On ground-state wavefunctions for Sutherland-Calogero systems in an external field

Inozemtsev, V. I.; Meshcheryakov, D. V.

doi:10.1016/0375-9601(84)90898-3

Cited by 7 publications

(14 citation statements)

References 7 publications

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“…In principle, according to [12], (2.1) should not correspond to an exactly solvable model satisfying (2.3) (like Calogero model for example). A careful treatment of (2.3) with a generic wavefunction 4) shows that there are more exactly solvable models than has been recognized hitherto.…”

Section: D Exactly Solvable Modelmentioning

confidence: 99%

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Chern–simons Theory, Exactly Solvable Models and Free Fermions at Finite Temperature

Tierz

2009

Mod. Phys. Lett. A

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We show that matrix models in Chern-Simons theory admit an interpretation as 1D exactly solvable models, paralleling the relationship between the Gaussian model and the Calogero model. We compute the corresponding Hamiltonians, ground-state wavefunctions and ground-state energies and point out that the models can be interpreted as quasi-1D Coulomb plasmas. We also study the relationship between Chern-Simons theory on S 3 and a system of N one-dimensional fermions at finite temperature with harmonic confinement. In particular we show that the Chern-Simons partition function can be described by the density matrix of the free fermions in a very particular, crystalline, configuration. For this, we both use the Brownian motion and the matrix model description of Chern-Simons theory and find several common features with c=1 theory at finite temperature. Finally, using the exactly solvable model result, we show that the finite temperature effect can be described with a specific two-body interaction term in the Hamiltonian, with 1D Coulombic behavior at large separations.

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Section: D Exactly Solvable Modelmentioning

confidence: 99%

“…In the finite temperature case, it was found in [32] (see [33] for a modern and direct treatment) that (3.10) is replaced by: 12) where δk 0 is the momentum width where the Fermi surface is no longer infinitely sharp, but has a smooth variation with an energy width of k B T . More precisely, we have:…”

Section: D Exactly Solvable Modelmentioning

confidence: 99%

Chern–simons Theory, Exactly Solvable Models and Free Fermions at Finite Temperature

Tierz

2009

Mod. Phys. Lett. A

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“…However, for integrable particle systems there might be a much deeper connection between the classical and quantum dynamics due to the common symmetry of the Hamiltonians having the same group-theoretical grounds. The aim of the present work is demonstration of the existence of such a connection for integrable cases of the motion of the CalogeroÄMoser particle systems [1,2] in the externalˇeld discovered by mine [3,4]. These systems of arbitrary number of particles N with the mutual two-particle interaction given by the potential V (q) are supposed to move in the externalˇeld with the potential W (q) and are deˇned by the quantum Hamiltonian…”

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confidence: 92%

“…The case of A = B = C = 0 corresponds to the usual CalogeroÄMoser hyperbolic systems [1,2]. It was shown in the paper [4] that, under the condition…”

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confidence: 99%

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Factorization of wave functions of the quantum integrable particle systems

Inozemtsev

2011

Phys. Part. Nuclei Lett.

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The relation between the characteristics of the equilibrium conˇgurations of the classical CalogeroÄ Moser integrable systems and properties of the ground state of their quantum analogs is found. It is shown that, under the condition of factorization of the wave function of these systems, the coordinates of classical particles at equilibrium are zeroes of the polynomial solutions of the second-order linear differential equation. It turns out that, under these conditions, the dependence of classical and quantum minimal energies on the parameters of the interaction potential is the same.

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Jastrow-like ground states for quantum many-body potentials with near-neighbors interactions

et al. 2018

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We completely solve the problem of classifying all one-dimensional quantum potentials with nearest-and next-to-nearest-neighbors interactions whose ground state is Jastrowlike, i.e., of Jastrow type but depending only on differences of consecutive particles. In particular, we show that these models must necessarily contain a three-body interaction term, as was the case with all previously known examples. We discuss several particular instances of the general solution, including a new hyperbolic potential and a model with elliptic interactions which reduces to the known rational and trigonometric ones in appropriate limits.Here x ≡ (x 1 , . . . , x N ) andfor the (harmonic) Calogero model, whilefor the Sutherland model (with ω > 0 and a > −1/2). As first noted by Sutherland [2, 23], this property makes it possible to compute in closed form certain correlation functions of the latter models by exploiting their connection with random matrix theory. Indeed, for Calogero's model ψ 2 coincides with the joint probability density of the eigenvalues of the Gaussian orthogonal, unitary and symplectic ensembles respectively for 2ω = 2a = 1, 2, 4, while the same relation holds for the ground state of the Sutherland model and Dyson's circular unitary ensembles (with eigenvalues parametrized as e 2ix k ) [24]. Later on, Dyson [25] showed how to construct analogues of the Gaussian ensembles with eigenvalues distributed according to ψ 2 in Eq. (1.1), with essentially arbitrary ρ and χ(x) = |x| β/2 (as usual, β will be assumed to take the values 1, 2, 4 for the orthogonal, unitary and symplectic ensembles, respectively). In view of these results, it is natural to look for the most general quantum Hamiltonian of Calogero-Sutherland (CS) type (i.e, with one-and two-body long-range interactions) whose ground state is of the form (1.1). A restricted version of this problem (with ρ = 1) was already formulated by Sutherland himself [2], who later found a solution thereof with an elliptic two-body interaction potential [26,27]. Shortly afterwards, Calogero [28] showed that this is in fact the most general solution of this restricted problem. The general problem (with ρ not necessarily equal to 1) was tackled by Inozemtsev and Meshcheryakov [29], who claimed to have found a complete solution. A decade later, however, Forrester [30] found a model of CS type whose ground state, which exhibits longrange crystalline order in the thermodynamic limit, is of the factorized form (1.1) and yet did not appear in the classification of Ref. [29]. The latter classification was finally completed several years later by Koprucki and Wagner [31], who obtained Forrester's model as a particular case.The probability distribution p β (s) of the (normalized) spacing s between two consecutive eigenvalues of the Gaussian β-ensembles is approximately given by Wigner's surmise p β (s) = A β s β e −c β s 2 , where the positive parameters A β , c β are fixed by normalization and the condition that the mean spacing be equal to 1 (see, e.g., Refs. [32,33]). By contra...

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On ground-state wavefunctions for Sutherland-Calogero systems in an external field

Cited by 7 publications

References 7 publications

Chern–simons Theory, Exactly Solvable Models and Free Fermions at Finite Temperature

Chern–simons Theory, Exactly Solvable Models and Free Fermions at Finite Temperature

Factorization of wave functions of the quantum integrable particle systems

Jastrow-like ground states for quantum many-body potentials with near-neighbors interactions

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