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arXiv:cond-mat/9901073

A Unified Algebraic Approach to Few and Many-Body Hamiltonians having Linear Spectra

Nakka Gurappa, Prasanta Kumar Panigrahi, Thokala Soloman Raju

Published Jan 9, 1999• 4 citations
arXivHTMLPDF

Abstract

We develop an algebraic approach for finding the eigenfunctions of a large class of few and many-body Hamiltonians, in one and higher dimensions, having linear spectra. The method presented enables one to exactly map these interacting Hamiltonians to decoupled oscillators, thereby, giving a precise correspondence between the oscillator eigenspace and the wavefunctions of these quantum systems. The symmetries behind the degeneracy structure and the commuting constants of motion responsible for the quantum integrability of some of these models are made transparent. Apart from analysing a number of well-known dynamical systems like planar oscillators with commensurate frequencies, both with or without singular inverse-square terms and generalized Calogero-Sutherland type models, we also point out a host of other examples where the present approach can be profitably employed. We further study Hamiltonians having Laughlin wavefunction as the ground-state and establish their equivalence to free oscillators. This reveals the underlying linear $W_{1+\infty}$ symmetry algebra unambiguously and establishes the Laughlin wavefunction as the highest weight vector of this algebra.

DOI
10.48550/arxiv.cond-mat/9901073

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We develop an algebraic approach for finding the eigenfunctions of a large class of few and many-body Hamiltonians, in one and higher dimensions, having linear spectra. The method presented enables one to exactly map these interacting Hamiltonians to decoupled oscillators, thereby, giving a precise correspondence between the oscillator eigenspace and the wavefunctions of these quantum systems. The symmetries behind the degeneracy structure and the commuting constants of motion responsible for the quantum integrability of some of these models are made transparent. Apart from analysing a number of well-known dynamical systems like planar oscillators with commensurate frequencies, both with or without singular inverse-square terms and generalized Calogero-Sutherland type models, we also point out a host of other examples where the present approach can be profitably employed. We further study Hamiltonians having Laughlin wavefunction as the ground-state and establish their equivalence to free oscillators. This reveals the underlying linear W1+∞W_{1+\infty}W1+∞​ symmetry algebra unambiguously and establishes the Laughlin wavefunction as the highest weight vector of this algebra.
The present paper is an extended version of another paper [Phys. Rev. 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 solvable interacting models is outlined. We then point out the inadequacies of the present technique and make use of a 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 an 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 ${W}_{1+\ensuremath{\infty}}$ symmetry in this approach.
A similarity transformation is constructed, through which a system of particles interacting with inverse-square two-body and harmonic potentials in one dimension can be mapped identically to a set of free harmonic oscillators. This equivalence provides a straightforward method to find the complete set of eigenfunctions, the exact constants of motion, and a linear ${W}_{1+\ensuremath{\infty}}$ algebra associated with this model. It is also demonstrated that a large class of models with long-range interactions, both in one and higher dimensions can be made equivalent to decoupled oscillators.
There is developed a current algebra representation scheme for reconstructing algebraically factorized quantum Hamiltonian and symmetry operators in the Fock type space and its application to quantum Hamiltonian and symmetry operators in case of quantum integrable spatially many- and one-dimensional dynamical systems. As examples, we have studied in detail the factorized structure of Hamiltonian operators, describing such quantum integrable spatially many- and one-dimensional models as generalized oscillatory, Calogero-Sutherland, Coulomb type and nonlinear Schrödinger dynamical systems of spinless bose-particles.
We study a multispecies one-dimensional Calogero model with two- and three-body interactions. Using an algebraic approach (Fock space analysis), we construct ladder operators and find infinitely many, but not all, exact eigenstates of the model Hamiltonian. Besides the ground state energy, we deduce energies of the excited states. It turns out that the spectrum is linear in quantum numbers and that the higher-energy levels are degenerate. The dynamical symmetry responsible for degeneracy is SU(2). We also find the universal critical point at which the model exhibits singular behaviour. Finally, we make contact with some special cases mentioned in the literature.
This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.
Quantum integrable particle systems with inverse square interactions in one-dimensional space are investigated by use of the quantum inverse scattering method. For the case of particles with $\mathrm{SU}(\ensuremath{\nu})$ spins, a hidden symmetry is the $\mathrm{SU}(\ensuremath{\nu})\ensuremath{\bigotimes}\mathrm{U}(1)\ensuremath{\simeq}\mathrm{U}(\ensuremath{\nu})$ current algebra. The system has a higher symmetry, the ${W}_{1+\ensuremath{\infty}}$ algebra. This WWW algebra structure unifies the Calogero type (1x2\frac{1}{{x}^{2}}x21​ interactions) and the Sutherland type (1sin2x\frac{1}{{sin}^{2}x}sin2x1​ interactions). The energy spectra of the systems are obtained in a simple manner based on the fermion representation of the algebra.