BoundarySmatrix of an open XXZ spin chain with nondiagonal boundary terms

Abstract

Using a recently proposed solution for an open antiferromagnetic spin-1/2 XXZ quantum spin chain with N (even) spins and two arbitrary boundary parameters at roots of unity, we compute the boundary scattering amplitudes for one-hole states. We also deduce the relations between the lattice boundary parameters appearing in the spin-chain Hamiltonian and the IR (infrared) parameters that appear in the boundary sine-Gordon S matrix. 1 β 2 = 8(π − µ) 2 Very recently, similar relations were found for the open XXZ spin chain with diagonal-nondiagonal boundary terms in [16]. 3 In fact, there is a change ξ ± → −ξ ± in the Bethe equation for the diagonal case [11].

Non-diagonal reflection for the non-criticalXXZmodel

“…Notice that the term depending on the boundary parameters (2.16) is 'double' compared to the diagonal case studied in [6,7]. Analogous phenomenon occurs in the open critical XXZ chain [14,15] and the sine-Gordon model [13]. It is straightforward to see that in the diagonal limit we recover the results of [6,7].…”

“…Such configurations have been utilized in deriving the kink-antikink scattering amplitudes in the bulk XXZ model (see e.g. [7]) as well as in open XXZ chain with the most general boundary conditions [15], where the 'duality' p ± → −p ± cannot be implemented for the derivation of the second eigenvalue of the boundary S-matrix. Notice that the term depending on the boundary parameters (2.16) is 'double' compared to the diagonal case studied in [6,7].…”

“…An alternative way to extract the second eigenvalue is instead of the 'kink' state with S = − 1 2 , -after implementing the duality transformation-to consider the anti-kink state consisting of a hole and a two-string state. Such configurations have been utilized in deriving the kink-antikink scattering amplitudes in the bulk XXZ model (see, e.g., [7]) as well as in open XXZ chain with the most general boundary conditions [15], where the 'duality' p ± → −p ± cannot be implemented for the derivation of the second eigenvalue of the boundary S-matrix. Note that the term depending on the boundary parameters (2.16) is 'double' compared to the diagonal case studied in [6,7].…”

“…Similarly, diagonal reflection matrices were derived in [11,12] for the critical XXZ model, corresponding to the sine-Gordon boundary S-matrix for 'fixed' boundary conditions [13]. After the derivation of the exact spectrum and Bethe equations for the XXZ chain with non-diagonal boundaries the generic boundary S-matrix for the critical XXZ chain was computed in [14,15], corresponding to the boundary S-matrix of sine-Gordon model [13] for 'free' boundary conditions. A relevant discussion on the generic breather boundary S-matrix within the XXZ framework may be also found in [14].…”