41 research outputs found
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
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
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
Generalized isotropic Lipkin-Meshkov-Glick models: ground state entanglement and quantum entropies
We introduce a new class of generalized isotropic Lipkin-Meshkov-Glick models
with su spin and long-range non-constant interactions, whose
non-degenerate ground state is a Dicke state of su type. We evaluate in
closed form the reduced density matrix of a block of spins when the whole
system is in its ground state, and study the corresponding von Neumann and
R\'enyi entanglement entropies in the thermodynamic limit. We show that both of
these entropies scale as when tends to infinity, where the
coefficient is equal to in the ground state phase with
vanishing su magnon densities. In particular, our results show that none
of these generalized Lipkin-Meshkov-Glick models are critical, since when
their R\'enyi entropy becomes independent of the parameter
. We have also computed the Tsallis entanglement entropy of the ground state
of these generalized su Lipkin-Meshkov-Glick models, finding that it can
be made extensive by an appropriate choice of its parameter only when
. Finally, in the su case we construct in detail the phase
diagram of the ground state in parameter space, showing that it is determined
in a simple way by the weights of the fundamental representation of su.
This is also true in the su case; for instance, we prove that the region
for which all the magnon densities are non-vanishing is an -simplex in
whose vertices are the weights of the fundamental representation
of su.Comment: Typeset with LaTeX, 32 pages, 3 figures. Final version with
corrections and additional reference
Critical behavior of su(1|1) supersymmetric spin chains with long-range interactions
We introduce a general class of su supersymmetric spin chains with
long-range interactions which includes as particular cases the su
Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We
show that this class of models can be fermionized with the help of the
algebraic properties of the su permutation operator, and take advantage
of this fact to analyze their quantum criticality when a chemical potential
term is present in the Hamiltonian. We first study the low energy excitations
and the low temperature behavior of the free energy, which coincides with that
of a -dimensional conformal field theory (CFT) with central charge
when the chemical potential lies in the critical interval , being the dispersion relation. We also analyze the
von Neumann and R\'enyi ground state entanglement entropies, showing that they
exhibit the logarithmic scaling with the size of the block of spins
characteristic of a one-boson -dimensional CFT. Our results thus show
that the models under study are quantum critical when the chemical potential
belongs to the critical interval, with central charge . From the analysis
of the fermion density at zero temperature, we also conclude that there is a
quantum phase transition at both ends of the critical interval. This is further
confirmed by the behavior of the fermion density at finite temperature, which
is studied analytically (at low temperature), as well as numerically for the
su elliptic chain.Comment: 13 pages, 6 figures, typeset in REVTe