Ruijsenaars-Schneider Model Research Articles

R-Matrix Quantization of the Elliptic Ruijsenaars-Schneider Model

It is shown that the classical L-operator algebra of the elliptic Ruijsenaars-Schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions. It is governed by two dynamical r and ?r-matrices satisfying a closed system of equations. The corresponding quantum R and ?R-matrices are found as solutions to quantum analogs of these equations. We present the quantum L-operator algebra and show that the system of equations on R and ?R arises as the compatibility condition for this algebra. It turns out that the R-matrix is twist-equivalent to the Felder elliptic RF-matrix with ?R playing the role of the twist. The simplest representation of the quantum L-operator algebra corresponding to the elliptic Ruijsenaars-Schneider model is obtained. The connection of the quantum L-operator algebra to the fundamental relation RLL=LLR with Belavin's elliptic R matrix is established. As a byproduct of our construction, we find a new N-parameter elliptic solution to the classical Yang-Baxter equation.

Quantum Dynamical R-Matrices and Quantum Frobenius Group

We propose an algebraic scheme for quantizing the rational Ruijsenaars-Schneider model in the R-matrix formalism. We introduce a special parametrization of the cotangent bundle over . In new variables the standard symplectic structure is described by a classical (Frobenius) r-matrix and by a new dynamical -matrix. Quantizing both of them we find the quantum L-operator algebra and construct its particular representation corresponding to the rational Ruijsenaars-Schneider system. Using the dual parametrization of the cotangent bundle we also derive the algebra for the L-operator of the hyperbolic Calogero-Moser system.

Elliptic Ruijsenaars–Schneider model from the cotangent bundle over the two-dimensional current group

It is shown that the elliptic Ruijsenaars–Schneider model can be obtained from the cotangent bundle over the two-dimensional current group by means of the Hamiltonian reduction procedure.

Integrable discretizations of the spin Ruijsenaars–Schneider models

Integrable discretizations are introduced for the rational and hyperbolic spin Ruijsenaars–Schneider models. These discrete dynamical systems are demonstrated to belong to the same integrable hierarchies as their continuous-time counterparts. Explicit solutions are obtained for arbitrary flows of the hierarchies, including the discrete time ones.

Elliptic Ruijsenaars - Schneider model via the Poisson reduction of the affine Heisenberg double

It is shown that the elliptic Ruijsenaars-Schneider model can be obtained from the affine Heisenberg Double by means of the Poisson reduction procedure. The dynamical $r$-matrix naturally appears in the construction.

R-Matrix quantization of the elliptic Ruijsenaars-Schneider model

It is shown that the classical L-operator algebra of the elliptic Ruijsenaars-Schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions. It is governed by two dynamical r and $\bar{r}$-matrices satisfying a closed system of equations. The corresponding quantum R and $\overline{R}$-matrices are found as solutions to quantum analogs of these equations. We present the quantum L-operator algebra and show that the system of equations on R and $\overline{R}$ arises as the compatibility condition for this algebra. It turns out that the R-matrix is twist-equivalent to the Felder elliptic R^F-matrix with $\overline{R}$ playing the role of the twist. The simplest representation of the quantum L-operator algebra corresponding to the elliptic Ruijsenaars-Schneider model is obtained. The connection of the quantum L-operator algebra to the fundamental relation RLL=LLR with Belavin's elliptic R matrix is established. As a byproduct of our construction, we find a new N-parameter elliptic solution to the classical Yang-Baxter equation.

The Ruijsenaars-Schneider Model

We seek to clarify some of the physical aspects of the Ruijsenaars-Schneider models. This important class of models was presented as a relativistic generalisation of the Calogero-Moser models but, as we shall argue, this description is misleading. It is far better to simply view the models as a one-parameter generalisation of Calogero-Moser models. By viewing the models as describing certain eigenvalue motions we can appreciate the generic nature of the models.

Why is the Ruijsenaars-Schneider hierarchy governed by the same R-operator as the Calogero-Moser one?

We demonstrate that in a certain gauge the Lax matrix of the Ruijsenaars-Schneider (RS) model has a quadratic r-matrix structure independent of the relativistic parameter of the model. In particular, the dynamical r-matrix object governing the hierarchy of Lax equations connected with the RS model, turns out to be identical with the r-matrix of the Calogero-Moser (CM) model. In the degenerate cases (hyperbolic and rational) this phenomenon is partly explained by a geometric derivation of Lax equations for arbitrary flows of both the RS and the CM hierarchies.

Dynamical correlation functions and finite-size scaling in the Ruij senaars-Schneider model

The trigonometric Ruijsenaars-Schneider model is diagonalized by means of the Macdonald symmetric functions. We evaluate the dynamical density-density correlation function and the one-particle retarded Green function as well as their thermodynamic limit. Based on these results and finite-size scaling analysis, we show that the low-energy behavior of the model is described by the C = 1 Gaussian conformal field theory under a new fractional selection rule for the quantum numbers labefing the critical exponents.

Dynamicalr-matrix for the elliptic Ruijsenaars - Schneider system

The classical r-matrix structure for the generic elliptic Ruijsenaars-Schneider model is presented. It makes the integrability of this model as well as of its discrete-time version that was constructed in a recent paper manifest.

Affine Toda solitons and systems of Calogero-Moser type

The solitons of affine Toda field theory are related to the spin-generalised Ruijsenaars-Schneider (or relativistic Calogero-Moser) models. This provides the sought after extension of the correspondence between the sine-Gordon solitons and the Ruijsenaars-Schneider model.