Calogero-Moser Model Research Articles

Generalized model of interacting integrable tops

We introduce a family of classical integrable systems describing dynamics of M interacting glN integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the GLNR-matrix satisfying the associative Yang-Baxter equation. The obtained systems can be considered as extensions of the spin type Calogero-Moser models with (the classical analogues of) anisotropic spin exchange operators given in terms of the R-matrix data. In N = 1 case the spin Calogero-Moser model is reproduced. Explicit expressions for glNM -valued Lax pair with spectral parameter and its classical dynamical r-matrix are obtained. Possible applications are briefly discussed.

Harmonically Confined Particles with Long-Range Repulsive Interactions.

We study an interacting system of N classical particles on a line at thermal equilibrium. The particles are confined by a harmonic trap and repel each other via pairwise interaction potential that behaves as a power law ∝∑[under i≠j][over N]|x_{i}-x_{j}|^{-k} (with k>-2) of their mutual distance. This is a generalization of the well-known cases of the one-component plasma (k=-1), Dyson's log gas (k→0^{+}), and the Calogero-Moser model (k=2). Because of the competition between harmonic confinement and pairwise repulsion, the particles spread over a finite region of space for all k>-2. We compute exactly the average density profile for large N for all k>-2 and show that while it is independent of temperature for sufficiently low temperature, it has a rich and nontrivial dependence on k with distinct behavior for -2<k<1, k>1 and k=1.

Some Connections Between the Classical Calogero–Moser Model and the Log-Gas

In this work we discuss connections between a one-dimensional system of $N$ particles interacting with a repulsive inverse square potential and confined in a harmonic potential (Calogero-Moser model) and the log-gas model which appears in random matrix theory. Both models have the same minimum energy configuration, with the particle positions given by the zeros of the Hermite polynomial. Moreover, the Hessian describing small oscillations around equilibrium are also related for the two models. The Hessian matrix of the Calogero-Moser model is the square of that of the log-gas. We explore this connection further by studying finite temperature equilibrium properties of the two models through Monte-Carlo simulations. In particular, we study the single particle distribution and the marginal distribution of the boundary particle which, for the log-gas, are respectively given by the Wigner semi-circle and the Tracy-Widom distribution. For particles in the bulk, where typical fluctuations are Gaussian, we find that numerical results obtained from small oscillation theory are in very good agreement with the Monte-Carlo simulation results for both the models. For the log-gas, our findings agree with rigorous results from random matrix theory.

Extended supersymmetric multiparticle Euler–Calogero–Moser model

We review the construction of supersymmetric extension of the n-particle Euler–Calogero–Moser model within the Hamiltonian approach [1]. The main feature of the proposed supersymmetrization method is that it is automatically adapted for a model with an arbitrary even number of supersymmetries. It is shown that the number of fermions that must be used in this construction is . We demonstrate that the resulting supersymmetric system is dynamically invariant with respect to the superconformal group and give the explicit realization of its generators in terms of the conserved currents. For the simplest case of the supersymmetric n-particle Euler–Calogero–Moser model we provide its description in superspace by using the corresponding constrained superfields.

Time discretization of the spin Calogero-Moser model and the semi-discrete matrix KP hierarchy

We introduce the discrete time version of the spin Calogero-Moser system. The equations of motion follow from the dynamics of poles of rational solutions to the matrix Kadomtsev-Petviashvili hierarchy with discrete time. The dynamics of poles is derived using the auxiliary linear problem for the discrete flow.

Supersymmetric many-body Euler–Calogero–Moser model

We explicitly construct a supersymmetric so(n) spin-Calogero model with an arbitrary even number N of supersymmetries. It features 12Nn(n+1) rather than Nn fermionic coordinates and a very simple structure of the supercharges and the Hamiltonian. The latter, together with additional conserved currents, form an osp(N|2) superalgebra. We provide a superspace description for the simplest case, namely N=2 supersymmetry. The reduction to an N-extended supersymmetric goldfish model is also discussed.

Calogero–Moser Model and R-Matrix Identities

We discuss properties of R-matrix-valued Lax pairs for the elliptic Calogero-Moser model. In particular, we show that the family of Hamiltonians arising from this Lax representation contains only known Hamiltonians and no others. We review the relation of R-matrix-valued Lax pairs to Hitchin systems on bundles with nontrivial characteristic classes over elliptic curves and also to quantum long-range spin chains. We prove a general higher-order identity for solutions of the associative Yang–Baxter equation.

Quantum SU(2|1) supersymmetric Calogero-Moser spinning systems

SU(2|1) supersymmetric multi-particle quantum mechanics with additional semi-dynamical spin degrees of freedom is considered. In particular, we provide an mathcal{N}=4 supersymmetrization of the quantum U(2) spin Calogero-Moser model, with an intrinsic mass parameter coming from the centrally-extended superalgebra widehat{su}left(2Big|1right) . The full system admits an SU(2|1) covariant separation into the center-of-mass sector and the quotient. We derive explicit expressions for the classical and quantum SU(2|1) generators in both sectors as well as for the total system, and we determine the relevant energy spectra, degeneracies, and the sets of physical states.

R-matrix-valued Lax pairs and long-range spin chains

In this paper we discuss R-matrix-valued Lax pairs for slN Calogero–Moser model and their relation to integrable quantum long-range spin chains of the Haldane–Shastry–Inozemtsev type. First, we construct the R-matrix-valued Lax pairs for the third flow of the classical Calogero–Moser model. Then we notice that the scalar parts (in the auxiliary space) of the M-matrices corresponding to the second and third flows have form of special spin exchange operators. The freezing trick restricts them to quantum Hamiltonians of long-range spin chains. We show that for a special choice of the R-matrix these Hamiltonians reproduce those for the Inozemtsev chain. In the general case related to the Baxter's elliptic R-matrix we obtain a natural anisotropic extension of the Inozemtsev chain. Commutativity of the Hamiltonians is verified numerically. Trigonometric limits lead to the Haldane–Shastry chains and their anisotropic generalizations.

Realization of the Infinite-Dimensional 3-Algebras in the Calogero—Moser Model**Supported by the National Natural Science Foundation of China under Grant Nos 11375119 and 11031005, and the Beijing Municipal Commission of Education under Grant No KZ201210028032.

We investigate realization of the infinite-dimensional 3-algebras in the classical Calogero—Moser model. In terms of the Lax matrix of the Calogero—Moser model and the Nambu 3-brackets in which the variables are the coordinates qi, and canonically conjugate momenta pi and the coupling parameter β are an extra auxiliary phase-space parameter, we present the realization of the Virasoro—Witt, w∞ and SDiff(T2) 3-algebras, respectively.

The quantum angular Calogero-Moser model

The rational Calogero-Moser model of n one-dimensional quantum particles with inverse-square pairwise interactions (in a confining harmonic potential) is reduced along the radial coordinate of R^n to the `angular Calogero-Moser model' on the sphere S^{n-1}. We discuss the energy spectrum of this quantum system, its degeneracies and the eigenstates. The spectral flow with the coupling parameter yields isospectrality for integer increments. Decoupling the center of mass before effecting the spherical reduction produces a `relative angular Calogero-Moser model', which is analyzed in parallel. We generalize our considerations to the Calogero-Moser models associated with Coxeter groups. Finally, we attach spin degrees of freedom to our particles and extend the results to the spin-Calogero system.

Rational Calogero-Moser Model: Explicit Form and r-Matrix of the Second Poisson Structure

We compute the full expression of the second Poisson bracket structure for N=2 and N=3 site rational classical Calogero-Moser model. We propose an r-matrix formulation for N=2. It is identified with the classical limit of the second dynamical boundary algebra previously built by the authors.

A 5d/3d duality from relativistic integrable system

We propose and prove a new exact duality between the F-terms of supersymmetric gauge theories in five and three dimensions with adjoint matter fields. The theories are compactified on a circle and are subject to the Omega deformation. In the limit proposed by Nekrasov and Shatashvili, the supersymmetric vacua become isolated and are identified with the eigenstates of a quantum integrable system. The effective twisted superpotentials are the Yang-Yang functional of the relativistic elliptic Calogero-Moser model. We show that they match on-shell by deriving the Bethe ansatz equation from the saddle point of the five-dimensional partition function. We also show that the Chern-Simons terms match and extend our proposal to the elliptic quiver generalizations.

Additional Constants of Motion for a Discretization of the Calogero–Moser Model

The maximal super-integrability of a discretization of the Calogero–Moser model introduced by Nijhoff and Pang is presented. An explicit formula for the additional constants of motion is given.

A CURVATURE DEPENDENT BOUND FOR ENTANGLEMENT CHANGE IN CLASSICALLY CHAOTIC SYSTEMS

Using the Calogero–Moser model and the Nakamura–Lakshmanan equations of motion for eigenvalues and eigenfunctions associated with a multi-partite quantum system, we prove an inequality between the mean bi-partite entanglement rate of change under the variation of a critical parameter and the level-curvature. This provides an upper bound for the rate of production or destruction of entanglement induced dynamically. We then investigate the dependence of the upper bound on the degree of chaos of the system, which in turn, through the inequality, gives a measure of the stability of the entangled state. Our analytical results are supported by extensive numerical calculations.

DIMENSIONAL REDUCTION ON A SPHERE

The question of the dimensional reduction of two-dimensional (2d) quantum models on a sphere to one-dimensional (1d) models on a circle is addressed. A possible application is to look at a relation between the 2d anyon model and the 1d Calogero–Sutherland model, which would allow for a better understanding of the connection between 2d anyon exchange statistics and Haldane exclusion statistics. The latter is realized microscopically in the 2d LLL anyon model and in the 1d Calogero model. In a harmonic well of strength ω or on a circle of radius R — both parameters ω and R have to be viewed as long distance regulators — the Calogero spectrum is discrete. It is well known that by confining the anyon model in a 2d harmonic well and projecting it on a particular basis of the harmonic well eigenstates, one obtains the Calogero–Moser model. It is then natural to consider the anyon model on a sphere of radius R and look for a possible dimensional reduction to the Calogero–Sutherland model on a circle of radius R. First, the free one-body case is considered, where a mapping from the 2d sphere to the 1d chiral circle is established by projection on a special class of spherical harmonics. Second, the N-body interacting anyon model is considered: it happens that the standard anyon model on the sphere is not adequate for dimensional reduction. One is thus led to define a new spherical anyon-like model deduced from the Aharonov–Bohm problem on the sphere where each flux line pierces the sphere at one point and exits it at its antipode.

Integrable Model of Interacting Elliptic Tops

We suggest a method for constructing a system of interacting elliptic tops. It is integrable and symplectomorphic to the Calogero-Moser model by construction.

Invariant Classification of Orthogonally Separable Hamiltonian Systems in Euclidean Space

The problem of the invariant classification of the orthogonal coordinate webs defined in Euclidean space is solved within the framework of Felix Klein's Erlangen Program. The results are applied to the problem of integrability of the Calogero-Moser model.

Universality of Calogero-Moser Model

In this review we explain interrelations between the Elliptic Calogero-Moser model, the integrable Elliptic Euler-Arnold top, monodromy preserving equations and the Knizhnik-Zamolodchikov-Bernard equation on a torus.

The Perturbation of the Quantum¶Calogero-Moser-Sutherland System¶and Related Results

The Hamiltonian of the trigonometric Calogero-Sutherland model coincides with some limit of the Hamiltonian of the elliptic Calogero-Moser model. In other words the elliptic Hamiltonian is a perturbed operator of the trigonometric one. In this article we show the essential self-adjointness of the Hamiltonian of the elliptic Calogero-Moser model and the regularity (convergence) of the perturbation for the arbitrary root system. We also show the holomorphy of the joint eigenfunctions of the commuting Hamiltonians w.r.t the variables (x_1, >..., x_N) for the A_{N-1}-case. As a result, the algebraic calculation of the perturbation is justified.