Baxter operators for arbitrary spin

We construct Baxter operators for the homogeneous closed XXX spin chain with the quantum space carrying infinite- or finite-dimensional s?₂ representations. All algebraic relations of Baxter operators and transfer matrices are deduced uniformly from Yang-Baxter relations of the local building blocks of these operators. This results in a systematic and very transparent approach where the cases of finite- and infinite-dimensional representations are treated in analogy. Simple relations between the Baxter operators of both cases are obtained. We represent the quantum spaces by polynomials and build the operators from elementary differentiation and multiplication operators. We present compact explicit formulae for the action of Baxter operators on polynomials.     

Universal R operator with deformed conformal symmetry

We study the general solution of the Yang–Baxter equation with deformed sl(2) symmetry. The universal R operator acting on tensor products of arbitrary representations is obtained in spectral decomposition and in integral forms. The results for eigenvalues, eigenfunctions and integral kernel appear as deformations of the ones in the rational case. They provide a basis for the construction of integrable quantum systems generalizing the XXZ spin models to the case of arbitrary not necessarily finite-dimensional representations on the sites.     

New rational solutions of Yang-Baxter equation and deformed Yangians

In this paper a new class of quantum groups, deformed Yangians, is used to obtain new matrix rational solutions of the Yang-Baxter equation (YBE). The deformed Yangians arise from rational solutions of the classical Yang-Baxter equation of the form c ₂/u + const. The image of the universal quantum R-matrix for the deformed Yangian in finite-dimensional representations gives these new matrix rational solutions of YBE.     

Baxter Operator Formalism for Macdonald Polynomials

We develop basic constructions of the Baxter operator formalism for the Macdonald polynomials associated with root systems of type A. Precisely, we construct a bispectral pair of mutually commuting Baxter operators such that the Macdonald polynomials are their common eigenfunctions. The bispectral pair of Baxter operators is closely related to the bispectral pair of recursive operators for Macdonald polynomials leading to various families of their integral representations. We also construct the Baxter operator formalism for the q-deformed ${\mathfrak{gl}_{\ell+1}}$ -Whittaker functions and the Jack polynomials obtained by degenerations of the Macdonald polynomials associated with the type A _? root system. This note provides a generalization of our previous results on the Baxter operator formalism for the Whittaker functions. It was demonstrated previously that Baxter operator formalism for the Whittaker functions has deep connections with representation theory. In particular, the Baxter operators should be considered as elements of appropriate spherical Hecke algebras and their eigenvalues are identified with local Archimedean L-factors associated with admissible representations of reductive groups over ${\mathbb{R}}$ . We expect that the Baxter operator formalism for the Macdonald polynomials has an interpretation in representation theory over higher-dimensional local/global fields.     

Regular Basis and R-Matrices for the ^(su)(n)k Knizhnik–Zamolodchikov Equation

Dynamical R-matrix relations are derived for the group-valued chiral vertex operators in the SU(n) WZNW model from the KZ equation for a general four-point function including two step operators. They fit the exchange relations of the U _q (sl _n ) covariant quantum matrix algebra derived previously by solving the dynamical Yang–Baxter equation. As a byproduct, we extend the regular basis introduced earlier for SU(2) chiral fields to SU(n) step operators and display the corresponding triangular matrix representation of the braid group.     

Spin Operator Matrix Elements in the Superintegrable Chiral Potts Quantum Chain

We derive spin operator matrix elements between general eigenstates of the superintegrable ℤ _N -symmetric chiral Potts quantum chain of finite length. Our starting point is the extended Onsager algebra recently proposed by Baxter. For each pair of spaces (Onsager sectors) of the irreducible representations of the Onsager algebra, we calculate the spin matrix elements between the eigenstates of the Hamiltonian of the quantum chain in factorized form, up to an overall scalar factor. This factor is known for the ground state Onsager sectors. For the matrix elements between the ground states of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters. For the Ising quantum chain in a transverse field (N=2 case) the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the Separation of Variables method.     

Group-Theoretical Approach to Extended Conformal Supersymmetry: Function Space Realizations and Invariant Differential Operators

We give function space realizations of all representations of the conformal superalgebra su(2,2/N) and of the supergroup SU(2, 2 /N) induced from irreducible finite-dimensional Lorentz and SU(N) representations realized without spin and isospin indices. We use the lowest weight module structure of our su(2,2/N) representations to present a general procedure (adapted from the semisimple Lie algebra case) for the canonical construction of invariant differential operators closely related to the reducible (indecomposable) representations. All conformal supercovariant derivatives are obtained in this way. Examples of higher order invariant differential operators are given.     

Development of the Hamiltonian and transition moment operators of symmetric top molecules using the O (3) ⊃ C∞v ⊃ C3v group chain

We present a development of the Hamiltonian, dipole moment, and polarizability operators for XY₃Z molecules. These rovibrational operators are written with the aid of a tensorial formalism derived from the one already used in Dijon and adapted to the XY₃Z symmetric tops in a recent paper [A. El Hilali, V. Boudon, M. Loëte, J. Mol. Spectrosc. 234 (2005) 166-174]. We use the O (3) ⊃ C_∞v ⊃ C_3v group chain. Expressions for the matrix elements are derived for these operators.     

Baxter operators with deformed symmetry

Baxter operators are constructed for quantum spin chains with deformed _s?2${\mathrm{s?}}_2$ symmetry. The parallel treatment of Yang–Baxter operators for the cases of undeformed, trigonometrically and elliptically deformed symmetries presented earlier and relying on the factorization regarding parameter permutations is extended to the global chain operators following the scheme worked out recently in the undeformed case.     

Quantum affine algebras and holonomic difference equations

We derive new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra representations of levelk. We study the connection opertors between the solutions with different asymptotics and show that they are given by products of elliptic theta functions. We prove that the connection operators automatically provide elliptic solutions of Yang-Baxter equations in the face formulation for any type of Lie algebra and arbitrary finite-dimensional representations of. We conjecture that these solutions of the Yang-Baxter equations cover all elliptic solutions known in the contexts of IRF models of statistical mechanics. We also conjecture that in a special limit whenq1 these solutions degenerate again into solutions with . We also study the simples examples of solutions of our holonomic difference equations associated to and find their expressions in terms of basic (orq–)-hypergeometric series. In the special case of spin –1/2 representations, we demonstrate that the connection matrix yields a famous Baxter solution of the Yang-Baxter equation corresponding to the solid-on-solid model of statistical mechanics.     

An all-silicon linear chain NMR quantum computer

An updated version of our all-silicon quantum computing scheme [T.D. Ladd, J.R. Goldman, F. Yamaguchi, Y. Yamamoto, E. Abe, K.M. Itoh, Phys. Rev. Lett. 89 (2002) 017901. [3]] and the experimental progress towards its realization are discussed. We emphasize the importance of revisiting a wide range of isotope effects which have been explored over the past several decades for the construction of solid-state silicon quantum computers. Using RF decoupling techniques [T.D. Ladd, D. Maryenko, Y. Yamamoto, E. Abe, K.M. Itoh, Phys. Rev. B. 71 (2005) 014401] phase decoherence times T₂=25 s of ²⁹Si nuclear spins in single-crystal Si have been obtained at room temperature. We show that a linear chain of ²⁹Si stable isotopes with nuclear spin I=1/2 embedded in a spin free ²⁸Si stable isotope matrix can form an ideal building block for solid-state quantum information processors, especially, in the form of a quantum memory which requires a large number of operations within T₂ for the continuous error correction.     

Aspects of the functional renormalisation group

We discuss structural aspects of the functional renormalisation group. Flows for a general class of correlation functions are derived, and it is shown how symmetry relations of the underlying theory are lifted to the regularised theory. A simple equation for the flow of these relations is provided. The setting includes general flows in the presence of composite operators and their relation to standard flows, an important example being NPI quantities. We discuss optimisation and derive a functional optimisation criterion. Applications deal with the interrelation between functional flows and the quantum equations of motion, general Dyson-Schwinger equations. We discuss the combined use of these functional equations as well as outlining the construction of practical renormalisation schemes, also valid in the presence of composite operators. Furthermore, the formalism is used to derive various representations of modified symmetry relations in gauge theories, as well as to discuss gauge-invariant flows. We close with the construction and analysis of truncation schemes in view of practical optimisation.     

Multiparameter deformed and non-standard Y(glM) Yangian symmetry in a novel class of spin Calogero-Sutherland models

It is well known through a recent work of Bernard, Gaudin, Haldane and Pasquier (BGHP) that the usual spin Calogero-Sutherland (CS) model, containing particles with M internal degrees of freedom, respects the Y(gl_M) Yangian symmetry. By following and suitably modifying the approach of BGHP, in this article we construct a novel class of spin CS models which exhibit multiparameter deformed or ‘non-standard’ variants of Y(gl_M) Yangian symmetry. An interesting feature of such CS Hamiltonians is that they contain many-body spin-dependent interactions, which can be calculated directly from the associated rational solutions of the Yang-Baxter equation. Moreover, these spin-dependent interactions often lead to ‘anyon-like’ representations of the permutation algebra on the combined internal space of all particles. We also find the general forms of conserved quantities as well as Lax pairs for the above-mentioned class of spin CS models, and describe the method of constructing their exact wave functions.     

Boundary Quantum Group Generators of Type A

We construct boundary quantum group generators which, through linear intertwining relations, determine nondiagonal solutions of the boundary Yang–Baxter equation for vector representations of A _n-1 ⁽¹⁾ and A ₂ ⁽²⁾ .     

Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors

In this paper we consider the spin-1/2 highest weight representations for the 6-vertex Yang–Baxter algebra on a finite lattice and analyze the integrable quantum models associated to the antiperiodic transfer matrix. For these models, which in the homogeneous limit reproduces the XXZ spin-1/2 quantum chains with antiperiodic boundary conditions, we obtain in the framework of Sklyanin?s quantum separation of variables (SOV) the following results: I) The complete characterization of the transfer matrix spectrum (eigenvalues/eigenstates) and the proof of its simplicity. II) The reconstruction of all local operators in terms of Sklyanin?s quantum separate variables. III) One determinant formula for the scalar products of separates states, the elements of the matrix in the scalar product are sums over the SOV spectrum of the product of the coefficients of the states. IV) The form factors of the local spin operators on the transfer matrix eigenstates by one determinant formulae given by simple modifications of the scalar product formulae.     

Nekrasov prepotential with fundamental matter from the quantum spin chain

Nekrasov functions were conjectured in Mironov and Morozov (2009) [1] to be related to exact Bohr-Sommerfeld periods of quantum integrable systems. This statement was thoroughly checked for the case of the pure SU(Nc) gauge theory in Mironov and Morozov (2009) [2] and Popolitov (2010) [3]. Here we successfully perform a set of checks in the case of gauge group SU(Nc) with additional Nf fundamental hypermultiplets. We show that the Baxter equation for the spin chain gives the same quantum periods as the one for the Gaudin system in this case.     

Yang-Baxter and other relations for free-fermion and Ising models

Eight-vertex, free fermion, and Ising models are formulated using a convention that emphasizes the algebra of the local transition operators that arise in the quantum inverse method. Equivalent classes of models, are investigated, with particular emphasis on the role of the star-triangle relations. Using these results, a natural and symmetrical parametrization is introduced and Yang-Baxter relations are constructed in an elementary way. The paper concludes with a consideration of duality, which links the present work to a recent paper of Baxter on the free fermion model.     

Systematic classical continuum limits of integrable spin chains and emerging novel dualities

We examine certain classical continuum long wave-length limits of prototype integrable quantum spin chains. We define the corresponding construction of classical continuum Lax operators. Our discussion starts with the XXX chain, the anisotropic Heisenberg model and their generalizations and extends to the generic isotropic and anisotropic gln$g{l}_n$ magnets. Certain classical and quantum integrable models emerging from special “dualities” of quantum spin chains, parametrized by c-number matrices, are also presented.     

Q operators for the simple quantum relativistic Toda Chain

We investigate the simple quantum relativistic Toda chain. The ultralocal simple Weyl algebra pair is associated with each site of the chain. Weyl’s q is considered to be inside a unit circle. Both independent Baxter operators Q are constructed explicitly as series in local Weyl generators. The operator-valued Wronskian of Q-s is also calculated.