Algebraic Bethe Ansatz for U(1) Invariant Integrable Models: The Method and General Results

TL;DR

This paper develops a comprehensive algebraic Bethe ansatz framework for U(1) invariant integrable models, providing general tools, recurrence relations, and explicit eigenvalue and eigenvector structures applicable to models with arbitrary weights and states.

Contribution

It introduces a new method for constructing eigenstates using recurrence relations and generalizes the algebraic Bethe ansatz to models with arbitrary weights and multiple edge states.

Findings

Derived fundamental commutation relations from linear systems.

Constructed transfer matrix eigenstates via a new recurrence relation.

Explicitly formulated eigenvalues, Bethe equations, and eigenvector structures.

Abstract

In this work we have developed the essential tools for the algebraic Bethe ansatz solution of integrable vertex models invariant by a unique U(1) charge symmetry. The formulation is valid for arbitrary statistical weights and respective number $N$ of edge states. We show that the fundamental commutation rules between the monodromy matrix elements are derived by solving linear systems of equations. This makes possible the construction of the transfer matrix eigenstates by means of a new recurrence relation depending on $N-1$ distinct types of creation fields. The necessary identities to solve the eigenvalue problem are obtained exploring the unitarity property and the Yang-Baxter equation satisfied by the $R$-matrix. The on-shell and off-shell properties of the algebraic Bethe ansatz are explicitly presented in terms of the arbitrary $R$-matrix elements. This includes the transfer matrix eigenvalues, the Bethe ansatz equations and the structure of the vectors not parallel to the eigenstates.