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arXiv:0810.3135 · Annales Henri Poincaré

Bethe Ansatz for the Universal Weight Function

L. Frappat, S. Khoroshkin, Stanislav Zdislavovich Pakuliak, E. Ragoucy

Published May 20, 2009• 24 citations
arXivHTMLPDF
DOI
10.1007/s00023-009-0416-x

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The eigenfunctions and eigenvalues of the master equation for zero-range process with totally asymmetric dynamics on a ring are found exactly using the Bethe ansatz weighted with the stationary weights of particle configurations. The Bethe ansatz applicability requires the rates of hopping of particles out of a site to be the q numbers [n](q). This is a generalization of the rates of hopping of noninteracting particles equal to the occupation number n of a site of departure. The noninteracting case can be restored in the limit q-->1. The limiting cases of the model for q=0, infinity correspond to the totally asymmetric exclusion process, and the drop-push model, respectively. We analyze the partition function of the model and apply the Bethe ansatz to evaluate the generating function of the total distance traveled by particles at large time in the scaling limit. In case of nonzero interaction, q not equal 1, the generating function has the universal scaling form specific for the Kardar-Parizi-Zhang universality class.
We consider the Temperley–Lieb (TL) open quantum spin chain with “free” boundary conditions associated with the spin- s representation of quantum-deformed s l ( 2 ) . We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an off-shell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an appendix, we briefly consider the closed TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be improved so as to obtain the complete (albeit non-universal) spectrum.