Solving the fusion hierarchies of D and E lattice models

Yu‐Kui Zhou, Paul A. Pearce, Uwe Grimm

Anonymous, Philipp Vojta, Zohar Nussinov +2 more

Paul A. Pearce

André M Grabinski, Holger Frahm

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 non-diagonal 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.

Alexi Morin-Duchesne, Paul A Pearce, Jørgen Rasmussen

Abstract The family of models on the square lattice includes a dilute loop model, a -vertex model and, at roots of unity, a family of RSOS models. The fused transfer matrices of the general loop and vertex models are shown to satisfy -type fusion hierarchies. We use these to derive explicit - and -systems of functional equations. At roots of unity, we further derive closure identities for the functional relations and show that the universal -system closes finitely. The RSOS models are shown to satisfy the same functional and closure identities but with finite truncation.

T. Miwa