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

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 correspond 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.

Relativistic Toda Chain with Boundary Interaction at Root of Unity

“…In this section we recall briefly the subject of the model called the quantum Relativistic Toda Chain (RTC) chain at root of unity [11].…”

“…For any point p = (x, y) of Fermat curve x N + y N = 1, we define w p (s), s ∈ Z N , by In the case of homogeneous RTC these relations can be solved explicitly [11] (see also [8]). We will use the notation |γ n ∈ V 1 ⊗ · · · ⊗ V n for the eigenvectors of the operator A n (λ) of the open RTC with n particles.…”

“…These operators are intertwined by the six-vertex R-matrix at a root of unity. The cyclic L-operator of the BBS model reduces (at special values of parameters) to the L-operator of the quantum Relativistic Toda Chain (RTC) at a root of unity [11].…”

The Bazhanov–Stroganov model from 3D approach

“…Intertwining through the whole BS chain leads to isospectrality transforms of the transfer matrix. A special case of the BS model is the relativistic Toda chain, for which isospectral transforms have been constructed already in [15]. An important advantage of the 3D approach to 2D problems is the flexibility regarding the choice of parametrization.…”

The Modified Tetrahedron Equation and Its Solutions

“…Another application which can be made right now is to use these 3D models in an asymmetric geometry and to reduce them to new 2D integrable models [24]. Since Fermat curves are involved, these will be useful relatives of the chiral Potts model and of the relativistic Toda model [34]. The large variety of parameterizations which the MTE allows (in contrast to the Yang-Baxter equation) should help to overcome the serious parameterization problems which have prevented progress to finding more analytic results for the chiral Potts model.…”