Baxter Equation Research Articles

Nonsinglet pentagons and NMHV amplitudes

Scattering amplitudes in maximally supersymmetric gauge theory receive a dual description in terms of the expectation value of the super Wilson loop stretched on a null polygonal contour. This makes the analysis amenable to nonperturbative techniques. Presently, we elaborate on a refined form of the operator product expansion in terms of pentagon transitions to compute twist-two contributions to NMHV amplitudes. To start with, we provide a novel derivation of scattering matrices starting from Baxter equations for flux-tube excitations propagating on magnon background. We propose bootstrap equations obeyed by pentagon form factors with nonsinglet quantum numbers with respect to the R-symmetry group and provide solutions to them to all orders in 't Hooft coupling. These are then successfully confronted against available perturbative calculations for NMHV amplitudes to four-loop order.

Measured quantum groupoids associated to proper dynamical quantum groups

Dynamical quantum groups were introduced by Etingof and Varchenko in connection with the dynamical quantum Yang–Baxter equation, and measured quantum groupoids were introduced by Enock, Lesieur and Vallin in their study of inclusions of type II _1 factors. In this article, we associate to suitable dynamical quantum groups, which are purely algebraic objects, Hopf C* -bimodules and measured quantum groupoids on the level of von Neumann algebras. Assuming invariant integrals on the dynamical quantum group, we first construct a fundamental unitary which yields Hopf bimodules on the level of C* -algebras and von Neumann algebras. Next, we assume properness of the dynamical quantum group and lift the integrals to the operator algebras. In a subsequent article, this construction shall be applied to the dynamical SU _q (2) studied by Koelink and Rosengren.

Vector bundles on plane cubic curves and the classical Yang–Baxter equation

In this article, we develop a geometric method to construct solutions of the classical Yang-Baxter equation, attaching to the Weierstrass family of plane cubic curves and a pair of coprime positive integers, a family of classical r-matrices. It turns out that all elliptic r-matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin's classification and prove that they are degenerations of the corresponding elliptic solutions.

A q-difference Baxter operator for the Ablowitz–Ladik chain

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.

The anisotropic λ-deformed SU(2) model is integrable

The all-loop anisotropic Thirring model interpolates between the WZW model and the non-Abelian T-dual of the anisotropic principal chiral model. We focus on the SU(2) case and we prove that it is classically integrable by providing its Lax pair formulation. We derive its underlying symmetry current algebra and use it to show that the Poisson brackets of the spatial part of the Lax pair, assume the Maillet form. In this way we procure the corresponding r and s matrices which provide non-trivial solutions to the modified Yang–Baxter equation.

Yang–Baxter sigma models based on the CYBE

It is known that Yang–Baxter sigma models provide a systematic way to study integrable deformations of both principal chiral models and symmetric coset sigma models. In the original proposal and its subsequent development, the deformations have been characterized by classical r-matrices satisfying the modified classical Yang–Baxter equation (mCYBE). In this article, we propose the Yang–Baxter sigma models based on the classical Yang–Baxter equations (CYBE) rather than the mCYBE. This generalization enables us to utilize various kinds of solutions of the CYBE to classify integrable deformations. In particular, it is straightforward to realize partial deformations of the target space without loss of the integrability of the parent theory.

Applicability of Cassie–Baxter equation for superhydrophobic fluoropolymer–silica composite films

In this study, surface topographies and wettabilities of flat and composite rough films of perfluoro-styrene (TM/S) random copolymers with silica particles were investigated. The water and oil repellencies of flat TM/S copolymer and rough silica-TM/S composite surfaces were studied with varying perfluoro and silica contents. Increase in silica content resulted in an increase in water contact angle values; water contact angle values increased from 113° up to 170°, and the resultant surfaces were showing extremely superhydrophobic behavior depending on their silica contents. However a decrease in hexadecane contact angles from 61° to 25° depending on both silica %wt and perfluoro content of the TM/S copolymer was seen. Applicability of Cassie–Baxter equation was also investigated for TM/S copolymer films with silica particles. The results showed that Cassie–Baxter equation cannot be used for superhydrophobic surfaces, however, can be applied to oleophilic surfaces with caution. The oil repellencies of TM/S flat films in the terms of contact angle hysteresis was also compared with perfluoroethyl alkyl methacrylate–methyl methacrylate copolymer (TM/MMA) by using oil drops with varying surface tensions between 20.9mN/m and 26.9mN/m. The surface oleophobicities were changed according to the characteristics of the functional groups of MMA and S. The dependency of work of adhesion results of TM/S flat surfaces on wt.% of perfluoroalkyl content was also examined.

Baxter Q-operator and functional relations

We obtain the Baxter Q-operators in the Uq(slˆ2) invariant integrable models as a special limits of the quantum transfer matrices corresponding to different spins in the auxiliary space. We derive the Baxter equation from the well-known fusion relations for the transfer matrices. Our method is valid for an arbitrary integrable model corresponding to the quantum group Uq(slˆ2), for example for the XXZ-spin chain.

Azumaya Monads and Comonads

The definition of Azumaya algebras over commutative rings \(R\) requires the tensor product of modules over \(R\) and the twist map for the tensor product of any two \(R\)-modules. Similar constructions are available in braided monoidal categories, and Azumaya algebras were defined in these settings. Here, we introduce Azumaya monads on any category \(\mathbb{A}\) by considering a monad \((F,m,e)\) on \(\mathbb{A}\) endowed with a distributive law \(\lambda: FF\to FF\) satisfying the Yang–Baxter equation (BD%please define -law). This allows to introduce an opposite monad \((F^\lambda,m\cdot \lambda,e)\) and a monad structure on \(FF^\lambda\). The quadruple \((F,m,e,\lambda)\) is called an Azumaya monad, provided that the canonical comparison functor induces an equivalence between the category \(\mathbb{A}\) and the category of \(FF^\lambda\)-modules. Properties and characterizations of these monads are studied, in particular for the case when \(F\) allows for a right adjoint functor. Dual to Azumaya monads, we define Azumaya comonads and investigate the interplay between these notions. In braided categories (V\(,\otimes,I,\tau)\), for any V-algebra \(A\), the braiding induces a BD-law \(\tau_{A,A}:A\otimes A\to A\otimes A\), and \(A\) is called left (right) Azumaya, provided the monad \(A\otimes-\) (resp. \(-\otimes A\)) is Azumaya. If \(\tau\) is a symmetry or if the category V admits equalizers and coequalizers, the notions of left and right Azumaya algebras coincide.

On integrable matrix product operators with bond dimension D = 4

We construct and study a two-parameter family of matrix product operators of bond dimension D = 4. The operators M(x, y) act on , i.e. the space of states of a spin-1/2 chain of length N. For the particular values of the parameters: x = 1/3 and , the operator turns out to be proportional to the square root of the reduced density matrix of the valence-bond-solid state on a hexagonal ladder. We show that M(x, y) has several interesting properties when (x, y) lies on the unit circle centered at the origin: x2 + y2 = 1. In this case, we find that M(x, y) commutes with the Hamiltonian and all the conserved charges of the isotropic spin-1/2 Heisenberg chain. Moreover, M(x1, y1) and M(x2, y2) are mutually commuting if for both i = 1 and 2. These remarkable properties of M(x, y) are proved as a consequence of the Yang–Baxter equation.

Colored Quantum Algebra and Its Bethe State

We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation.

Exotic cluster structures on SL5

A conjecture by Gekhtman, Shapiro and Vainshtein suggests a correspondence between the Belavin–Drinfeld (BD) classification of solutions of the classical Yang–Baxter equation and cluster structures on simple Lie groups. This paper confirms the conjecture for SL5. Given a BD class, we construct the corresponding cluster structure in , and show that it satisfies all parts of the conjecture.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Cluster algebras in mathematical physics’.

The classical Yang–Baxter equation and the associated Yangian symmetry of gauged WZW-type theories

We construct the Lax-pair, the classical monodromy matrix and the corresponding solution of the Yang–Baxter equation, for a two-parameter deformation of the Principal chiral model for a simple group. This deformation includes as a one-parameter subset, a class of integrable gauged WZW-type theories interpolating between the WZW model and the non-Abelian T-dual of the principal chiral model. We derive in full detail the Yangian algebra using two independent methods: by computing the algebra of the non-local charges and alternatively through an expansion of the Maillet brackets for the monodromy matrix. As a byproduct, we also provide a detailed general proof of the Serre relations for the Yangian symmetry.

Parameter-dependent associative Yang–Baxter equations and Poisson brackets

We discuss associative analogues of classical Yang–Baxter equation (CYBE) meromorphically dependent on parameters. We discover that such equations enter in a description of a general class of parameter-dependent Poisson structures and double Lie and Poisson structures in sense of Van den Bergh. We propose a classification of all solutions for one-dimensional associative Yang–Baxter equations (AYBE).

On left-symmetric conformal bialgebras

The notion of left-symmetric bialgebra was introduced in [C. M. Bai, Left-symmetric bialgebra and an analogue of the classical Yang–Baxter equation, Commun. Contemp. Math. 10(2) (2008) 221–260] which is equivalent to a parakähler Lie algebra which is the Lie algebra of a Lie group G with a G-invariant parakähler structure. In this paper, we study a conformal analog of left-symmetric bialgebras. The notions of left-symmetric conformal coalgebra and bialgebra are introduced. Moreover, the constructions of matched pairs of Lie conformal algebras and left-symmetric conformal algebras are presented. We show that a finite left-symmetric conformal bialgebra which is free as a ℂ[∂]-module is equivalent to a parakähler Lie conformal algebra. We also obtain a conformal analog of the S-equation (see [C. M. Bai, Left-symmetric bialgebras and an analogue of the classical Yang–Baxter equation, Commun. Contemp. Math. 10(2) (2008) 221–260]), and give a construction of the conformal symplectic double.

Construction of quasi-linear left cycle sets

In this paper, we produce a method to construct quasi-linear left cycle sets A with Rad (A) ⊆ Fix (A). Moreover, among these cycle sets, we give a complete description of those for which Fix (A) = Soc (A) and the underlying additive group is cyclic. Using such cycle sets, we obtain left non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation which are different from those obtained in [P. Etingof, T. Schedler and A. Soloviev, Set-theoretical solutions to the quantum Yang–Baxter equation, Duke Math. J. 100 (1999) 169–209; P. Etingof, A. Soloviev and R. Guralnick, Indecomposable set-theoretical solutions to the quantum Yang–Baxter equation on a set with a prime number of elements, J. Algebra 242 (2001) 709–719].

Wettability versus roughness: Multi-scales approach

These authors experimentally investigate the influence of multi-scale roughness on contact angle measurements, and they propose a model combining the Wenzel and Cassie–Baxter equations with the 3D roughness parameters defined by ISO25178. To do this, a generic methodology based on a statistical method (bootstrap and correlation coefficient) is developed and applied to a polypropylene textured femtosecond laser (55 surfaces, ablation depths=5–45μm and depth/diameter=0.07–0.53). A mixed model is proposed according to the correlations obtained between the contact angle of a plane surface, θ0, and a textured surface, θ, with the roughness parameters as the developed surface, Sdr, closed hills area, Sha and the closed dales area, Sda, with regard to the models of Wenzel and Cassie–Baxter.

Generalizations of Poisson Structures Related to Rational Gaudin Model

The Poisson structure arising in the Hamiltonian approach to the rational Gaudin model looks very similar to the so-called modified Reflection Equation Algebra. Motivated by this analogy, we realize a braiding of the mentioned Poisson structure, i.e. we introduce a “braided Poisson” algebra associated with an involutive solution to the quantum Yang–Baxter equation. Also, we exhibit another generalization of the Gaudin type Poisson structure by replacing the first derivative in the current parameter, entering the so-called local form of this structure, by a higher order derivative. Finally, we introduce a structure, which combines both generalizations. Some commutative families in the corresponding braided Poisson algebra are found.

Geometric study of transparent superhydrophobic surfaces of molded and grid patterned polydimethylsiloxane (PDMS)

Herein we describe an economical method to fabricate a transparent superhydrophobic surface that uses grid patterning, and we report on the effects of grid geometry in determining the wettability and transparency of the fabricated surfaces. A polymer casting method was utilized because of its applicability to economical manufacturing and mass production; the material polydimethylsiloxane (PDMS) was selected because of its moldability and transparency. PDMS was replicated from a laser textured mold fabricated by a UV nanosecond pulsed laser. Sapphire wafer was used for the mold because it has very low surface roughness (Ra ≤0.3 nm) and adequate mechanical properties. To study geometric effects, grid patterns of a series of step sizes were fabricated. The maximum water droplet contact angle (WDCA) observed was 171°. WDCAs depended on the wetting area and the wetting state. The experimental results of WDCA were analyzed with Wenzel and Cassie–Baxter equations. The designed grid pattern was suitably transparent and structurally stable. Transmittance of the optimal transparent superhydrophobic surface was measured by using a spectrophotometer. Transmittance loss due to the presence of the grid was around 2–4% over the wavelength region measured (300–1000 nm); the minimum transmittance observed was 83.1% at 300 nm. This study also demonstrates the possibility of using a nanosecond pulsed laser for the surface texturing of a superhydrophobic surface.

Yang–Baxter equations and quantum entanglements

In this paper some results associated with a new type of Yang---Baxter equation (YBE) are reviewed. The braiding matrix of Kauffman---Lomonaco has been extended to the solution (called type-II) of Yang---Baxter equation (YBE) and the related chain Hamiltonian is given. The Lorentz additivity for spectral parameters is found, rather than the Galilean rule for the familiar solutions (called type-I) of YBE associated with the usually exact solvable models. Based on the topological basis, the N-dimensional solution of YBE is found to be the Wigner D-functions. The explicit examples for spin-$$\frac{1}{2}$$12 and spin-1 have been shown. The extremes of $$\ell _1$$l1-norm of $$D$$D-functions are introduced to distinguish the type-I from type-II of braiding matrices that also correspond to those of von Neumann entropy for quantum information.