Pijush K. Ghosh

Super-Calogero–Moser–Sutherland systems and free super-oscillators: a mapping

We show that the supersymmetric rational Calogero–Moser–Sutherland (CMS) model of AN+1-type is equivalent to a set of free super-oscillators, through a similarity transformation. We prescribe methods to construct the complete eigenspectrum and the associated eigenfunctions, both in supersymmetry-preserving as well as supersymmetry-breaking phases, from the free super-oscillator basis. Further we show that a wide class of super-Hamiltonians realizing dynamical OSp(2|2) supersymmetry, which also includes all types of rational super-CMS as a small subset, are equivalent to free super-oscillators. We study BCN+1-type super-CMS model in some detail to understand the subtleties involved in this method.

Integrable nonlocal vector nonlinear Schrödinger equation with self-induced parity-time-symmetric potential

Abstract A two component nonlocal vector nonlinear Schrodinger equation (VNLSE) is considered with a self-induced parity-time-symmetric potential. It is shown that the system possess a Lax pair and an infinite number of conserved quantities and hence integrable. Some of the conserved quantities like number operator, Hamiltonian etc. are found to be real-valued, in spite of the corresponding charge densities being complex. The soliton solution for the same equation is obtained through the method of inverse scattering transformation and the condition of reduction from nonlocal to local case is mentioned.

Supersymmetry, shape invariance, and solvability of A N-1 and BC N Calogero-Sutherland model

Using the ideas of supersymmetry and shape invariance we rederive the spectrum of the A N-1 and BC N Calogero-Sutherland model. We briefly discuss how to obtain the corresponding eigenfunctions. We also discuss the difficulties involved in extending this approach to the trigonometric models.

Supersymmetric Many-particle Quantum Systems with Inverse-square Interactions

The development in the study of supersymmetric many-particle quantum systems with inverse-square interactions is reviewed. The main emphasis is on quantum systems with dynamical OSp(2|2) supersymmetry. Several results related to the exactly solved supersymmetric rational Calogero model, including shape invariance, equivalence to a system of free superoscillators and non-uniqueness in the construction of the Hamiltonian, are presented in some detail. This review also includes a formulation of pseudo-Hermitian supersymmetric quantum systems with a special emphasis on the rational Calogero model. There are quite a few number of many-particle quantum systems with inverse-square interactions which are not exactly solved for a complete set of states in spite of the construction of infinitely many exact eigenfunctions and eigenvalues. The Calogero–Marchioro model with dynamical SU(1, 1|2) supersymmetry and a quantum system related to the short-range Dyson model belong to this class and certain aspects of these models are reviewed. Several other related and important developments are briefly summarized.The development in the study of supersymmetric many-particle quantum systems with inverse-square interactions is reviewed. The main emphasis is on quantum systems with dynamical OSp(2|2) supersymmetry. Several results related to exactly solved supersymmetric rational Calogero model, including shape invariance, equivalence to a system of free superoscillators and non-uniqueness in the construction of the Hamiltonian, are presented in some detail. This review also includes a formulation of pseudo-hermitian supersymmetric quantum systems with a special emphasis on rational Calogero model. There are quite a few number of many-particle quantum systems with inverse-square interactions which are not exactly solved for a complete set of states in spite of the construction of infinitely many exact eigen functions and eigenvalues. The Calogero-Marchioro model with dynamical SU(1, 1|2) supersymmetry and a quantum system related to short-range Dyson model belong to this class and certain aspects of these models are reviewed. Several other related and important developments are briefly summarized.

A note on the topological insulator phase in non-Hermitian quantum systems

Examples of non-Hermitian quantum systems admitting a topological insulator phase are presented in one, two and three space dimensions. All of these non-Hermitian Hamiltonians have entirely real bulk eigenvalues and unitarity is maintained with the introduction of appropriate inner products in the corresponding Hilbert spaces. The topological invariant characterizing a particular phase is shown to be identical for a non-Hermitian Hamiltonian and its Hermitian counterpart, to which it is related through a non-unitary similarity transformation. A classification scheme for topological insulator phases in pseudo-Hermitian quantum systems is suggested.

The exactly solvable quasi-Hermitian transverse Ising model

A non-Hermitian deformation of the one-dimensional transverse Ising model is shown to have the property of quasi-hermiticity. The transverse Ising chain is obtained from the starting non-Hermitian Hamiltonian through a similarity transformation. Consequently, both the models have identical eigen spectra, although the eigenfunctions are different. The metric in the Hilbert space, which makes the non-Hermitian model unitary and ensures the completeness of states, has been constructed explicitly. Although the longitudinal correlation functions are identical for both the non-Hermitian and the Hermitian Ising models, the difference shows up in the transverse correlation functions, which have been calculated explicitly and are not always real. A proper set of Hermitian spin operators in the Hilbert space of the non-Hermitian Hamiltonian has been identified, in terms of which all the correlation functions of the non-Hermitian Hamiltonian become real and identical to that of the standard transverse Ising model. Comments on the quantum phase transitions in the non-Hermitian model have been made.

Extended superconformal symmetry and Calogero-Marchioro model

We show that the two-dimensional Calogero-Marchioro model (CMM) without the harmonic confinement can naturally be embedded into an extended SU(1,1|2) superconformal Hamiltonian. We study the quantum evolution of the superconformal Hamiltonian in terms of suitable compact operators of the = 2 extended de Sitter superalgebra with central charge and discuss the pattern of supersymmetry breaking. We also study the arbitrary D-dimensional CMM having dynamical OSp(2|2) supersymmetry and point out the relevance of this model in the context of the low energy effective action of the dimensionally reduced Yang-Mills theory.

Calogero-Sutherland type models in higher dimensions

Abstract We construct two different Calogero-Sutherland type models with only two-body interactions in arbitrary dimensions. We obtain some exact wave functions, including the ground states, of these two models for an arbitrary number of spinless nonrelativistic particles.

The Kronig-Penney model on a generalized Fibonacci lattice

Abstract The Kronig-Penney model on a class of one-dimensional quasiperiodic lattices is studied. A unified trace map and the Landauer resistance are obtained. Numerical calculations of the energy spectra of such lattices reveal interesting features.

On the construction of a pseudo-Hermitian quantum system with a pre-determined metric in the Hilbert space

A class of pseudo-Hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators that are Hermitian with respect to a pre-determined positive-definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-Hermitian quantum systems admitting entirely real spectra and unitary time evolution. The examples considered include simple harmonic oscillators with complex angular frequencies, Stark (Zeeman) effect with non-Hermitian interaction, non-Hermitian general quadratic form of N boson (fermion) operators, symmetric and asymmetric XXZ spin chain in the complex magnetic field, non-Hermitian Haldane–Shastry spin chain and Lipkin–Meshkov–Glick model.