Truncation identities for the small polaron fusion hierarchy

Abstract

We study a one-dimensional lattice model of interacting spinless fermions. This model is integrable for both periodic and open boundary conditions; the latter case includes the presence of Grassmann valued nondiagonal boundary fields breaking the bulk U (1) symmetry of the model. Starting from the embedding of this model into a graded Yang-Baxter algebra, an infinite hierarchy of commuting transfer matrices is constructed by means of a fusion procedure. For certain values of the coupling constant related to anisotropies of the underlying vertex model taken at roots of unity, this hierarchy is shown to truncate giving a finite set of functional equations for the spectrum of the transfer matrices. For generic coupling constants, the spectral problem is formulated in terms of a functional (or TQ-)equation which can be solved by Bethe ansatz methods for periodic and diagonal open boundary conditions. Possible approaches for the solution of the model with generic non-diagonal boundary fields are discussed.

Small polaron with generic open boundary conditions: Exact solution via the off-diagonal Bethe ansatz

“…where the parity [|i ] is equal to zero (one) for bosonic (fermionic) indices. (For the details about the algorithm of super tensor product we refer the reader to [25,13]. )…”

Bethe ansatz solution of the small polaron with nondiagonal boundary terms

“…To keep the presentation of our results self-contained we begin by recalling the results of Ref. [19], namely the infinite hierarchy of fused transfer matrices and its truncation for particular values of the anisotropy parameter η. Then we use the truncation identity to rewrite the fusion hierarchy as a higher order functional relation for the transfer matrix (2.4) alone.…”

“…, is has been shown that the fused transfer matrices at level n + 1 and n − 1 are related by the truncation identity [19] 6) where the functions φ id n (u), φ τ n (u) are given by the following expressions…”

“…Inspired by the structure of the h-functions in the XXZ case [10], as well as the functions found in [19], we consider the following expressions 13) where the boundary information is contained in the functions g ± (u). Substituting the above expressions into (3.12), the following condition arises regarding the functions g ± (u):…”

“…[19]. Based on these findings we proceed to derive higher order functional equations for the transfer matrix of the small polaron model for particular values of the anisotropy parameter which can be formulated as a vanishing determinant condition.…”