1,730 research outputs found
On the Real Spectra of Calogero Model with Complex Coupling
We study the eigenvalue problem of the rational Calogero model with the
coupling of the inverse-square interaction as a complex number. We show that
although this model is manifestly non-invariant under the combined parity and
time-reversal symmetry , the eigenstates corresponding to the zero
value of the generalized angular momentum have real energies.Comment: revtex4 8 pages, 2 figure
Realisations of quantum group and its coloured extension through a novel Hopf algebra with five generators
A novel Hopf algebra , depending on two deformation
parameters and five generators, has been constructed. This
Hopf algebra might be considered as some quantisation of classical group, which contains the standard quantum group
(with ) as a Hopf subalgebra. However, we interestingly observe
that the two parameter deformed quantum group can also be
realised through the generators of this algebra, provided
the sets of deformation parameters and are related to each other
in a particular fashion. Subsequently we construct the invariant noncommutative
planes associated with algebra and show how the two well
known Manin planes corresponding to quantum group can easily be
reproduced through such construction. Finally we consider the `coloured'
extension of quantum group as well as corresponding Manin planes
and explore their intimate connection with the `coloured' extension of Hopf structure.Comment: 24 page
Low energy properties of the SU(m|n) supersymmetric Haldane-Shastry spin chain
The ground state and low energy excitations of the SU(m|n) supersymmetric
Haldane-Shastry spin chain are analyzed. In the thermodynamic limit, it is
found that the ground state degeneracy is finite only for the SU(m|0) and
SU(m|1) spin chains, while the dispersion relation for the low energy and low
momentum excitations is linear for all values of m and n. We show that the low
energy excitations of the SU(m|1) spin chain are described by a conformal field
theory of m non-interacting Dirac fermions which have only positive energies;
the central charge of this theory is m/2. Finally, for n \ge 1, the partition
functions of the SU(m|n) Haldane-Shastry spin chain and the SU(m|n)
Polychronakos spin chain are shown to be related in a simple way in the
thermodynamic limit at low temperatures.Comment: 40 pages including 2 figures; added some references; this version
will appear in Nuclear Physics
Fractional statistics in some exactly solvable Calogero-like models with PT invariant interactions
Here we review a method for constructing exact eigenvalues and eigenfunctions
of a many-particle quantum system, which is obtained by adding some
nonhermitian but PT invariant (i.e., combined parity and time reversal
invariant) interaction to the Calogero model. It is shown that such extended
Calogero model leads to a real spectrum obeying generalised exclusion
statistics. It is also found that the corresponding exchange statistics
parameter differs from the exclusion statistics parameter and exhibits a
`reflection symmetry' provided the strength of the PT invariant interaction
exceeds a critical value.Comment: 8 pages, Latex, Talk given at Joint APCTP-Nankai Symposium, Tianjin
(China), Oct. 200
Exact partition function of SU(m|n) supersymmetric Haldane-Shastry spin chain
By taking the freezing limit of a spin Calogero-Sutherland model containing
`anyon like' representation of the permutation algebra, we derive the exact
partition function of SU(m|n) supersymmetric Haldane-Shastry (HS) spin chain.
This partition function allows us to study global properties of the spectrum
like level density distribution and nearest neighbour spacing distribution. It
is found that, for supersymmetric HS spin chains with large number of lattice
sites, continuous part of the energy level density obeys Gaussian distribution
with a high degree of accuracy. The mean value and standard deviation of such
Gaussian distribution can be calculated exactly. We also conjecture that the
partition function of supersymmetric HS spin chain satisfies a duality relation
under the exchange of bosonic and fermionic spin degrees of freedom.Comment: Latex, 32 pages, 4 figures, minor typos corrected, to be published in
Nucl. Phys.
Thermodynamics of spin chains of Haldane-Shastry type and one-dimensional vertex models
We study the thermodynamic properties of spin chains of Haldane-Shastry type
associated with the A_{N-1} root system in the presence of a uniform external
magnetic field. To this end, we exactly compute the partition function of these
models for an arbitrary finite number of spins. We then show that these chains
are equivalent to a suitable inhomogeneous classical Ising model in a spatially
dependent magnetic field, generalizing the results of Basu-Mallick et al. for
thezero magnetic field case. Using the standard transfer matrix approach, we
are able to compute in closed form the free energy per site in the
thermodynamic limit. We perform a detailed analysis of the chains'
thermodynamics in a unified way, with special emphasis on the zero field and
zero temperature limits. Finally, we provide a novel interpretation of the
thermodynamic quantities of spin chains of Haldane-Shastry type as weighted
averages of the analogous quantities over an ensemble of classical Ising
models.Comment: LaTeX, 39 pages, 13 figure
Algebraic aspect and construction of Lax operators in quantum integrable systems
An algebraic construction more general and intimately connected with that of
Faddeev, along with its application for generating different classes of
quantum integrable models are summarised to complement the recent results of
ref. 1 ( L.D. Faddeev, {\it Int. J. Mod. Phys. } {\bf A10}, 1845 (1995) ).Comment: 8 pages, plain TEX, no figure
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