B. Basu-Mallick

Inequivalent quantizations of the rational Calogero model

We show that the rational Calogero model with suitable boundary condition admits quantum states with non-equispaced energy levels. Such a spectrum generically consists of infinitely many positive energy states and a single negative energy state. The new states appear for arbitrary number of particles and for specific ranges of the coupling constant. These states owe their existence to the self-adjoint extensions of the corresponding Hamiltonian, which are labelled by a real parameter z. Each value of z corresponds to a particular spectrum, leading to inequivalent quantizations of the model.

Bound states in one-dimensional quantum N-body systems with inverse square interaction

Abstract We investigate the existence of bound states in N-body Calogero model without the confining term. The effective Hamiltonian of this system admits a one-parameter family of self-adjoint extensions and the bound states occur when most general boundary conditions are considered. Such states are found to exist only when N=3,4 and for certain values of the system parameters.

Exact partition function of SU(m|n) supersymmetric Haldane-Shastry spin chain

Abstract 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.

Bound and Scattering States of Extended Calogero Model with an Additional PT Invariant Interaction

Here we discuss two many-particle quantum systems, which are obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the Calogero model with and without confining potential. It is shown that the energy eigenvalues are real for both of these quantum systems. For the case of extended Calogero model with confining potential, we obtain discrete bound states satisfying generalised exclusion statistics. On the other hand, the extended Calogero model without confining term gives rise to scattering states with continuous spectrum. The scattering phase shift for this case is determined through the exchange statistics parameter. We find that, unlike the case of usual Calogero model, the exclusion and exchange statistics parameters differ from each other in the presence of PT invariant interaction.

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 ngreater-or-equal, slanted1, 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.

Boson–fermion duality in SU(m|n) supersymmetric Haldane–Shastry spin chain

Abstract By using the Y ( gl ( m | n ) ) super Yangian symmetry of the SU ( m | n ) supersymmetric Haldane–Shastry spin chain, we show that the partition function of this model satisfies a duality relation under the exchange of bosonic and fermionic spin degrees of freedom. As a byproduct of this study of the duality relation, we find a novel combinatorial formula for the super Schur polynomials associated with some irreducible representations of the Y ( gl ( m | n ) ) Yangian algebra. Finally, we reveal an intimate connection between the global SU ( m | n ) symmetry of a spin chain and the boson–fermion duality relation.

Exactly solvable DN-type quantum spin models with long-range interaction

Abstract We derive the spectra of the D N -type Calogero (rational) su ( m ) spin model, including the degeneracy factors of all energy levels. By taking the strong coupling limit of this model, in which its spin and dynamical degrees of freedom decouple, we compute the exact partition function of the su ( m ) Polychronakos–Frahm spin chain of D N type. In particular, we show that this partition function cannot be obtained as a limiting case of its BC N counterpart. With the help of the partition function we study several statistical properties of the chains spectrum, such as the density of energy levels and the distribution of spacings between consecutive levels.

PHASE SHIFT ANALYSIS OF PT-SYMMETRIC NONHERMITIAN EXTENSION OF AN-1 CALOGERO MODEL WITHOUT CONFINING INTERACTION

We discuss a manyparticle quantum system, which is obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the AN-1 rational Calogero model without confining potential. This model gives rise to scattering states with continuous real spectrum. The scattering phase shift is determined through the exchange statistics parameter. We find that, unlike the case of the usual Calogero model, the exclusion and exchange statistics parameter differ from each other in the presence of PT invariant interaction.

Algebraic Bethe ansatz for a quantum integrable derivative nonlinear Schrödinger model

Abstract We find that the quantum monodromy matrix associated with a derivative nonlinear Schrodinger (DNLS) model exhibits U (2) or U (1,1) symmetry depending on the sign of the related coupling constant. By using a variant of quantum inverse scattering method which is directly applicable to field theoretical models, we derive all possible commutation relations among the operator valued elements of such monodromy matrix. Thus, we obtain the commutation relation between creation and annihilation operators of quasi-particles associated with DNLS model and find out the S -matrix for two-body scattering. We also observe that, for some special values of the coupling constant, there exists an upper bound on the number of quasi-particles which can form a soliton state for the quantum DNLS model.

Spectral properties of supersymmetric Polychronakos spin chain associated with AN―1 root system

Abstract By using the exact partition function of su ( m | n ) Polychronakos spin chain associated with A N − 1 root system, we numerically study some statistical properties of the related spectrum. It appears that the corresponding energy level density satisfies the Gaussian distribution and the cumulative distribution of spacing between consecutive energy levels obeys a certain ‘square root of a logarithm’ law.