Quantum incompressibility and Razumov Stroganov type conjectures

We establish a correspondence between polynomial representations of the Temperley and Lieb algebra and certain deformations of the Quantum Hall Effect wave functions. When the deformation parameter is a third root of unity, the representation degenerates and the wave functions coincide with the domain wall boundary condition partition function appearing in the conjecture of A.V. Razumov and Y.G. Stroganov. In particular, this gives a proof of the identification of the sum of the entries of a O(n) transfer matrix eigenvector and a six vertex-model partition function, alternative to that of P. Di Francesco and P. Zinn-Justin.Comment: latex ihp.tex, 2 files, 1 figure, 28 pages (http://www-spht.cea.fr/articles/T05/087

Incompressible representations of the Birman-Wenzl-Murakami algebra

We construct a representation of the Birman-Wenzl-Murakami algebra acting on a space of polynomials in n variables vanishing when three points coincide. These polynomials are closely related to the Pfaffian state of the Quantum Hall Effect and to the components the transfer matrix eigenvector of a O(n) crossing loop model.Comment: latex bmw.tex, 1 file, 20 pages (http://www-spht.cea.fr/articles/T05/121

Quantum transition in bilayer states

I study the possible phase transitions when two layers at filling factor $\nu_t=1$ are gradually separated. In the bosonic case the system should undergo a pairing transition from a Fermi liquid to an incompressible state. In the Fermionic case, the state evolves from an incompressible $(1,1,1)$ state to a Fermi liquid. I speculate that there is an intermediate phase involving charge two quasiparticles.Comment: Text modification

Dipoles at ν=1\nu =1

We consider the problem of Bosonic particles interacting repulsively in a strong magnetic field at the filling factor $\nu =1.$ We project the system in the Lowest Landau Level and map the dynamics into an interacting Fermion system. We study the resulting Hamiltonian in the Hartree--Fock approximation in the case of a $\delta$ repulsive potential. The physical picture which emerges is in agreement with the proposal of N. Read that the composite Fermions behave as a gas of dipoles. We argue that the consequence of this is that the composite Fermions interact with screened short range interactions. We develop a Landau theory which we also expect to describe the physical $\nu =1/2$ Fermionic state. The Form factor, the effective mass and the conductivity are analised in this model.Comment: flatex_new vincent.tex, 4 files Proceedings on Composite Fermions and Confinement Moriond, France 1999-03-01 1999-03-06 March 1-6, 199

Bethe Ansatz and Q-operator for the open ASEP

In this paper, we look at the asymmetric simple exclusion process with open boundaries with a current-counting deformation. We construct a two-parameter family of transfer matrices which commute with the deformed Markov matrix of the system. We show that these transfer matrices can be factorised into two commuting matrices with one parameter each, which can be identified with Baxter's Q-operator, and that for certain values of the product of those parameters, they decompose into a sum of two commuting matrices, one of which is the Bethe transfer matrix for a given dimension of the auxiliary space. Using this, we find the T-Q equation for the open ASEP, and, through functional Bethe Ansatz techniques, we obtain an exact expression for the dominant eigenvalue of the deformed Markov matrix.Comment: 46 pages. New version: references updated and typos correcte