1,434 research outputs found
The twisted XXZ chain at roots of unity revisited
The symmetries of the twisted XXZ spin-chain (alias the twisted six-vertex
model) at roots of unity are investigated. It is shown that when the twist
parameter is chosen to depend on the total spin an infinite-dimensional
non-abelian symmetry algebra can be explicitly constructed for all spin
sectors. This symmetry algebra is identified to be the upper or lower Borel
subalgebra of the sl_2 loop algebra. The proof uses only the intertwining
property of the six-vertex monodromy matrix and the familiar relations of the
six-vertex Yang-Baxter algebra.Comment: 10 pages, 2 figures. One footnote and some comments in the
conclusions adde
Children's Databases - Safety and Privacy
This report describes in detail the policy background, the systems that are being built, the problems with them, and the legal situation in the UK. An appendix looks at Europe, and examines in particular detail how France and Germany have dealt with these issues. Our report concludes with three suggested regulatory action strategies for the Commissioner: one minimal strategy in which he tackles only the clear breaches of the law, one moderate strategy in which he seeks to educate departments and agencies and guide them towards best practice, and finally a vigorous option in which he would seek to bring UK data protection practice in these areas more in line with normal practice in Europe, and indeed with our obligations under European law
A Q-operator for the twisted XXX model
Taking the isotropic limit in a recent representation theoretic construction
of Baxter's Q-operators for the XXZ model with quasi-periodic boundary
conditions we obtain new results for the XXX model. We show that quasi-periodic
boundary conditions are needed to ensure convergence of the Q-operator
construction and derive a quantum Wronskian relation which implies two
different sets of Bethe ansatz equations, one above the other below the
"equator" of total spin zero. We discuss the limit to periodic boundary
conditions at the end and explain how this construction might be useful in the
context of correlation functions on the infinite lattice. We also identify a
special subclass of solutions to the quantum Wronskian for chains up to a
length of 10 sites and possibly higher.Comment: 19 page
Quantum cohomology via vicious and osculating walkers
We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder, so-called vicious and osculating walkers. These lattice paths form exactly solvable statistical mechanics models and are obtained from solutions to the Yang–Baxter equation. The eigenvectors of the transfer matrices of these models yield the idempotents of the Verlinde algebra of the gauged u^(n)k -WZNW model. The latter is known to be closely related to the small quantum cohomology ring of the Grassmannian. We establish further that the partition functions of the vicious and osculating walker model are given in terms of Postnikov’s toric Schur functions and can be interpreted as generating functions for Gromov–Witten invariants. We reveal an underlying quantum group structure in terms of Yang–Baxter algebras and use it to give a generating formula for toric Schur functions in terms of divided difference operators which appear in known representations of the nil-Hecke algebra
Baxter Q-operators of the XXZ chain and R-matrix factorization
We construct Baxter operators as generalized transfer matrices being traces
of products of generic matrices. The latter are shown to factorize into
simpler operators allowing for explicit expressions in terms of functions of a
Weyl pair of basic operators. These explicit expressions are the basis for
explicit expression for Baxter Q-operators and for investigating their
properties.Comment: 19 pages LaTex, references adde
XXZ Bethe states as highest weight vectors of the loop algebra at roots of unity
We show that every regular Bethe ansatz eigenvector of the XXZ spin chain at
roots of unity is a highest weight vector of the loop algebra, for some
restricted sectors with respect to eigenvalues of the total spin operator
, and evaluate explicitly the highest weight in terms of the Bethe roots.
We also discuss whether a given regular Bethe state in the sectors generates an
irreducible representation or not. In fact, we present such a regular Bethe
state in the inhomogeneous case that generates a reducible Weyl module. Here,
we call a solution of the Bethe ansatz equations which is given by a set of
distinct and finite rapidities {\it regular Bethe roots}. We call a nonzero
Bethe ansatz eigenvector with regular Bethe roots a {\it regular Bethe state}.Comment: 40pages; revised versio
Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz
We connect two alternative concepts of solving integrable models, Baxter's
method of auxiliary matrices (or Q-operators) and the algebraic Bethe ansatz.
The main steps of the calculation are performed in a general setting and a
formula for the Bethe eigenvalues of the Q-operator is derived. A proof is
given for states which contain up to three Bethe roots. Further evidence is
provided by relating the findings to the six-vertex fusion hierarchy. For the
XXZ spin-chain we analyze the cases when the deformation parameter of the
underlying quantum group is evaluated both at and away from a root of unity.Comment: 32 page
Auxiliary matrices for the six-vertex model at roots of 1 and a geometric interpretation of its symmetries
The construction of auxiliary matrices for the six-vertex model at a root of
unity is investigated from a quantum group theoretic point of view. Employing
the concept of intertwiners associated with the quantum loop algebra
at a three parameter family of auxiliary matrices
is constructed. The elements of this family satisfy a functional relation with
the transfer matrix allowing one to solve the eigenvalue problem of the model
and to derive the Bethe ansatz equations. This functional relation is obtained
from the decomposition of a tensor product of evaluation representations and
involves auxiliary matrices with different parameters. Because of this
dependence on additional parameters the auxiliary matrices break in general the
finite symmetries of the six-vertex model, such as spin-reversal or spin
conservation. More importantly, they also lift the extra degeneracies of the
transfer matrix due to the loop symmetry present at rational coupling values.
The extra parameters in the auxiliary matrices are shown to be directly related
to the elements in the enlarged center of the quantum loop algebra
at . This connection provides a geometric
interpretation of the enhanced symmetry of the six-vertex model at rational
coupling. The parameters labelling the auxiliary matrices can be interpreted as
coordinates on a three-dimensional complex hypersurface which remains invariant
under the action of an infinite-dimensional group of analytic transformations,
called the quantum coadjoint action.Comment: 52 pages, TCI LaTex, v2: equation (167) corrected, two references
adde
- …