Universal Lax pairs for spin Calogero–Moser models and spin exchange models

Abstract

For any root system ∆ and an irreducible representation R of the reflection (Weyl) group G ∆ generated by ∆, a spin Calogero-Moser model can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member µ of R, to be called a "site", we associate a vector space V µ whose element is called a "spin". Its dynamical variables are the canonical coordinates {q j , p j } of a particle in R r , (r = rank of ∆), and spin exchange operators {P ρ } (ρ ∈ ∆) which exchange the spins at the sites µ and s ρ (µ). Here s ρ is the reflection generated by ρ. For each ∆ and R a spin exchange model can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by "freezing" the canonical variables at the equilibrium point of the corresponding classical CalogeroMoser model. For ∆ = A r and R = vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials.

Global properties of the spectrum of the Haldane-Shastry spin chain

“…By contrast, the spin chains associated with the spin models of BC N type [20][21][22][23][24][25][26] have received comparatively little attention. This is in part due to the fact that, unlike their A N counterparts, the BC N -type spin chains depend nontrivially on free parameters (one in the rational case and two in the trigonometric or hyperbolic cases).…”

Symmetries of Spin Calogero Models

“…It should be noted that one systematic method, based on [27], always yields the same symmetry algebra (here the half-loop algebra for Calogero type models) for all Coxeter groups. However, there seems to be no systematic understanding of how to produce other possibilities such as those of Section 5.1 for B L and Sections 6.2, 6.3 for G 2 .…”

“…Another general result holds: the construction of the spin representation described in Sections 5.2 and 6.1 applies to any finite Coxeter group W as explained in [27]. So fixing µ and denoting {µ 1 , .…”

The properties of the first equation of the Vlasov chain of equations

“…The first equation of the Vlasov chain of equations [3,7] is written down for the probability density function ( ) 1 , f r t and has the following form:…”

“…The aim of physics is not only obtaining the equations that describe the processes but also the study of its properties, interrelation with other equations, and the analysis of the properties of these equations. We can see it by the example of the Painleve analysis, research of the differential equations group properties, and its properties with the help of Lax pair, methods of the inverse scattering theory and many others [1,2]. This paper contains an analysis of the first equation of the Vlasov infinite chain of equations [3].The second equation of the Vlasov chain equations has been widely applied in accelerator physics, plasma physics, thermonuclear fusion problems, solid state physics, and crystals [4][5][6].…”