The integrable quantum group invariant [formula omitted] and [formula omitted] open spin chains

We consider open spin chains based on osp(m|2n) Yangians. We solve the reflection equations for some classes of reflection matrices, including the diagonal ones. Having then integrable open spin chains, we write the analytical Bethe Ansatz equations. More details and references can be found in [1,2].

Evolution equations for quark–gluon distributions in multi-color QCD and open spin chains

Motivated by a study of the crossing symmetry of the asymmetric twin or ‘gemini’representation of the affine Hecke algebra we give a construction for crossing tensor spacerepresentations of ordinary Hecke algebras. These representations build solutions to theYang–Baxter equation satisfying the crossing condition (that is, integrable quantum spinchains). We show that every crossing representation of the Temperley–Lieb algebra appearsin this construction, and in particular that this construction builds new representations. Weextend these to new representations of the blob algebra, which build new solutions to theboundary Yang–Baxter equation (i.e. open spin chains with integrable boundaryconditions).We prove that the open spin chain Hamiltonian derived from Sklyanin’s commutingtransfer matrix using such a solution can always be expressed as the representation of anelement of the blob algebra, and determine this element. We determine the representationtheory (irreducible content) of the new representations and hence show that all suchHamiltonians have the same spectrum up to multiplicity, for any given value of thealgebraic boundary parameter. (A corollary is that our models have the samespectrum as the open XXZ chain with nondiagonal boundary—despite differingfrom this model in having reference states.) Using these multiplicity data, andother ideas, we investigate the underlying quantum group symmetry of the newHamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.

The authors extend Sklyanin's formalism for constructing integrable open spin chains to the case of 'non-symmetric' (PT-invariant) R-matrices. They use this formalism to show that the Hamiltonian of an integrable open chain is invariant under gauge transformations of the R-matrix.

The authors have generalized Sklyanin's approach of constructing open integrable quantum spin chains to the case of PT-invariant R matrices. They formulate a fusion procedure for such chains. In particular, they show that the fused transfer matrix can be expressed in terms of products of the original transfer matrix and products of certain quantum determinants which can be explicitly evaluated. Applications of these results include constructing open integrable higher-spin chains, as well as obtaining functional equations for transfer-matrix eigenvalues, which may be solved by an analytical Bethe ansatz.

We argue that the multi-particle scattering amplitudes in N = 4 SUSY at large Nc and in the multi-Regge kinematics for some physical regions have the high energy behavior corresponding to the contribution of the Mandelstam cuts in the corresponding t-channel partial waves. The Mandelstam cuts correspond to gluon composite states in the adjoint representation of the gauge group SU(Nc). The Hamiltonian for these states in the leading logarithmic approximation coincides with the local Hamiltonian of an integrable open spin chain. We construct the corresponding wavefunctions using the integrals of motion and the Baxter–Sklyanin approach.

For Abstract see ChemInform Abstract in Full Text.

This is a pedagogical account on reaction-diffusion systems and their relationship with integrable quantum spin chains. Reaction-diffusion systems are paradigmatic examples of non-equilibrium systems. Their long-time behaviour is strongly influenced through fluctuation effects in low dimensions which renders the habitual mean-field cinetic equations inapplicable. Starting from the master equation rewritten as a Schr\"odinger equation with imaginary time, the associated quantum hamiltonian of certain one-dimensional reaction-diffusion models is closely related to integrable magnetic chains. The relationship with the Hecke algebra and its quotients allows to identify integrable reaction-diffusion models and, through the Baxterization procedure, relate them to the solutions of Yang-Baxter equations which can be solved via the Bethe ansatz. Methods such as spectral and partial integrability, free fermions, similarity transformations or diffusion algebras are reviewed, with several concrete examples treated explicitly. An outlook on how the recently-introduced concept of local scale invariance might become useful in the description of non-equilibrium ageing phenomena is presented, with particular emphasis on the kinetic Ising model with Glauber dynamics.

We review the dynamics after quantum quenches in integrable quantum spin chains. We give a pedagogical introduction to relaxation in isolated quantum systems, and discuss the description of the steady state by (generalized) Gibbs ensembles. We then turn to general features in the time evolution of local observables after the quench, using a simple model of free fermions as an example. In the second part we present an overview of recent progress in describing quench dynamics in two key paradigms for quantum integrable models, the transverse field Ising chain and the anisotropic spin-1/2 Heisenberg chain.

Integrable quantum spin chains display distinctive physical properties making them a laboratory to test and assess different states of matter. The study of the finite temperature properties is possible by use of the thermodynamic Bethe ansatz, however at the expense of dealing with non-linear integral equations for, in general, infinitely many auxiliary functions. The definition of an alternative finite set of auxiliary functions allowing for the complete description of their thermodynamic properties at finite temperature and fields has been elusive. Indeed, in the context of sln-invariant models satisfactory auxiliary functions have been established only for n≤4. In this paper we take a step further by proposing a systematic approach to generate finite sets of auxiliary functions for sln-invariant models. We refer to this construction as the canonical formulation. The numerical efficiency is illustrated for n=5, for which we present some of the thermodynamic properties of the corresponding spin chain.

In this contribution we briefly review recent developments in the theory of long-range integrable spin chains. These spin chains constitute a natural generalisation of the well-studied integrable nearest-neighbour chains and are of particular relevance to the integrability in the AdS/CFT correspondence since the dilatation operator in the asymptotic region is conjectured to be a Hamiltonian of an integrable long-range psu spin chain.

We consider the Temperley–Lieb (TL) open quantum spin chain with “free” boundary conditions associated with the spin-s representation of quantum-deformed sl(2). We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an off-shell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an appendix, we briefly consider the closed TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be improved so as to obtain the complete (albeit non-universal) spectrum.

The exact solution of an integrable anisotropic Heisenberg spin chain with nearest-neighbor, next-nearest-neighbor and scalar chirality couplings is studied, where the boundary condition is the antiperiodic one. The detailed construction of Hamiltonian and the proof of integrability are given. The antiperiodic boundary condition breaks the U(1)-symmetry of the system and we use the off-diagonal Bethe Ansatz to solve it. The energy spectrum is characterized by the inhomogeneous T−Q relations and the contribution of the inhomogeneous term is studied. The ground state energy and the twisted boundary energy in different regions are obtained. We also find that the Bethe roots at the ground state form the string structure if the coupling constant J=−1 although the Bethe Ansatz equations are the inhomogeneous ones.

Viscoelastic Kelvin–Voigt model on Ulam–Hyer’s stability and T-controllability for a coupled integro fractional stochastic systems with integral boundary conditions via integral contractors

By employing upper and lower solutions method together with maximal principle, we establish a necessary and sufficient condition for the existence of pseudo- as well as positive solutions for fourth-order singular -Laplacian differential equations with integral boundary conditions. Our nonlinearity may be singular at , , and . The dual results for the other integral boundary condition are also given.