15,367 research outputs found
The exactly solvable spin Sutherland model of B_N type and its related spin chain
We compute the spectrum of the su(m) spin Sutherland model of B_N type,
including the exact degeneracy of all energy levels. By studying the large
coupling constant limit of this model and of its scalar counterpart, we
evaluate the partition function of their associated spin chain of
Haldane-Shastry type in closed form. With the help of the formula for the
partition function thus obtained we study the chain's spectrum, showing that it
cannot be obtained as a limiting case of its BC_N counterpart. The structure of
the partition function also suggests that the spectrum of the Haldane-Shastry
spin chain of B_N type is equivalent to that of a suitable vertex model, as is
the case for its A_{N-1} counterpart, and that the density of its eigenvalues
is normally distributed when the number of sites N tends to infinity. We
analyze this last conjecture numerically using again the explicit formula for
the partition function, and check its validity for several values of N and m.Comment: Typeset in LaTeX (24 pages, 4 figures). arXiv admin note: text
overlap with arXiv:0909.296
Worse than a big rip?
We show that a generalised phantom Chaplygin gas can present a future
singularity in a finite future cosmic time. Unlike the big rip singularity,
this singularity happens for a finite scale factor, but like the big rip
singularity, it would also take place at a finite future cosmic time. In
addition, we define a dual of the generalised phantom Chaplygin gas which
satisfies the null energy condition. Then, in a Randall-Sundrum 1 brane-world
scenario, we show that the same kind of singularity at a finite scale factor
arises for a brane filled with a dual of the generalised phantom Chaplygin gas.Comment: 6 pages, 4 figures, RevTeX 4. Discussion expanded and references
added. Version to appear in PL
Quasi-Exactly Solvable N-Body Spin Hamiltonians with Short-Range Interaction Potentials
We review some recent results on quasi-exactly solvable spin models
presenting near-neighbors interactions. These systems can be understood as
cyclic generalizations of the usual Calogero-Sutherland models. A nontrivial
modification of the exchange operator formalism is used to obtain several
infinite families of eigenfunctions of these models in closed form.Comment: This is a contribution to the Proc. of workshop on Geometric Aspects
of Integrable Systems (July 17-19, 2006; Coimbra, Portugal), published in
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
A Haldane-Shastry spin chain of BC_N type in a constant magnetic field
We compute the spectrum of the trigonometric Sutherland spin model of BC_N
type in the presence of a constant magnetic field. Using Polychronakos's
freezing trick, we derive an exact formula for the partition function of its
associated Haldane-Shastry spin chain.Comment: LaTeX, 13 page
Rational quantum integrable systems of D_N type with polarized spin reversal operators
We study the spin Calogero model of D_N type with polarized spin reversal
operators, as well as its associated spin chain of Haldane-Shastry type, both
in the antiferromagnetic and ferromagnetic cases. We compute the spectrum and
the partition function of the former model in closed form, from which we derive
an exact formula for the chain's partition function in terms of products of
partition functions of Polychronakos-Frahm spin chains of type A. Using a
recursion relation for the latter partition functions that we derive in the
paper, we are able to numerically evaluate the partition function, and thus the
spectrum, of the D_N-type spin chain for relatively high values of the number
of spins N. We analyze several global properties of the chain's spectrum, such
as the asymptotic level density, the distribution of consecutive spacings of
the unfolded spectrum, and the average degeneracy. In particular, our results
suggest that this chain is invariant under a suitable Yangian group, and that
its spectrum coincides with that of a Yangian-invariant vertex model with
linear energy function and dispersion relation.Comment: 26 pages, 5 figures, typeset in LaTe
The Berry-Tabor conjecture for spin chains of Haldane-Shastry type
According to a long-standing conjecture of Berry and Tabor, the distribution
of the spacings between consecutive levels of a "generic'' integrable model
should follow Poisson's law. In contrast, the spacings distribution of chaotic
systems typically follows Wigner's law. An important exception to the
Berry-Tabor conjecture is the integrable spin chain with long-range
interactions introduced by Haldane and Shastry in 1988, whose spacings
distribution is neither Poissonian nor of Wigner's type. In this letter we
argue that the cumulative spacings distribution of this chain should follow the
"square root of a logarithm'' law recently proposed by us as a characteristic
feature of all spin chains of Haldane-Shastry type. We also show in detail that
the latter law is valid for the rational counterpart of the Haldane-Shastry
chain introduced by Polychronakos.Comment: LaTeX with revtex4, 6 pages, 6 figure
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