Abstract
We study the scalar products between Bethe states in the XXZ spin chain with anisotropy |Δ| > 1 in the semi-classical limit where the length of the spin chain and the number of magnons tend to infinity with their ratio kept finite and fixed. Our method is a natural yet non-trivial generalization of similar methods developed for the XXX spin chain. The final result can be written in a compact form as a contour integral in terms of Faddeev’s quantum dilogarithm function, which in the isotropic limit reduces to the classical dilogarithm function.
Highlights
We study the scalar products between Bethe states in the XXZ spin chain with anisotropy |∆| > 1 in the semi-classical limit where the length of the spin chain and the number of magnons tend to infinity with their ratio kept finite and fixed
We investigated scalar products of Bethe states of the type on-shell/off-shell in the semi-classical limit for the XXZ spin chain with anisotropy |∆| > 1
We show the scalar product of the type on-shell/off-shell can be written in terms of the A -functional
Summary
We compute the scalar product of a generic off-shell Bethe state with a vacuum descendant state for the XXZ Heisenberg spin chain. The vacuum descendant state is defined by acting with generators of Uq(sl(2)) on the pseudovacuum state. In the XXX case, this scalar product gives rise to the so-called A -functional, which plays an important role in computing the semi-classical limit of other scalar products [22]. We define a q-deformed version of the A -functional, which is subsequently used for obtaining scalar products of more general states. Before we proceed to define the A -functional, we briefly review the Algebraic Bethe Ansatz. This will serve to set up our notations and conventions
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