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arXiv:1805.04632 · Nuclear Physics B

Series of the solutions to Yang–Baxter equations: Hecke type matrices and descendant R-, L-operators

Shahane A. Khachatryan

Published Sep 19, 2018• 2 citations
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Abstract

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.

DOI
10.1016/j.nuclphysb.2018.09.012

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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.
Abstract Supersymmetry algebras can be used to obtain algebraic expressions for constant Yang–Baxter solutions, also known as braid group generators. This was done for non-invertible braid operators in Maity et al (2025 J. High Energy Phys 5 1–31) In this work we extend this construction for the invertible ones. The resulting expressions are then shown to obey relations analogous to those satisfied by quotients of braid groups. Examples of the latter include the Iwahori–Hecke algebra and the Birman–Murakami–Wenzl algebra. As a result, we can Baxterize the constant Yang–Baxter solutions to yield spectral parameter dependent R -matrices. The regularity of these R -matrices depends on the representation of SUSY generators. In some cases they are regular in the usual sense and in the remaining they are ‘almost’ regular. In the latter case they are also non-invertible. Nevertheless, we show that they can still help us construct integrable models in all dimensions of the local Hilbert space. These models can be described by Hamiltonian densities that are either local or non-local, depending on the representation chosen for the SUSY generators. We demonstrate this for all constant 4 × 4 invertible Yang–Baxter solutions. Apart from reproducing nearest-neighbor interaction spin 1 2 systems, we also find their higher spin analogs due to the algebraic [representation independent] approach.
An operator defoemed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its q → 1 limit seem to be the most general Yang-Baxter algebra underlying quantum integrable systems. Three different directions of application of this algebra in integrable systems depending on different sets of values of deforming operators are identified. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, while the associated Lax operators generate and classify them in an unified way. Variable values construct a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect. PACS numbers 02.30.Ik, 02.20.Uw, 05.50+q, 03.65.Fd 1
We obtain two series of spectral parameter dependent solutions to the generalized Yang-Baxter equations (GYBE), for definite types of N12×N22 matrices with general dimensions N1 and N2. Appropriate extensions are presented for the inhomogeneous GYBEs. The first series of the solutions includes as particular cases the X-shaped trigonometric braiding matrices. For construction of the second series the colored and graded permutation operators are defined, and multi-spectral parameter Yang-Baxterization is performed. For some examples the corresponding integrable models are discussed. The unitary solutions existing in these two series can be considered as generalizations of the multipartite Bell matrices in the quantum information theory.
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.
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.