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 ${\cal{PT}}$, the eigenstates corresponding to the zero value of the generalized angular momentum have real energies.Comment: revtex4 8 pages, 2 figure

Realisations of GLp,q(2)GL_{p,q}(2) quantum group and its coloured extension through a novel Hopf algebra with five generators

A novel Hopf algebra $( {\tilde G}_{r,s} )$, depending on two deformation parameters and five generators, has been constructed. This ${\tilde G}_{r,s}$ Hopf algebra might be considered as some quantisation of classical $GL(2) \otimes GL(1)$ group, which contains the standard $GL_q(2)$ quantum group (with $q=r^{-1}$) as a Hopf subalgebra. However, we interestingly observe that the two parameter deformed $GL_{p,q}(2)$ quantum group can also be realised through the generators of this ${\tilde G}_{r,s}$ algebra, provided the sets of deformation parameters $p,~q$ and $r,~s$ are related to each other in a particular fashion. Subsequently we construct the invariant noncommutative planes associated with ${\tilde G}_{r,s}$ algebra and show how the two well known Manin planes corresponding to $GL_{p,q}(2)$ quantum group can easily be reproduced through such construction. Finally we consider the `coloured' extension of $GL_{p,q}(2)$ quantum group as well as corresponding Manin planes and explore their intimate connection with the `coloured' extension of ${\tilde G}_{r,s}$ 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$^1$, 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