\leqslant2$ particles are labelled by their fermionic mode numbers, with a prime for the two eigenstates resulting from diagonalising a $2\times 2$ block of $\mathrm{H}$, also indicated. Global symmetry and parity are shown as in Fig. \ref{['fg:N=3']}. Dotted lines like ' ' represent the action of extended-symmetry generators $\widehat{\mathrm{F}}_1^\pm$, joining together representations of the global symmetry algebra. Right. The corresponding spectrum.
• Figure 4: The spectrum for $N=7$ in the sectors of the Fock space with $\leqslant 2$ particles.