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arXiv:nlin/0205037 · International Journal of Mathematics and Mathematical Sciences

Quantum relativistic Toda chain at root ofunity: isospectrality, modified Q‐operator, and functional Bethe ansatz

Stanislav Zdislavovich Pakuliak, С. М. Сергеев

Published Jan 1, 2002• 9 citations

Abstract

We investigate an N‐state spin model called quantum relativistic Toda chain and based on the unitary finite‐dimensional representations of the Weyl algebra with q being Nth primitive root of unity. Parameters of the finite‐dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter′s Q‐operators. The classical counterpart of the modified Q‐operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota‐type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modified Q‐operators.

DOI: 10.1155/s0161171202105059

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We investigate an N‐state spin model called quantum relativistic Toda chain and based on the unitary finite‐dimensional representations of the Weyl algebra with q being Nth primitive root of unity. Parameters of the finite‐dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter′s Q‐operators. The classical counterpart of the modified Q‐operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota‐type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modified Q‐operators.

We conjecture a new way to construct eigenstates of integrable XXX quantum spin chains with SU(N) symmetry. The states are built by repeatedly acting on the vacuum with a single operator B good(u) evaluated at the Bethe roots. Our proposal serves as a compact alternative to the usual nested algebraic Bethe ansatz. Furthermore, the roots of this operator give the separated variables of the model, explicitly generalizing Sklyanin’s approach to the SU(N) case. We present many tests of the conjecture and prove it in several special cases. We focus on rational spin chains with fundamental representation at each site, but expect many of the results to be valid more generally.

Investigated is the relativistic periodic Toda chain, to each site of which the ultra-local Weyl algebra is associated. Weyl's

q

we are considering here is restricted to be inside the unit circle. Quantum Lax operators of the model are intertwined by six vertex

R

-matrix. Both independent Baxter's

Q

-operators are constructed explicitly as seria over local Weyl generators. The operator-valued wronskian of

Q

-s is also calculated.

An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on small sites (i.e. N = 2) check, it is conjectured that the homogeneous limit of the eigenstates exists, which gives rise to the corresponding eigenstates of the homogenous model.