Abstract
We study Slavnov's inner products for open Temperley-Lieb chains and their relations with the KP hierarchy. We show that when s=1/2 the quantum group invariant XXZ spin chain has Slavnov products given by the quotient of tau functions. The Schur polynomials expansion and a matrix construction for the Baker-Akhiezer functions are also briefly considered.
Highlights
Classical and quantum integrable systems share an extensive set of mathematical commonalities
Using the Jacobi-Trudi identities and the Schur polynomials expansions – important components in their work – the authors have shown that the inner product between on-shell and off-shell Bethe states can be represented in terms of a solution of an integrable hierarchy
In this work we argue that the Slavnov product for the spin-s Temperley-Lieb (TL) open chain [10,11] defines two KP tau functions
Summary
Classical and quantum integrable systems share an extensive set of mathematical commonalities. Using the Jacobi-Trudi identities and the Schur polynomials expansions – important components in their work – the authors have shown that the inner product between on-shell and off-shell Bethe states can be represented in terms of a solution of an integrable hierarchy This duality might allow us to study quantum spin chain from another perspective, but one might ask first whether the results of [6,7] are beautiful occurrences of the XXZ spin-1/2 chain with periodic boundary conditions, or if there is something more interesting behind the curtains. Slavnov products are complex objects to construct, see [3], and there is no general protocol teaching us how to build them; it is safe to say that we are essentially confined to the case-by-case analysis at the time of writing Despite these difficulties, some new results have recently appeared in the literature [10,11] allowing us to advance this program a bit further.
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