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arXiv:nlin/0203030

Baxter Q-operators for integrable DST chain

A. E. Kovalsky, G. P. Pronko

Published Mar 14, 2002• 6 citations
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Abstract

Following the procedure, described in the paper nlin.SI/0003002, for the integrable DST chain we construct Baxter Q-operators as the traces of monodromy of some M-operators, that act in quantum and auxiliary spaces. Within this procedure we obtain two basic M-operators and derive some functional relations between them such as intertwining relations and wronskian-type relations between two basic Q-operators.

DOI
10.48550/arxiv.nlin/0203030

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Following the procedure, described in the paper nlin.SI/0003002, for the integrable DST chain we construct Baxter Q-operators as the traces of monodromy of some M-operators, that act in quantum and auxiliary spaces. Within this procedure we obtain two basic M-operators and derive some functional relations between them such as intertwining relations and wronskian-type relations between two basic Q-operators.
In the present paper we describe the procedure of the Q-operators construction for the q-deformed model, described by the Lax operator, which is important to formulate the Bethe ansatz for the Sin-Gordon model. This Lax operator can also be considered as some massless limit of the Lax operator of SG model. We constructed two R-operators which are the universal intertwiners for the Lax operators. The traces of its monodromies over the auxiliary space are Baxter operators i.e. the operator solutions of T-Q equation. We also found the intertwining relations which imply the mutual commutativity of the corresponding Q-operators.
The master T-operator is a generating function for transfer matrices and a tau-function of the modified KP (mKP) hierarchy. It is conventionally introduced through a Schur-function expansion whose coefficients are fused transfer matrices satisfying the Cherednik--Bazhanov--Reshetikhin determinant formula. For rational inhomogeneous gl(M) spin chains, we give an alternative definition: we realize gl(M) on an auxiliary Fock space generated by finitely many families of Schwinger bosons, form a monodromy matrix from the resulting L-operator, and define the master T-operator as its trace over the Fock space. Howe duality then reproduces the Schur-function expansion. This realization relates two constructions of Baxter Q-operators. In the construction of [arXiv:1112.3310] (see also [arXiv:1010.4022]), the Q-operators are obtained from residues of the master T-operator with respect to selected Miwa variables. In the construction of [arXiv:1010.3699], they are defined using degenerate Yangian L-operators in oscillator form. When selected Miwa variables approach the inverses of boundary-twist eigenvalues, the defining trace develops poles. After a suitable rescaling of this L-operator, only the terms contributing to the highest-order pole survive in the normalized limit; before the trace is taken, they form the degenerate Yangian L-operator of the second construction. The trace then gives an explicit residue formula for the corresponding Q-operator. Independently, a Holstein--Primakoff-type large-occupation-number contraction of this L-operator on subspaces with fixed total occupation numbers yields the same degenerate Yangian L-operator.
We construct the Baxter operator and the corresponding Baxter equation for a quantum version of the Ablowitz–Ladik model. The result is achieved in two different ways: by using the well-known Bethe ansatz technique and by looking at the quantum analogue of the classical Bäcklund transformations. General results about integrable models governed by the same r -matrix algebra will be given. Baxter's equation comes out to be a q-difference equation involving both the trace and the quantum determinant of the monodromy matrix. The spectrality property of the classical Bäcklund transformations gives a trace formula representing the classical analogue of Baxter's equation. A q-integral representation of the Baxter operator is discussed.
Baxter's Q -operator is generally believed to be the most powerful tool for the exact diagonalization of integrable models. Curiously, it has hitherto not yet been properly constructed in the simplest such system, the compact spin- Heisenberg–Bethe XXX spin chain. Here we attempt to fill this gap and show how two linearly independent operatorial solutions to Baxter's TQ equation may be constructed as commuting transfer matrices if a twist field is present. The latter are obtained by tracing over infinitely many oscillator states living in the auxiliary channel of an associated monodromy matrix. We furthermore compare our approach to and differentiate it from earlier articles addressing the problem of the construction of the Q -operator for the XXX chain. Finally we speculate on the importance of Q -operators for the physical interpretation of recent proposals for the Y -system of AdS/CFT.