Unified algebraic approach to few- and many-body correlated systems

Abstract

The present article is an extended version of the paper Phys. Rev. B 59, R2490 (1999, where, we have established the equivalence of the CalogeroSutherland model to decoupled oscillators. Here, we first employ the same approach for finding the eigenstates of a large class of Hamiltonians, dealing with correlated systems. A number of few and many-body interacting models are studied and the relationship between their respective Hilbert spaces, with that of oscillators, is found. This connection is then used to obtain the spectrum generating algebras for these systems and make an algebraic statement about correlated systems. The procedure to generate new solvable interacting models is outlined. We then point out the inadequacies of the present technique and make use of a novel method for solving linear differential equations to diagonalize the Sutherland model and establish a precise connection between this correlated system's wave functions, with those of the free particles on a circle. In the process, we obtain a new expression for the Jack polynomials. In two dimensions, we analyze the Hamiltonian having Laughlin wave function as the ground-state and point out the natural emergence of the underlying linear W 1+∞ symmetry in this approach.

Quantum supremacy of many-particle thermal machines

“…General results for the many-particle engine B.1. Preliminary Although the Calogero-Sutherland model described by the Hamiltonian in equation (7) is a truly interacting many-body system, to account for its thermodynamics it is useful to exploit its mapping to effectively free harmonic oscillators [28,35,44,45]. .…”

Long-range interactions in the quantum many-body problem in one dimension: Ground state

“…Due to the presence of long range interaction, the PCF shows a departure from the CSM. Theoretical understanding of the ground state properties of complex many body systems have received considerable attention in recent years [1,2,3,4,5,6,7,8,9,10,11,12,13]. In this context, we study rigorously a wide class of 1-dimensional N -body systems having different densities, nature and strength of a 2-body interaction.…”

Aspects of Generalized Calogero Model

“…In a case of a single-species model in D dimensions, some exact eigenstates (including the ground state) are known but the complete solution of the problem is still lacking. Some progress has been achieved only recently for a class of 2D models [8]. Usually, the inevitable appearance of the three-body interaction in D > 1 is the main obstacle which makes any analysis of such a model(s) highly nontrivial.…”