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arXiv:solv-int/9902015 · Journal of Physics A Mathematical and General

Baxter'sQ-operator for the homogeneousXXXspin chain

Sergey E. Derkachov

Published Jan 1, 1999• 96 citations
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Abstract

Applying the Pasquier-Gaudin procedure we construct Baxter's Q operator for the homogeneous XXX model as an integral operator in the standard representation of SL (2). The connection between the Q operator and the local Hamiltonians is discussed. We show that Lipatov's duality symmetry operator arises naturally as the leading term of the asymptotic expansion of the Q operator for large values of the spectral parameter.

DOI
10.1088/0305-4470/32/28/309

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Applying the Pasquier-Gaudin procedure we construct Baxter's Q operator for the homogeneous XXX model as an integral operator in the standard representation of SL (2). The connection between the Q operator and the local Hamiltonians is discussed. We show that Lipatov's duality symmetry operator arises naturally as the leading term of the asymptotic expansion of the Q operator for large values of the spectral parameter.
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We study an integrable non-compact superspin chain model that emerged in recent studies of the dilatation operator in the super-Yang–Mills theory. It was found that the latter can be mapped into a homogeneous Heisenberg magnet with the quantum space in all sites corresponding to infinite dimensional representations of the SL (2|1) group. We extend the method of the Baxter Q -operator to spin chains with supergroup symmetry and apply it to determine the eigenspectrum of the model. Our analysis relies on a factorization property of the -operators acting on the tensor product of two generic infinite dimensional SL (2|1) representations. It allows us to factorize an arbitrary transfer matrix into a product of three ‘elementary’ transfer matrices which we identify as Baxter Q -operators. We establish functional relations between transfer matrices and use them to derive the T – Q relations for the Q -operators. The proposed construction can be generalized to integrable models based on supergroups of higher rank and, as distinct from the Bethe ansatz, it is not sensitive to the existence of the pseudovacuum state in the quantum space of the model.
A new type of Yang-Baxter equation (YBE) for RRR-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the ZNZ_NZN​ symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for N>2N>2N>2, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated RRR-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter RRR-matrix---realized as an intertwiner of cyclic representations of Uq(sl2)U_q(\mathfrak{sl}_2)Uq​(sl2​) at a root of unity---and Shastry's two-parameter RRR-operator obtained from the decorated YBE.
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.
We have constructed series of the spectral parameter dependent solutions to the Yang–Baxter equations defined on the tensor product of reducible representations with a symmetry of quantum (super)algebra. These series are produced as descendant solutions from the s l q ( 2 ) -invariant Hecke type R r r ( u ) -matrices. The analogues of the matrices of Hecke type with the symmetry of the quantum super-algebra o s p q ( 1 | 2 ) are obtained precisely. For the homogeneous solutions R r 2 − 1 r 2 − 1 there are constructed Hamiltonian operators of the corresponding one-dimensional quantum integrable models, which describe rather intricate interactions between different kind of spin states. Centralizer operators defined on the products of the composite states are discussed. The inhomogeneous series of the operators R r R ( u ) , extended Lax operators of Hecke type, also are suggested.