# University of Leeds Workshop on Integrable Systems, 9th/10th May, 2007

## Available Abstracts

### Abstract

The recent interest in aspects common to quantum information and condensed matter has prompted a prosperous activity at the border of these disciplines that were far distant until few years ago. Numerous interesting questions have been addressed so far. Here we review an important part of this field, the properties of the entanglement in many-body systems. We discuss the zero and finite temperature properties of entanglement in interacting spin, fermion and boson model systems. Both bipartite and multipartite entanglement will be considered. In equilibrium we emphasize on how entanglement is connected to the phase diagram of the underlying model. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium we discuss how to generate and manipulate entangled states by means of many-body Hamiltonians.

### Abstract

Baxter's Q-operator is a central object in solving integrable lattice models in statistical mechanics and quantum spin-chains. It has recently seen a "renaissance" due to its application to a wide range of models and new construction methods. In my talk I will focus on the six-vertex model (a model describing ferroelectrics such as water/ice) and the associated quantum XXZ chain. In this example Q can be constructed from the representation theory of affine quantum algebras leading to a number of operator functional equations which allow one to solve the mentioned models, i.e. compute the spectra of the six-vertex transfer matrix and the XXZ quantum Hamiltonian.

### Abstract

This talk describes symmetries of all integrable difference equations that belong to the famous Adler-Bobenko-Suris (ABS) classification. For each equation, continuous symmetries are determined by a functional equation, which can be solved by reducing it to a system of partial differential equations. In this way we obtain all point symmetries and higher-order symmetries on up to five points. These include mastersymmetries, which allow one to construct infinite hierarchies of local symmetries. Each integrable equation in the ABS classification has at least one such hierarchy, provided that parameter changes are allowed. The consequences of this result are discussed.

### Abstract

Many integrable systems arise naturally in the classical differential geometry of surfaces in ordinary Euclidean space: for example, CMC surfaces or K-surfaces are described by integrable systems.

There is a similar story for submanifold geometry in the celestial n-spheres which arise as projective light cones in Minkowski (n+1,1)-space. The integrable theories of conformally-flat submanifolds, isothermic surfaces and Willmore surfaces fit naturally into this framework. In this talk I plan to outline a novel and systematic approach to this geometry developed with Fran Burstall.

### Abstract

I shall present a new and surprisingly simple Poisson piece of the Schrödinger / KdV correspondence. It gives rise to a Poisson-Lie group of symmetries acting naturally on the space of wave functions.

### Abstract

We study systems evolutionary odd order equations $$\label{sys0} \begin{array}{l} u_t=\lambda_1 u_n+ F_1(u_{n-1},v_{n-1},\ldots,u,v)\\ v_t=\lambda_2 v_n+ F_2(u_{n-1},v_{n-1},\ldots,u,v) \end{array}\quad n=2s+1, s\in \bbbn$$ which possess infinite hierarchies of local symmetries. We formulate a number of conditions on the right hand side of the system which are necessary for the existence of an infinite sequence of local symmetries.

We show that the requirement of existence of an infinite hierarchy of commuting flows imposes tight constraints on possible dispersion laws of the system. The dispersion law of an integrable system carries important information on the structure of hidden symmetries and conservation laws. To determine possible dispersion laws for integrable systems is an important and far non-trivial problem.