341 research outputs found
Exact solution and thermodynamics of a spin chain with long-range elliptic interactions
We solve in closed form the simplest (su(1|1)) supersymmetric version of
Inozemtsev's elliptic spin chain, as well as its infinite (hyperbolic)
counterpart. The solution relies on the equivalence of these models to a system
of free spinless fermions, and on the exact computation of the Fourier
transform of the resulting elliptic hopping amplitude. We also compute the
thermodynamic functions of the finite (elliptic) chain and their low
temperature limit, and show that the energy levels become normally distributed
in the thermodynamic limit. Our results indicate that at low temperatures the
su(1|1) elliptic chain behaves as a critical XX model, and deviates in an
essential way from the Haldane-Shastry chain.Comment: Typeset with LaTeX, 7 figures, 30 pages; considerably enlarged
version of previous submissio
A new perspective on the integrability of Inozemtsev's elliptic spin chain
The 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 well-established
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.Comment: Pdflatex, 35 pages, 6 figures. Minor changes, to appear in Annals of
Physic
Quasi-Exactly Solvable Potentials on the Line and Orthogonal Polynomials
In this paper we show that a quasi-exactly solvable (normalizable or
periodic) one-dimensional Hamiltonian satisfying very mild conditions defines a
family of weakly orthogonal polynomials which obey a three-term recursion
relation. In particular, we prove that (normalizable) exactly-solvable
one-dimensional systems are characterized by the fact that their associated
polynomials satisfy a two-term recursion relation. We study the properties of
the family of weakly orthogonal polynomials defined by an arbitrary
one-dimensional quasi-exactly solvable Hamiltonian, showing in particular that
its associated Stieltjes measure is supported on a finite set. From this we
deduce that the corresponding moment problem is determined, and that the -th
moment grows like the -th power of a constant as tends to infinity. We
also show that the moments satisfy a constant coefficient linear difference
equation, and that this property actually characterizes weakly orthogonal
polynomial systems.Comment: 22 pages, plain TeX. Please typeset only the file orth.te
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