Reply to comment on $lquot$Integrable Kondo impurity in one-dimensional q-deformed t$ndash$J models$rquot$

Abstract

This is a reply to the comment by P Schlottmann and A A Zvyagin. PACS numbers: 71.10.Fd, 71.27, 75.10.Jm In their comment [1], Schlottmann and Zvyagin raise several issues regarding the nature of integrable impurities in one-dimensional quantum lattice models, and claim to expose false statements in our recent paper [2]. In order to address these issues in a pedagogical manner, we feel that it is appropriate to discuss these questions in terms of models based on the gl(2|1) invariant solution of the Yang–Baxter equation. However, it is important from the outset to make it clear that the arguments we present below are general and apply to other classes of models. The first point that we would like to make is that it is claimed in [1] that there are two approaches to the algebraic Bethe ansatz. However, it appears to us that approach (i) described in point (2) of [1] is the coordinate Bethe ansatz and we do not understand why this should be referred to as an algebraic Bethe ansatz. In the coordinate Bethe ansatz approach, one starts with a prescribed Hamiltonian and then solves the Schrodinger equation to obtain the two-particle scattering matrices and the particle–impurity scattering matrix. Together these scattering matrices form the monodromy matrix. It is impossible to infer the Hamiltonian from such a monodromy matrix. In this context we do not feel that point (1) of [1] answers our query about the existence of the impurity monodromy matrix in the algebraic approach (ii) of point (2) in [1]. Hereafter, we focus our attention on this case. The solution of the Yang–Baxter equation R12(u− v)R13(u)R23(v) = R23(v)R13(u)R12(u− v) (1) associated with the Lie superalgebra gl(2|1) is an operator R(u) ∈ End (V ⊗ V ), where V is a three-dimensional Z2-graded space with one bosonic and two fermionic degrees of freedom. Explicitly, this operator takes the form R(u) = u · I ⊗ I + P (2) where P is the Z2-graded permutation operator. For the purposes of constructing integrable one-dimensional quantum systems on a closed lattice, it is usual to introduce the Yang–Baxter 0305-4470/02/296197+05$30.00 © 2002 IOP Publishing Ltd Printed in the UK 6197