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arXiv:hep-th/0307126 · Physics Letters B

A multispecies Calogero model

Stjepan Meljanac, Marijan Mileković, Andjelo Samsarov

Published Oct 1, 2003• 28 citations
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Abstract

We study a multispecies one-dimensional Calogero model with two- and three-body interactions. Using an algebraic approach (Fock space analysis), we construct ladder operators and find infinitely many, but not all, exact eigenstates of the model Hamiltonian. Besides the ground state energy, we deduce energies of the excited states. It turns out that the spectrum is linear in quantum numbers and that the higher-energy levels are degenerate. The dynamical symmetry responsible for degeneracy is SU(2). We also find the universal critical point at which the model exhibits singular behaviour. Finally, we make contact with some special cases mentioned in the literature.

DOI
10.1016/j.physletb.2003.08.029

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We study a multispecies one-dimensional Calogero model with two- and three-body interactions. Using an algebraic approach (Fock space analysis), we construct ladder operators and find infinitely many, but not all, exact eigenstates of the model Hamiltonian. Besides the ground state energy, we deduce energies of the excited states. It turns out that the spectrum is linear in quantum numbers and that the higher-energy levels are degenerate. The dynamical symmetry responsible for degeneracy is SU(2). We also find the universal critical point at which the model exhibits singular behaviour. Finally, we make contact with some special cases mentioned in the literature.
A multispecies model of Calogero type in D≥1D\geq 1D≥1 dimensions is constructed. The model includes harmonic, two-body and three-body interactions. Using the underlying conformal SU(1,1) algebra, we find the exact eigenenergies corresponding to a class of the exact global collective states. Analysing corresponding Fock space, we detect the universal critical point at which the model exhibits singular behaviour.
We construct a new many-body Hamiltonian with two- and three-body interactions in two space dimensions and obtain its exact many-body ground state for an arbitrary number of particles. This ground state has a novel pairwise correlation. A class of exact solutions for the excited states is also found. These excited states display an energy spectrum similar to the Calogero-Sutherland model in one dimension. The model reduces to an analog of the well-known trigonometric Sutherland model when projected on to a circular ring.
We develop an algebraic approach for finding the eigenfunctions of a large class of few and many-body Hamiltonians, in one and higher dimensions, having linear spectra. The method presented enables one to exactly map these interacting Hamiltonians to decoupled oscillators, thereby, giving a precise correspondence between the oscillator eigenspace and the wavefunctions of these quantum systems. The symmetries behind the degeneracy structure and the commuting constants of motion responsible for the quantum integrability of some of these models are made transparent. Apart from analysing a number of well-known dynamical systems like planar oscillators with commensurate frequencies, both with or without singular inverse-square terms and generalized Calogero-Sutherland type models, we also point out a host of other examples where the present approach can be profitably employed. We further study Hamiltonians having Laughlin wavefunction as the ground-state and establish their equivalence to free oscillators. This reveals the underlying linear W1+∞W_{1+\infty}W1+∞​ symmetry algebra unambiguously and establishes the Laughlin wavefunction as the highest weight vector of this algebra.
We study a one-dimensional model with F interacting families of Calogero-type particles. The model includes harmonic, two-body and three-body interactions. We emphasize the universal SU (1,1) structure of the model. We show how SU (1,1) generators for the whole system are composed of SU (1,1) generators of arbitrary subsystems. We find the exact eigenenergies corresponding to a class of the exact eigenstates of the F-family model. By imposing the conditions for the absence of the three-body interaction, we find certain relations between the coupling constants. Finally, we establish some relations of equivalence between two systems containing F families of Calogero-type particles.
Motivated by the concept of ideal mutual statistics, we study a multispecies Calogero-Sutherland model in which the interaction parameters and masses satisfy some specific relations. The ground state is exactly solvable if those relations hold, both on a circle and on a line with a simple harmonic potential. In the latter case, the one-particle densities can be obtained using a generalization of the Thomas-Fermi method. We calculate the second virial\ncoefficients in the high temperature expansion for the pressure. We show that the low-energy excitations are the same as those of a Gaussian conformal field theory. Finally, we discuss similar relations between the statistics parameters and charges for a multispecies anyon model in a magnetic field.\n\n
We construct the dynamical symmetry of the quantum Calogero model with particle exchange in a confining Coulomb field. This symmetry is governed by the algebra so(N+1,2)so(N+1,2)so(N+1,2), deformed by exchange (Dunkl) operators, with its invariant sector generated by the Dunkl angular momentum tensor and the modified Laplace-Runge-Lenz vector. The equidistant analogue of the Hamiltonian, with a linear spectrum, is expressed in terms of the conformal subalgebra so(1,2)so(1,2)so(1,2). In addition, the wave functions of the Calogero-Coulomb Hamiltonian are classified into infinite-dimensional lowest-weight so(1,2)so(1,2)so(1,2) multiplets.