A Unified Algebraic Approach to Few and Many-Body Correlated Systems

TL;DR

This paper develops an algebraic framework to analyze and solve a broad class of correlated many-body systems, connecting their spectra to oscillator models and introducing new solvable models and mathematical tools.

Contribution

It extends algebraic methods to a wide range of correlated systems, providing new solutions, spectrum generating algebras, and insights into their symmetries and wave functions.

Findings

Established equivalence of Calogero-Sutherland model to decoupled oscillators

Derived new expressions for Jack polynomials and analyzed Laughlin wave function symmetry

Proposed a novel method for diagonalizing complex correlated models

Abstract

The present article is an extended version of the paper {\it Phys. Rev.} {\bf B 59}, R2490 (1999), where, we have established the equivalence of the Calogero-Sutherland 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 $W_{1+\infty}$ symmetry in this approach.