Baxter Equation Research Articles

An elliptic parameterisation of the Zamolodchikov model

The Zamolodchikov model describes an exact relativistic factorized scattering theory of straight strings in (2+1)-dimensional space–time. It also defines an integrable 3D lattice model of statistical mechanics and quantum field theory. The three-string S-matrix satisfies the tetrahedron equation which is a 3D analog of the Yang–Baxter equation. Each S-matrix depends on three dihedral angles formed by three intersecting planes, whereas the tetrahedron equation contains five independent spectral parameters, associated with angles of an Euclidean tetrahedron. The vertex weights are given by rather complicated expressions involving square roots of trigonometric function of the spectral parameters, which is quite unusual from the point of view of 2D solvable lattice models. In this paper we consider a particular four-parameter specialisation of the tetrahedron equation when one of its vertices goes to infinity and the tetrahedron itself degenerates into an infinite prism. We show that in this limit all the vertex weights in the tetrahedron equation can be represented as meromorphic functions on an elliptic curve. Moreover we show that a special reduction of the tetrahedron equation in this case leads precisely to an example of the tetrahedral Zamolodchikov algebra, previously constructed by Korepanov. This algebra plays important role for a “layered” construction of the Shastryʼs R-matrix and the 2D S-matrix appearing in the problem of the ADS/CFT correspondence for N=4 SUSY Yang–Mills theory in four dimensions. Possible applications of our results in this field are briefly discussed.

On the multiparametric vertex model

In this paper we consider families of multiparametric R-matrices to make a systematic study of the boundary Yang–Baxter equations in order to discuss the corresponding families of multiparametric K-matrices. Our results are indeed a non-trivial generalization of the K-matrix solutions of the vertex model when distinct reflections and extra free parameters are admissible.

The classification of Wada-type representations of braid groups

We study a variant of the set-theoretical Yang–Baxter equation adapted to the free-product group structure, and give a classification of Wada-type representations of the braid groups. In particular, we prove Wada’s conjecture: There are only seven types of Wada-type representation up to certain natural symmetries.

Influence of long alkyl spacers in the elaboration of superoleophobic surfaces with short fluorinated chains

In order to produce extremely high oleophobic properties with a low fluorine content, long dodecyl spacers were introduced between a polymer and fluorinated chains in order to reduce the mobility of the fluorinated chains, especially short perfluorobutyl (F-butyl) chains. We report the synthesis and characterization of electrodeposited polymer surfaces obtained from original fluorinated 3,4-ethylenedioxypyrrole (EDOP) derivatives. Semi-fluorinated chains of three different lengths (F-octyl: PEDOPC12F8, F-hexyl: PEDOPC12F6 and F-butyl: PEDOPC12F4) were grafted to EDOP and separated with a n-dodecyl chain in order to reduce the mobility of F-butyl chains. For the first time, a surprisingly high contact angle of 138° was measured with hexadecane on PEDOPC12F4, although F-octyl tails are necessary to produce robust superoleophobic surfaces. By producing smooth surfaces, we successfully separate the influence of the intrinsic hydro- and oleophobicity and the surface structuration on the static contact angles of water and hexadecane. Here, the surface structuration increases the contact angle of hexadecane by 90° for PEDOPC12F4. The Cassie–Baxter equation predicts a porosity of 84.5% between the surface and a hexadecane droplet.

Development of superhydrophilic and superhydrophobic polyester fabric by growing Zinc Oxide nanorods

ZnO nanorods were grown on microfibers of Polyethylene terephthalate (PET) fabric by seeding method to develop hierarchical roughness structure. XRD and XPS analysis show the presence of crystalline ZnO and chemical Zn species at the fiber surface at each stage of the process. Five series of samples with different seed concentrations have been realized, and their surface morphology and topography were characterized by AFM and SEM. Increasing seed concentrations lead to samples with superhydrophilic properties. Not only the water contact angle at fabric surface tends to zero but also the water capillary diffusion inside fabric is faster. Nanostructuration affects the structure inside the fabric, and further experiments with decane liquid have been made to get a better understanding of this effect. To study the superhydrophobicity, nanorods treated samples were modified with octadecyltrimethoxysilane (ODS) by two method; solution deposition and vapor deposition. The superhydrophobicity was characterized by measuring the water contact angle and water sliding angle with 5μl water droplet. The samples modified with ODS by vapor deposition showed higher water contact angles and low water sliding angle than the ones modified with solution method. The lotus effect has been well correlated with the surface morphology of the nanorods structured fibers. The application of the Cassie–Baxter equation is discussed.

Integrability as a consequence of discrete holomorphicity: the ZN model

It has recently been established that imposing the condition of discrete holomorphicity on a lattice parafermionic observable leads to the critical Boltzmann weights in a number of lattice models. Remarkably, the solutions of these linear equations also solve the Yang–Baxter equations. We extend this analysis for the ZN model by explicitly considering the condition of discrete holomorphicity on two and three adjacent rhombi. For two rhombi this leads to a quadratic equation in the Boltzmann weights and for three rhombi a cubic equation. The two-rhombus equation implies the inversion relations. The star–triangle relation follows from the three-rhombus equation. We also show that these weights are self-dual as a consequence of discrete holomorphicity.This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

THE CENTER OF THE CATEGORY OF BIMODULES AND DESCENT DATA FOR NONCOMMUTATIVE RINGS

Let A be an algebra over a commutative ring k. We compute the center of the category of A-bimodules. There are six isomorphic descriptions: the center equals the weak center, and can be described as categories of noncommutative descent data, comodules over the Sweedler canonical A-coring, Yetter–Drinfeld type modules or modules with a flat connection from noncommutative differential geometry. All six isomorphic categories are braided monoidal categories: in particular, the category of comodules over the Sweedler canonical A-coring A ⊗ A is braided monoidal. We provide several applications: for instance, if A is finitely generated projective over k then the category of left End k(A)-modules is braided monoidal and we give an explicit description of the braiding in terms of the finite dual basis of A. As another application, new families of solutions for the quantum Yang–Baxter equation are constructed: they are canonical maps Ω associated to any right comodule over the Sweedler canonical coring A ⊗ A and satisfy the condition Ω3 = Ω. Explicit examples are provided.

New solutions to the [formula omitted]-invariant Yang–Baxter equations at roots of unity: Cyclic representations

We find all the solutions to the slq(2)-invariant multi-parametric Yang–Baxter equations (YBE) at q=i defined on the cyclic (semi-cyclic, nilpotent) representations of the algebra. We derive the solutions in form of the linear combinations over the slq(2)-invariant objects — projectors. The direct construction of the projector operators at roots of unity gives us an opportunity to consider all the possible cases, including also degenerated one, when the number of the projectors becomes larger, and various types of solutions are arising, and as well as the inhomogeneous case. We give full classification of the YBE solutions for the considered representations. A specific character of the solutions is the existence of the arbitrary functions.

Analytic solution of Bremsstrahlung TBA

We consider the quark-anti-quark potential on the three sphere or the generalized cusp anomalous dimension in planar $ \mathcal{N}=4 $ SYM. We concentrate on the vacuum potential in the near BPS limit with L units of R-charge. Equivalently, we study the anomalous dimension of a super-Wilson loop with L local fields inserted at a cusp. The system is described by a recently proposed infinite set of non-linear integral equations of the Thermodynamic Bethe Ansatz (TBA) type. That system of TBA equations is very similar to the one of the spectral problem but simplifies a bit in the near BPS limit. Using techniques based on the Y-system of functional equations we first reduced the infinite system of TBA equations to a Finite set of Nonlinear Integral Equations (FiNLIE). Then we solve the FiNLIE system analytically, obtaining a simple analytic result for the potential! Surprisingly, we find that the system has equivalent descriptions in terms of an effective Baxter equation and in terms of a matrix model. At L = 0, our result matches the one obtained before using localization techniques. At all other L’s, the result is new. Having a new parameter, L, allows us to take the large L classical limit. We use the matrix model description to solve the classical limit and match the result with a string theory computation. Moreover, we find that the classical string algebraic curve matches the algebraic curve arising from the matrix model.

The τ2-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin’s SOV method

The most general cyclic representations of the quantum integrable τ2-model are analyzed. The complete characterization of the τ2-spectrum (eigenvalues and eigenstates) is achieved in the framework of Sklyanin’s separation of variables (SOV) method by extending and adapting previous results of one of the authors: (i) the determination of the τ2-spectrum is reduced to the classification of the solutions of a given functional equation in a class of polynomials; (ii) the determination of the τ2-eigenstates is reduced to the classification of the solutions of an associated Baxter equation. These last solutions are proved to be polynomials for a quite general class of τ2-self-adjoint representations and the completeness of the associated Bethe ansatz type equations is derived. Finally, the following results are derived for the inhomogeneous chiral Potts model: (i) simplicity of the spectrum, for general representations; (ii) complete characterization of the chiral Potts spectrum (eigenvalues and eigenstates) and completeness of Bethe ansatz type equations, for the self-adjoint representations of the τ2-model on the chiral Potts algebraic curves.

A CATEGORICAL MODEL FOR THE VIRTUAL BRAID GROUP

This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this approach is to take pure virtual braids as primary. The generators of the pure virtual braid group are abstract solutions to the algebraic Yang–Baxter equation. This point of view illuminates representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter equation. In this categorical framework, the virtual braid group is a natural group associated with the structure of algebraic braiding. We then point out how the category SC is related to categories associated with quantum algebras and Hopf algebras and with quantum invariants of virtual links.

ON SOME LEFT-SYMMETRIC SUPERALGEBRAS

In this paper, we classify the left-symmetric superalgebras in dimensions two and three over the field of complex numbers. As a direct consequence, we obtain some explicit solutions of the graded classical Yang–Baxter equation in certain Lie superalgebras and their corresponding Lie bisuperalgebras.

R-Matrices and the Yang–Baxter equation on GNS representations on C∗-bialgebras

A new construction method of R-matrix is given. Let A be a C∗-bialgebra with a comultiplication Δ without the assumption of the quasi-cocommutativity. For two states ω and ψ of A which satisfy certain conditions, we construct a unitary R-matrix R(ω,ψ) of the C∗-bialgebra (A,Δ) on the tensor product of GNS representation spaces associated with ω and ψ. The set {R(ω,ψ):ω,ψ} satisfies a kind of Yang–Baxter equation. Furthermore, we show a nontrivial example of such R-matrices for a non-quasi-cocommutative C∗-bialgebra.

Hopf Algebra Symmetries of an Integrable Hamiltonian for Anyonic Pairing

Since the advent of Drinfel’d’s double construction, Hopf algebraic structures have been a centrepiece for many developments in the theory and analysis of integrable quantum systems. An integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries for particular values of its coupling parameters. While the integrable structure of the model relates to the well-known six-vertex solution of the Yang–Baxter equation, the Hopf algebra symmetries are not in terms of the quantum algebra Uq(sl(2)). Rather, they are associated with the Drinfel’d doubles of dihedral group algebras D(Dn).

ON CRYSTAL OPERATORS IN LUSZTIG'S PARAMETRIZATIONS AND STRING CONE DEFINING INEQUALITIES

AbstractLet w0 be a reduced expression for the longest element of the Weyl group, adapted to a quiver of type An. We compare Lusztig's and Kashiwara's (string) parametrizations on canonical basis associated with w0. Crystal operators act in a finite number of patterns in Lusztig's parametrization, which may be seen as vectors. We show that this set gives the system of defining inequalities of the string cone constructed by Gleizer and Postnikov (O. Gleizer and A Postnikov, Littlewood–Richardson coefficients via Yang–Baxter equation, IMRN14 (2000) 741–774). We use combinatorics of the Auslander–Reiten quivers, and as a by-product we get an alternative enumeration of a set of inequalities defining the string cone based on hammocks.

On Dynamical Galois Extensions

In the theory of dynamical Yang–Baxter equation, with any Hopf algebra H, a certain H-module, and H-comodule algebra L (base algebra) one can associate a monoidal category. Given an algebra A in that category, one can construct an associative algebra A ⋊ L, which is a generalization of the ordinary smash product when A is an ordinary H-algebra. We study this “dynamical smash product” and its modules induced from one-dimensional representations of the subalgebra L. In particular, we construct an analog of the Galois map A ⊗ A H A → A ⊗ H* and define dynamical Galois extensions.

Three-dimensional extensions of the Alday-Gaiotto-Tachikawa relation

An extension of the two-dimensional (2d) Alday-Gaiotto-Tachikawa (AGT) relation to three dimensions starts from relating the theory on the domain wall between some two S-dual supersymmetric Yang-Mills (SYM) models to the 3d Chern-Simons (CS) theory. The simplest case of such a relation would presumably connect traces of the modular kernels in 2d conformal theory with knot invariants. Indeed, the two quantities are very similar, especially if represented as integrals of quantum dilogarithms. But there are also various differences, especially in the “conservation laws” for the integration variables holding for the monodromy traces but not for the knot invariants. We also consider another possibility: interpreting knot invariants as solutions of the Baxter equations for the relativistic Toda system. This implies another AGT-like relation: between the 3d CS theory and the Nekrasov-Shatashvili limit of the 5d SYM theory.

Weak Lax pairs for lattice equations

We consider various 2D lattice equations and their integrability, from the point of view of 3D consistency, Lax pairs and Bäcklund transformations. We show that these concepts, which are associated with integrability, are not strictly equivalent. In the course of our analysis, we introduce a number of black and white lattice models, as well as variants of the functional Yang–Baxter equation.

𝒪-operators on associative algebras and associative Yang–Baxter equations

An O-operator on an associative algebra is a generalization of a Rota-Baxter operator that plays an important role in the Hopf algebra approach of Connes and Kreimer to the renormalization of quantum field theory. It is also the associative analog of an O-operator on a Lie algebra in the study of the classical Yang-Baxter equation. We introduce the concept of an extended O-operator on an associative algebra whose Lie algebra analog has been applied to generalized Lax pairs and PostLie algebras. We study algebraic structures coming from extended O-operators. Continuing the work of Aguiar deriving Rota-Baxter operators from the associative Yang-Baxter equation, we show that its solutions correspond to extended O-operators through a duality. We also establish a relationship of extended O-operators with the generalized associative Yang-Baxter equation.

GENERALIZED YANG–BAXTER EQUATIONS AND BRAIDING QUANTUM GATES

Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.