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
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.
In the present chapter, we discuss the Sutherland model for particles without internal degrees of freedom such as spin. We call this model the singlecomponent Sutherland model. For two particles, the eigenenergies and eigenstates of the Sutherland model (1.3) have been obtained explicitly in the previous chapter. The most striking feature of the Sutherland model is that one can derive not only the energy spectrum but also the dynamics for the many-particle case with an exact account of interaction effects. Thus, the Sutherland model provides an ideal framework to study a one-dimensional quantum liquid in detail. In Section 2.1, we derive the eigenenergies of eigenstates. In Section 2.2, we present different but equivalent physical pictures for the energy spectrum. Namely, the energy spectrum is naturally regarded as that of interacting bosons or fermions. The same spectrum can also be interpreted as that of free particles obeying nontrivial quantum statistics, i.e., free anyons in one dimension. The exclusion statistics proposed by Haldane will be explained on this occasion. Spectrum and statistics of elementary excitations are derived in Section 2.3. In Section 2.4, we discuss thermodynamic properties, which can be rewritten as those of free anyons. In Section 2.5, we identify the eigenfunctions with Jack symmetric polynomials, and discuss their basic properties. In Section 2.6, we consider dynamical correlation functions such as Green's functions and the density correlation function. These quantities are derived with the use of Jack polynomials, and are naturally interpreted in terms of elementary excitations with fractional charge. The Sutherland model is the simplest model to realize the Tomonaga–Luttinger liquid.
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.
The Sutherland model has a number of variants. One of them is the U( K ) Sutherland model [71, 85, 86, 132]. This model describes N particles moving along a circle of perimeter L , and each particle possesses an internal degree of freedom with K possible values. This corresponds to spin with K = 2, and more generally a color. In the U( K ) Sutherland model, all particles obey common statistics: bosonic or fermionic. We can generalize the model further. The U( K B , K F ) Sutherland model [177] consists of bosons having K B possible colors and fermions having K F colors. The multi-component Sutherland model has a degeneracy in energy levels which is described in terms of a Yangian. The Yangian is an algebra related to quantum groups [43, 44]. The Yangian is nicely realized by variants of Jack polynomials which are modified so as to conform to the internal symmetry. Elementary excitations in the multi-component Sutherland model are described in a few alternative ways: interacting bosonic or fermionic particles, or non-interacting particles obeying generalized exclusion statistics. Furthermore, the lattice models such as the Haldane–Shastry models [77, 161] and 1/ r 2 supersymmetric t – J model [119] are obtained in the strong coupling limit of U(2) and U(2, 1) Sutherland models, respectively. The Sutherland models in the continuum space are much more tractable mathematically than the corresponding lattice models. Hence, the mapping to lattice models turns out to be useful to derive the explicit results on thermodynamics and dynamics in lattice models. In the present chapter, we extend our treatment for the single-component Sutherland model in order to include the internal degrees of freedom. We shall discuss the energy spectrum, thermodynamics, and dynamical correlation functions.
This chapter examines the linear harmonic oscillator in the energy (Fock) representation, emphasizing its algebraic structure and spectral properties. It is shown that the energy spectrum is discrete and bounded from below. The energy raising and lowering (ladder) operators are introduced, and the energy eigenvalues are derived. The eigenstates are constructed algebraically from the ground state by repeated application of the raising operator, forming the Fock basis. These energy eigenstates are then transformed into the coordinate representation, where the corresponding wavefunctions are expressed in terms of Hermite polynomials. This establishes a connection between the algebraic and coordinate formulations of the quantum harmonic oscillator. Metadata/Keywords: linear harmonic oscillator, Fock representation, ladder operators, energy spectrum, energy eigenstates, ground state, Hermite polynomials, raising and lowering operators