Integrable boundary conditions for staggered vertex models

TL;DR

This paper explores integrable boundary conditions for staggered vertex models, revealing that only two staggering types produce local Hamiltonians with open boundary conditions and introducing a second hierarchy of conserved quantities.

Contribution

It demonstrates that specific staggering patterns yield integrable open boundary conditions and constructs a unified framework for obtaining the Hamiltonian and quasi momentum operator.

Findings

Only two staggering types yield local Hamiltonians with open boundaries.

A second hierarchy of commuting integrals of motion is identified.

Spectral flow between local cases is constructed for the staggered six-vertex model.

Abstract

Yang-Baxter integrable vertex models with a generic $\mathbb{Z}_2$-staggering can be expressed in terms of composite $\mathbb{R}$-matrices given in terms of the elementary $R$-matrices. Similarly, integrable open boundary conditions can be constructed through generalized reflection algebras based on these objects and their representations in terms of composite boundary matrices $\mathbb{K}^\pm$. We show that only two types of staggering yield a local Hamiltonian with integrable open boundary conditions in this approach. The staggering in the underlying model allows for a second hierarchy of commuting integrals of motion (in addition to the one including the Hamiltonian obtained from the usual transfer matrix), starting with the so-called quasi momentum operator. In this paper, we show that this quasi momentum operator can be obtained together with the Hamiltonian for both periodic and open models in a unified way from enlarged Yang-Baxter or reflection algebras in the composite picture. For the special case of the staggered six-vertex model, this allows constructing an integrable spectral flow between the two local cases.