Conjectures on hidden Onsager algebra symmetries in interacting quantum lattice models

Abstract

We conjecture the existence of hidden Onsager algebra symmetries in two interacting quantum integrable lattice models, i.e. spin-1/2 XXZ model and spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the anisotropy. The conjectures relate the Onsager generators to the conserved charges obtained from semi-cyclic transfer matrices. The conjectures are motivated by two examples which are spin-1/2 XX model and spin-1 U(1)-invariant clock model. A novel construction of the semi-cyclic transfer matrices of spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the anisotropy is carried out via the transfer matrix fusion procedure.

On the Q operator and the spectrum of the XXZ model at root of unity

“…The structure of descendant towers from Section 8 implies the existence of a hidden Onsager algebra at least for a part of the physical Hilbert space [86]. At the free-fermion point ∆ = 0, and at η = iπ 3 for the integrable spin-1 XXZ chain, the Onsager algebra plays a crucial role.…”

Generalised Onsager Algebra in Quantum Lattice Models

“…The algebra GC(r, N ) contains (r+1)(r+2) 2 operators that commute with any two of the generators A (s) that do not commute with each other. These operators are referred as "U (1)-invariant Hamiltonian" in the case of GC(1, N ) in [14,15], when considering the representation related to the transverse field Ising model; we will comment on that in Section 3.1.…”

“…q = exp(iπ/2) for spin 1/2 or q = exp(iπ/3) for spin 1, the Onsager algebra can be found explicitly for the spin-1/2 XXZ model or the spin-1 Zamolodchikov-Fateev model [14], where the Onsager generators consist of local operators. When we are at other roots of unity, the Onsager generators are expected to have quasi-local density [15]. present a different approach compared to the one in [26], analogous to the free fermionic eight-vertex model.…”

“…In addition, the Onsager algebra is closely related to quantum integrability [10] and Kramers-Wannier duality [11][12][13], offering many facets on understanding exactly solvable models to us. More recently, the Onsager algebra has been conjectured to be present in a series of quantum integrable systems at root of unity values of anisotropy [14][15][16], e.g. the spin-1/2 quantum XXZ model at root of unity with quasi-periodic boundary conditions 1 , hinting at an intriguing relation to the representation theory of quantum groups.…”

“…From the mathematical perspective, the Onsager algebra is an infinite-dimensional Lie algebra [19], which is a subalgebra of the sl 2 loop algebra [15,20]. There are few ways to extend the Onsager algebra.…”