On the solutions of 3-particle spin elliptic Calogero–Moser systems

Barba, J.C.; Inozemtsev, V. I.

doi:10.1016/j.physleta.2008.07.067

Cited by 2 publications

(1 citation statement)

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“…On the other hand, in spite of extensive work by many authors (see, e.g., [18][19][20][21][22]), there are no general explicit formulas for the eigenvalues and eigenfunctions of this model. Even less is known about its spin version, for which only the lowest integrals of motion [23] and a few particular solutions (in the three-particle case) have been found [24]. Notwithstanding this, it is natural to consider the spin chain obtained by applying Polychronakos's freezing trick to the elliptic CS model.…”

Section: Introductionmentioning

confidence: 99%

A new perspective on the integrability of Inozemtsev’s elliptic spin chain

Finkel

González‐López

2014

Annals of Physics

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h i g h l i g h t s• Construction of Inozemtsev's elliptic spin chain using Polychronakos's freezing trick.• Numerical evidence of the Gaussian character of the level density.• Exact computation and asymptotics of the mean and standard deviation of the spectrum. • Evidence of the chain's integrability from key statistical properties of its spectrum. • Exact evaluation of finite sums of powers of Weierstrass's elliptic function. a b s t r a c tThe aim of this paper is studying from an alternative point of view the integrability of the spin chain with long-range elliptic interactions introduced by Inozemtsev. Our analysis relies on some wellestablished conjectures characterizing the chaotic vs. integrable behavior of a quantum system, formulated in terms of statistical properties of its spectrum. More precisely, we study the distribution of consecutive levels of the (unfolded) spectrum, the power spectrum of the spectral fluctuations, the average degeneracy, and the equivalence to a classical vertex model. Our results are consistent with the general consensus that this model is integrable, and that it is closer in this respect to the Heisenberg chain than to its trigonometric limit (the Haldane-Shastry chain). On the other hand, we present some numerical and analytical evidence showing that the level density of Inozemtsev's chain is asymptotically Gaussian as the number of spins tends to infinity, as is the case with the Haldane-Shastry chain. We are also able to compute analytically the mean and the standard deviation of the spectrum, showing that their asymptotic behavior coincides with that of the Haldane-Shastry chain.

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