We algebraically construct the Fock space of the Sutherland model in terms of the eigenstates of the pseudomomenta as basis vectors. For this purpose, we derive the raising and lowering operators which increase and decrease eigenvalues of pseudomomenta. The operators exchanging eigenvalues of two pseudomomenta have been known. All the eigenstates are systematically produced by starting from the ground state and multiplying these operators by it.
We algebraically construct the Fock space of the Sutherland model in terms of the eigenstates of the pseudomomenta as basis vectors. For this purpose, we derive the raising and lowering operators which increase and decrease eigenvalues of pseudomomenta. The operators exchanging eigenvalues of two pseudomomenta have been known. All the eigenstates are systematically produced by starting from the ground state and multiplying these operators by it.
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 spectrum of the Sutherland model for fermions with spin has a degeneracy. The orthogonal set of the basis of the Fock space of degenerate states is usually related to a Lie algebra such as su 2 and su 3 . In our case, however, the degeneracy extends to states with different values of total spin, and the ordinary Lie algebra is insufficient. The relevant algebra is called Yangian [43, 44], and is the algebra controlling those operators satisfying the Yang–Baxter relation [118, 178]. In this chapter we describe how the Yangian naturally emerges in the many-particle states described by Jack polynomials, and constructs an orthogonal basis of degenerate states. We concentrate here on the simplest case of the gl 2 Yangian, corresponding to spin 1/2 particles. More general cases will be discussed in Chapter 9. We introduce a variant of the R-matrix in Section 8.1, and its standard form in Section 8.2. We then introduce a monodromy matrix and discuss its algebraic properties, such as the Yang–Baxter relation and the intertwining property. In Section 8.4, we consider the action of the monodromy matrix on the set of states constituting the degenerate eigenstates. We will see that an element of the monodromy matrix plays the role of a creation or annihilation operator of orthogonal eigenfunctions. This means that all eigenfunctions can be generated algebraically from the highest-weight state, for example, by successive action of an element of the monodromy matrix. Thus, the degeneracy beyond the Lie algebra is accounted for by the symmetry of the monodromy matrix, which is nothing but the Yangian. A demonstration of this construction will be given in Section 8.5 for the two-particle and three-particle systems.
We show that it is possible to give to any (linear and Hermitian) operator with discrete spectrum, acting on a Hilbert space, a canonical form in terms of the commutator and anticommutator of a pair of adequately constructed raising and lowering operators. This construction enables one to investigate in a very compact way the (discrete) spectrum of the given operator, which becomes determined by a set of consistency conditions. We apply the method to several important problems of quantum mechanics.
For a given eigenvalue of the square of the total spin operator, the spin eigenvectors with various eigenvalues of the z-component are obtained systematically by successive operations of the spin lowering operator on the eigenvector proposed in a previous paper for a one-dimensional many-fermion system with delta-function potential. The property of the spin eigenvector under the operation of the spin raising operator is investigated to clarify the mechanism how it has the given eigenvalue of the square of the total spin operator. It is shown that the spin eigenvector can be modified in various forms under some condition presented in the previous paper.
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.