Preface 1. Introduction Part I. Physical Properties: 2. Single-component Sutherland model 3. Multi-component Sutherland model 4. Spin chain with 1/r2 interactions 5. SU(K) spin chain 6. Supersymmetric t-J model with 1/r2 interaction Part II. Mathematics Related to 1/r2 Systems: 7. Jack polynomials 8. Yang-Baxter relations and orthogonal eigenbasis 9. SU(K) and supersymmetric Yangians 10. Uglov's theory Epilogue References List of symbols Index.
Preface 1. Introduction Part I. Physical Properties: 2. Single-component Sutherland model 3. Multi-component Sutherland model 4. Spin chain with 1/r2 interactions 5. SU(K) spin chain 6. Supersymmetric t-J model with 1/r2 interaction Part II. Mathematics Related to 1/r2 Systems: 7. Jack polynomials 8. Yang-Baxter relations and orthogonal eigenbasis 9. SU(K) and supersymmetric Yangians 10. Uglov's theory Epilogue References List of symbols Index.
One-dimensional quantum systems show fascinating properties beyond the scope of the mean-field approximation. However, the complicated mathematics involved is a high barrier to non-specialists. Written for graduate students and researchers new to the field, this book is a self-contained account of how to derive the exotic quasi-particle picture from the exact solution of models with inverse-square interparticle interactions. The book provides readers with an intuitive understanding of exact dynamical properties in terms of exotic quasi-particles which are neither bosons nor fermions. Powerful concepts, such as the Yangian symmetry in the Sutherland model and its lattice versions, are explained. A self-contained account of non-symmetric and symmetric Jack polynomials is also given. Derivations of dynamics are made easier, and are more concise than in the original papers, so readers can learn the physics of one-dimensional quantum systems through the simplest model.
One-dimensional quantum systems show fascinating properties beyond the scope of the mean-field approximation. However, the complicated mathematics involved is a high barrier to non-specialists. Written for graduate students and researchers new to the field, this book is a self-contained account of how to derive the exotic quasi-particle picture from the exact solution of models with inverse-square interparticle interactions. The book provides readers with an intuitive understanding of exact dynamical properties in terms of exotic quasi-particles which are neither bosons nor fermions. Powerful concepts, such as the Yangian symmetry in the Sutherland model and its lattice versions, are explained. A self-contained account of non-symmetric and symmetric Jack polynomials is also given. Derivations of dynamics are made easier, and are more concise than in the original papers, so readers can learn the physics of one-dimensional quantum systems through the simplest model.
We have discussed the physical and mathematical aspects of one-dimensional quantum particles with1/ r 2 interactions. On the physics side, we began with the simplest two-body system, and culminated in the exact dynamics of a supersymmetric electronic model. In this way we have tried to draw a comprehensive picture of one-dimensional quantum systems, taking the canonical systems with 1/ r 2 interactions. On the mathematical side, the most important subjects are first the Jack polynomials, and second the Yangians. These two subjects were originally developed independently, but are actually closely interrelated. Our basic viewpoint is that the 1/ r 2 systems provide the most natural working model to synthesize these two subjects, and to sharpen the concepts further. There are many topics, however, which we could not discuss in this book. This is firstly because of our insufficient understanding of the topics, and secondly because of our intention of making a reasonably sized book. We shall mention some of the omitted topics, hoping that they will be covered on another occasion. Examples of other canonical systems, which have escaped our discussion in this book, include the Calogero–Sutherland model with harmonic confinement. The spectrum of the model is equidistant as in the simple harmonic oscillator. Because of this feature the model can be compared with the chiral conformal field theory (CFT) with right-going particles only. We refer to a recent review [88] for the systems. In relation to the CFT, there are active studies to investigate the quasi-particle structure for the general case of the internal symmetry.
Recent developments in the theory of quantum integrable particle systems in one-dimension with inverse square interactions are reviewed. First the Yangian symmetry is introduced and the energy spectra of the related spin models are discussed. The character of the su(n)1 WZNW theory is shown to be closely related with the Rogers–Szegö polynomial. Second, the infinite dimensional representation for solutions of the Yang–Baxter equation and the reflection equation is given. Based on the representation, the Dunkl operators associated with the classical root systems are constructed. The Macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case. Finally, some results on short-range interacting systems are mentioned.
We now describe the properties of the SU( >K ) chain in more algebraic terms. The relevant symmetry is the SU( K ) Yangian, which is usually referred to as Y ( sl K ). On the other hand, the Yangian symmetry of the Sutherland model with spin 1/2 is given by Y ( gl 2 ). The difference between Y ( sl K ) and Y ( gl K ) is that the charge degrees of freedom U(1) is absent in Y ( sl K ), while the spin degrees of freedom is included in both Y ( sl K ) and Y ( sl K ). In Section 4.8.3, we have removed the U(1) component by means of the freezing trick, i.e., by taking the infinitely large repulsion parameter. In this chapter we take a more algebraic approach to study Y ( sl K ), which makes the relation between Y ( gl K ) and Y ( sl K ) clearer. In Section 9.1, we begin with the construction of the SU( K ) monodromy matrix as the product form. The commuting property of the Cherednik–Dunkl operators plays a crucial role here. The monodromy matrix is the basic building block of the Yangian algbebra, as discussed in Section 8.2. Because the Hamiltonian of the SU( K ) spin chain and the monodromy matrix commute, each eigenstate of the spin chain constitutes a basis for representations of the SU( K ) Yangian. In Section 9.2 and 9.3, we discuss the quantum determinant by comparing it with the ordinary determinant. It is demonstrated in Section 9.4 that the quantum determinant has a scalar nature that leads to a simple evaluation. Because of the scalar nature, however, the quantum determinant cannot distinguish different SU( K ) Yangian states.
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.