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arXiv:hep-th/0610332 · Journal of Statistical Mechanics Theory and Experiment

The BaxterQ-operator for the gradedSL(2|1) spin chain

A.V. Belitsky, S É Derkachov, Gregory P. Korchemsky, A N Manashov

Published Jan 4, 2007• 34 citations
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Abstract

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.

DOI
10.1088/1742-5468/2007/01/p01005

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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.
We develop an approach for constructing the Baxter -operators for generic sl ( N ) spin chains. The key element of our approach is the possibility of representing a solution of the Yang–Baxter equation in the factorized form. We prove that such a representation holds for a generic sl ( N ) invariant -operator and find the explicit expression for the factorizing operators. Taking trace of monodromy matrices constructed of the factorizing operators one defines a family of commuting (Baxter) operators on the quantum space of the model. We show that a generic transfer matrix factorizes into the product of N Baxter -operators and discuss an application of this representation for a derivation of functional relations for transfer matrices.
Abstract One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine sl2\mathfrak {sl}_2sl2​ sl 2 and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.
Inspired by the integrable structures appearing in weakly coupled planar N=4 super Yang-Mills theory, we study Q-operators and Yangian invariants of rational integrable spin chains. We review the quantum inverse scattering method along with the Yang-Baxter equation which is the key relation in this systematic approach to study integrable models. Our main interest concerns rational integrable spin chains and lattice models. We recall the relation among them and how they can be solved using Bethe ansatz methods incorporating so-called Q-functions. In order to remind the reader how the Yangian emerges in this context, an overview of its so-called RTT-realization is provided. The main part is based on the author's original publications. Firstly, we construct Q-operators whose eigenvalues yield the Q-functions for rational homogeneous spin chains. The Q-operators are introduced as traces over certain monodromies of R-operators. Our construction allows us to derive the hierarchy of commuting Q-operators and the functional relations among them. We study how the nearest-neighbor Hamiltonian and in principle also higher local charges can be extracted from the Q-operators directly. Secondly, we formulate the Yangian invariance condition, also studied in relation to scattering amplitudes of N=4 super Yang-Mills theory, in the RTT-realization. We find that Yangian invariants can be interpreted as special eigenvectors of certain inhomogeneous spin chains. This allows us to apply the algebraic Bethe ansatz and derive the corresponding Bethe equations that are relevant to construct the invariants. We examine the connection between the Yangian invariant spin chain eigenstates whose components can be understood as partition functions of certain 2d lattice models and tree-level scattering amplitudes of the four-dimensional gauge theory. Finally, we conclude and discuss some future directions.
The solution of the Heisenberg model ( XXX -spin chain) is reviewed to define main ingredients of the quantum inverse scattering method. The spectrum and eigenfunctions are constructed by the algebraic Bethe Ansatz. A dynamical symmetry algebra is identified with a quantum group —the sl (2) Yangian. The Yang-Baxter equation and its GL ( n )-invariant solutions ( R -matrices) are discussed. Integral equations describing the Heisenberg model in the limit of an infinite chain are given. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In this paper we continue the study of Q-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). Taking a special limit in this R-matrix we obtained new formulas for the Q-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct Q-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-)integer spins.