arXiv subjects

Denis Bernard

Publications and source records attributed to Denis Bernard.

137 records · Page 8Linked to original sources

arXiv1995

The Large N Random Phase sine-Gordon Model

At large distances and in the low temperature phase, the quenched correlation functions in the 2d random phase sine-Gordon model have been argued to be of the form~: $ \bar {\vev{~[φ(x)-φ(0)]^2~}}_* = A (\log|x|) + B \ep^2 (\log|x|)^2 $, with $\ep=(T-T_c)$. However, renormalization group computations predict $B\not=0$ while variational approaches (which are supposed to be exact for models with a large number of components) give $B=0$. We introduce a large $N$ version of the random phase sine-Gordon model. Using non-Abelian bosonization and renormalization group techniques, we show that the correlation functions of our models have the above form but with a coefficient $B$ suppressed by a factor $1/N^3$ compared to $A$.

hep-th↗

arXiv1995

On The Random Vector Potential Model In Two Dimensions

The random vector potential model describes massless fermions coupled to a quenched random gauge field. We study its abelian and non-abelian versions. The abelian version can be completely solved using bosonization. We analyse the non-abelian model using its supersymmetric formulation and show, by a perturbative renormalisation group computation, that it is asymptotically free at large distances. We also show that all the quenched chiral current correlation functions can be computed exactly, without using the replica trick or the supersymmetric formulation, but using an exact expression for the effective action for any sample of the random gauge field. These chiral correlation functions are purely algebraic.

hep-th↗

arXiv1994

The Yangian Deformation of the W-Algebras and and the Calogero-Sutherland model

The Yangian symmetry Y(su($n$)) of the Calogero-Sutherland-Moser spin model is reconsidered. The Yangian generators are constructed from two non-commuting su($n$)-loop algebras. The latters generate an infinite dimensional symmetry algebra which is a deformation of the $W_\infty$-algebra. We show that this deformed $W_\infty$-algebra contains an infinite number of Yangian subalgebras with different deformation parameters.

hep-th↗

arXiv1994

Some Simple (Integrable) Models of Fractional Statistics

In the first part, we introduce the notion of fractional statistics in the sense of Haldane. We illustrate it on simple models related to anyon physics and to integrable models solvable by the Bethe ansatz. In the second part, we describe the properties of the long-range interacting spin chains. We describe its infinite dimensional symmetry, and we explain how the fractional statistics of its elementary excitations is an echo of this symmetry. In the third part, we review recent results on the Yangian representation theory which emerged from the study of the integrable long-range interacting models.

hep-th↗

arXiv1994

A Note on Statistical Interactions and the Thermodynamic Bethe Ansatz

We show that the thermodynamic Bethe ansatz equations for one-dimensional integrable many-body systems can be reinterpreted in such a way that they only code the statistical interactions, in the sense of Haldane, between particles of identical or different momenta. Thus, the thermodynamic properties of these systems can be characterized by the generalized ideal gases recently proposed by one of us. For example, the Yang-Yang $δ$-function gas is a gas with specific statistical interactions between particles of different momenta, while the Calogero-Sutherland system provides a model for an ideal gas of particles with a fractional statistics.

cond-mat↗

arXiv1993

The Sine-Gordon Solitons as a N-Body Problem

We consider the N-soliton solutions in the sine-Gordon model as a N-body problem. This leads to a relativistic generalization of the Calogero model first introduced by Ruijsenaars. We show that the fundamental Poisson bracket of the Lax matrix is quadratic, and the $r$-matrix is a dynamical one. This is in contrast to the Calogero model where the fundamental Poisson bracket of the Lax matrix is linear.

hep-th↗

arXiv1992

An Introduction to Yangian Symmetries

We review some aspects of the quantum Yangians as symmetry algebras of two-dimensional quantum field theories. The plan of these notes is the following: 1 - The classical Heisenberg model: Non-Abelian symmetries; The generators of the symmetries and the semi-classical Yangians; An alternative presentation of the semi-classical Yangians; Digression on Poisson-Lie groups. 2 - The quantum Heisenberg chain: Non-Abelian symmetries and the quantum Yangians; The transfer matrix and an alternative presentation of the Yangians; Digression on the double Yangians. Talk given at the "Integrable Quantum Field Theories" conference held at Come, Italy , September 13-19, 1992.

hep-th↗

arXiv1992

From Form Factors to Correlation Functions: The Ising Model

Using exact expressions for the Ising form factors, we give a new very simple proof that the spin-spin and disorder-disorder correlation functions are governed by the Painlevé III non linear differential equation. We also show that the generating function of the correlation functions of the descendents of the spin and disorder operators is a $N$-soliton, $N\to\infty$, $τ$-function of the sinh-Gordon hierarchy. We discuss a relation of our approach to isomonodromy deformation problems, as well as further possible generalizations.

hep-th↗

arXiv1992

Affine Solitons: A Relation Between Tau Functions, Dressing and Bäcklund Transformations

We reconsider the construction of solitons by dressing transformations in the sine-Gordon model. We show that the $N$-soliton solutions are in the orbit of the vacuum, and we identify the elements in the dressing group which allow us to built the $N$-soliton solutions from the vacuum solution. The dressed $τ$-functions can be computed in two different ways~: either using adjoint actions in the affine Lie algebra $\hat {sl_2}$, and this gives the relation with the Bäcklund transformations, or using the level one representations of the affine Lie algebra $\widehat{sl_2}$, and this directly gives the formulae for the $τ$-functions in terms of vertex operators.

hep-th↗

arXiv1992

On Symmetries of Some Massless 2D Field Theories

We describe few aspects of the quantum symmetries of some massless two-dimensional field theories. We discuss their relations with recent proposals for the factorized scattering theories of the massless $PCM_1$ and $O(3)_{θ=π}$ sigma models. We use these symmetries to propose massless factorized S-matrices for the $su(2)$ sigma models with topological terms at any level, alias the $PCM_k$ models, and for the $su(2)$-coset massless flows.

hep-th↗

arXiv1991

Quantum Symmetries in 2D Massive Field Theories

We review various aspects of (infinite) quantum group symmetries in 2D massive quantum field theories. We discuss how these symmetries can be used to exactly solve the integrable models. A possible way for generalizing to three dimensions is shortly described.

hep-th↗