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

Abstract

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.

Yangian-invariant spin models and Fibonacci numbers

“…This shows that in these cases the chain (28) is not a Yangian-invariant spin model (see also Ref. [42] for the su(2) case). …”

The non-chiral intermediate Heisenberg ferromagnet equation

“…In the same way as the Inozemtsev spin model (3.7)-(3.8) can be obtained from the elliptic spin CM model (3.1)-(3.2) by freezing [31], we believe that the modified Inozemtsev model (2.11)-(2.12) can be obtained from the modified spin CM model (3.5)-(3.6) (it would be interesting to make this precise, but this goes beyond goes beyond the scope of the current paper). We now show that the ncIHF equation is related to this modified Inozemtsev spin chain in the same way as the HWM equation is related to the Haldane-Shastry spin chain [20,22].…”

“…, N , with J a real coupling constant; we are interested in the ferromagnetic case where J > 0. This model can be obtained from a quantum version of the elliptic spin CM model using Polychronakos's freezing trick [31], and it reduces to the Haldane-Shastry spin chain [32,33] in the limit δ → ∞. Zhou and Stone [20] proposed a classical version of the Haldane-Shastry spin chain, and they derived the HWM equation from this classical spin chain model in a continuum limit; see also [21].…”

A novel class of translationally invariant spin chains with long-range interactions

“…The unique global minimum of U in the open set A is the point whose coordinates x k = kπ/(2N ) ≡ θ k coincide with the sites of the chain (2.1), apart from an irrelevant overall translation [59]. Thus in the limit a → ∞ the particles' dynamic and spin degrees of freedom effectively decouple.…”