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arXiv:math-ph/0509010

Calogero-Sutherland Model with Anti-periodic Boundary Conditions: Eigenvalues and Eigenstates

Arindam Chakraborty, Subhankar Ray, J. Shamanna

Published Sep 6, 2005• 0 citations
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Abstract

The U(1) Calogero Sutherland Model with anti-periodic boundary condition is studied. The Hamiltonian is reduced to a convenient form by similarity transformation. The matrix representation of the Hamiltonian acting on a partially ordered state space is obtained in an upper triangular form. Consequently the diagonal elements become the energy eigenvalues. The eigenstates are constructed using Young diagram and represented in terms of Jack symmetric polynomials. The eigenstates so obtained are orthonormalized.

DOI
10.48550/arxiv.math-ph/0509010

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The U(1) Calogero Sutherland Model with anti-periodic boundary condition is studied. The Hamiltonian is reduced to a convenient form by similarity transformation. The matrix representation of the Hamiltonian acting on a partially ordered state space is obtained in an upper triangular form. Consequently the diagonal elements become the energy eigenvalues. The eigenstates are constructed using Young diagram and represented in terms of Jack symmetric polynomials. The eigenstates so obtained are orthonormalized.
An algebraic construction of the eigenstates for the quantum Calogero model is investigated. Extending the method of Lapointe and Vinet, we construct the eigenstates for the second conserved operator of the quantum Calogero model. All the eigenstates can be factorized into symmetric polynomials which we call “Hi-Jack symmetric polynomials” and the ground state wave function. The conjectured formula for the eigenvalue of the second conserved operator is confirmed. The Hi-Jack polynomials are strong candidates for the orthogonal basis of the quantum Calogero model.
The 40-year-old Calogero-Sutherland (CS) model remains a source of inspirations for understanding 1d interacting fermions. At β=1,or0β=1, \text{or}0β=1,or0, the CS model describes a free non-relativistic fermion, or boson theory, while for generic βββ, the system can be interpreted either as interacting fermions or bosons, or free anyons depending on the context. However, we shall show in this letter that the fermionic picture is advantageous in diagonalizing the CS Hamiltonian. Comparing to the previously known multi-integral representation or the Dunkl operator formalism for the CS wave functions, our method depends on the (upper or lower) triangular nature of the fermion interaction, which is resolved in perturbation theory of the second quantized form. The eigenstate is constructed from a multiplet of unperturbed states and the perturbation is of finite order. The full construction is a similarity transformation from the free fermion theory, in the same spirit as the Landau Fermi liquid theory and the 1d Luttinger liquid theory. That means quasi-particles or anyons can be represented in terms of free fermion modes (or bosonic modes via bosonization). The method is applicable to other (higher than one space dimension) systems for which the adiabatic theorem applies.
We algebraically construct the Fock space of the Sutherland model in terms of the eigenstates of the pseudomomenta as basis vectors. We introduce the following characteristic operators and systematically produce all the eigenstates starting from the ground state by multiplying the operators. The braid-exclusion operators exchange eigenvalues of pseudomomenta; they follow the algebra of a braid group and also include effects of the exclusion statistics. The raising and lowering operators increase and decrease eigenvalues of pseudomomenta. The Sutherland model is a solvable quantum many-body system with inversesquare interaction on a circumference [1]. The ground-state wave function is of the Jastrow type and excited states are represented as polynomials multiplied by the ground state. Among the polynomials symmetric ones are the Jack polynomials [2, 3, 4], while the others are called nonsymmetric Jack polynomials. These states can be taken as eigenstates of the pseudomomenta or Dunkel
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.