2,232 research outputs found
On the quantum inverse scattering problem
A general method for solving the so-called quantum inverse scattering problem
(namely the reconstruction of local quantum (field) operators in term of the
quantum monodromy matrix satisfying a Yang-Baxter quadratic algebra governed by
an R-matrix) for a large class of lattice quantum integrable models is given.
The principal requirement being the initial condition (R(0) = P, the
permutation operator) for the quantum R-matrix solving the Yang-Baxter
equation, it applies not only to most known integrable fundamental lattice
models (such as Heisenberg spin chains) but also to lattice models with
arbitrary number of impurities and to the so-called fused lattice models
(including integrable higher spin generalizations of Heisenberg chains). Our
method is then applied to several important examples like the sl(n) XXZ model,
the XYZ spin-1/2 chain and also to the spin-s Heisenberg chains.Comment: Latex, 20 page
Analytic Bethe Ansatz and Baxter equations for long-range psl(2|2) spin chain
We study the largest particle-number-preserving sector of the dilatation
operator in maximally supersymmetric gauge theory. After exploring one-loop
Bethe Ansatze for the underlying spin chain with psl(2|2) symmetry for simple
root systems related to several Kac-Dynkin diagrams, we use the analytic Bethe
Anzats to construct eigenvalues of transfer matrices with finite-dimensional
atypical representations in the auxiliary space. We derive closed Baxter
equations for eigenvalues of nested Baxter operators. We extend these
considerations for a non-distinguished root system with FBBF grading to all
orders of perturbation theory in 't Hooft coupling. We construct generating
functions for all transfer matrices with auxiliary space determined by Young
supertableaux (1^a) and (s) and find determinant formulas for transfer matrices
with auxiliary spaces corresponding to skew Young supertableaux. The latter
yields fusion relations for transfer matrices with auxiliary space
corresponding to representations labelled by square Young supertableaux. We
derive asymptotic Baxter equations which determine spectra of anomalous
dimensions of composite Wilson operators in noncompact psl(2|2) subsector of
N=4 super-Yang-Mills theory.Comment: 32 pages, 2 figure
Quantum Groups
These notes correspond rather accurately to the translation of the lectures
given at the Fifth Mexican School of Particles and Fields, held in Guanajuato,
Gto., in December~1992. They constitute a brief and elementary introduction to
quantum symmetries from a physical point of view, along the lines of the
forthcoming book by C. G\'omez, G. Sierra and myself.Comment: 37 pages, plain.te
Integrability and Fusion Algebra for Quantum Mappings
We apply the fusion procedure to a quantum Yang-Baxter algebra associated
with time-discrete integrable systems, notably integrable quantum mappings. We
present a general construction of higher-order quantum invariants for these
systems. As an important class of examples, we present the Yang-Baxter
structure of the Gel'fand-Dikii mapping hierarchy, that we have introduced in
previous papers, together with the corresponding explicit commuting family of
quantum invariants.Comment: 26 page
Baxterization, dynamical systems, and the symmetries of integrability
We resolve the `baxterization' problem with the help of the automorphism
group of the Yang-Baxter (resp. star-triangle, tetrahedron, \dots) equations.
This infinite group of symmetries is realized as a non-linear (birational)
Coxeter group acting on matrices, and exists as such, {\em beyond the narrow
context of strict integrability}. It yields among other things an unexpected
elliptic parametrization of the non-integrable sixteen-vertex model. It
provides us with a class of discrete dynamical systems, and we address some
related problems, such as characterizing the complexity of iterations.Comment: 25 pages, Latex file (epsf style). WARNING: Postscript figures are
BIG (600kB compressed, 4.3MB uncompressed). If necessary request hardcopy to
[email protected] and give your postal mail addres
Correlation functions of the XXZ Heisenberg spin-1/2 chain in a magnetic field
Using the algebraic Bethe ansatz method, and the solution of the quantum
inverse scattering problem for local spins, we obtain multiple integral
representations of the -point correlation functions of the XXZ Heisenberg
spin- chain in a constant magnetic field. For zero magnetic field,
this result agrees, in both the massless and massive (anti-ferromagnetic)
regimes, with the one obtained from the q-deformed KZ equations (massless
regime) and the representation theory of the quantum affine algebra together with the corner transfer matrix approach (massive
regime).Comment: Latex2e, 26 page
Quantum 2+1 evolution model
A quantum evolution model in 2+1 discrete space - time, connected with 3D
fundamental map R, is investigated. Map R is derived as a map providing a zero
curvature of a two dimensional lattice system called "the current system". In a
special case of the local Weyl algebra for dynamical variables the map appears
to be canonical one and it corresponds to known operator-valued R-matrix. The
current system is a kind of the linear problem for 2+1 evolution model. A
generating function for the integrals of motion for the evolution is derived
with a help of the current system. The subject of the paper is rather new, and
so the perspectives of further investigations are widely discussed.Comment: LaTeX, 37page
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