Quantum Many-Body Systems in One Dimension

The following sections are included:

• Preface

• Contents

The following sections are included:

• Models solved by Bethe-ansatz

• Models with long-range interaction
  • Inverse-square exchange models
  • Hyperbolic and elliptic generalization

• Linearized models

The following sections are included:

• Elementary Features of the Heisenberg Spin Chain

• Bethe-Ansatz Solutions
  • Bethe-ansatz: Method I
  • Bethe-ansatz: Method II

• Transfer Matrix Formalism

• Strings in the Heisenberg Chain

• Thermodynamic Bethe-Ansatz Equations

The following sections are included:

• Introduction

• Bethe-Ansatz Equations

• Numerical Demonstration of the Strings

• Thermodynamic Bethe-ansatz Equations

• Calculation of the Thermodynamic Potential

• Verification of the Expansion for the Thermodynamic Potential

• Investigation of Other Thermodynamic Quantities

• Conclusion

• How to Solve the Equations for η and η'

The following sections are included:

• Introduction

• Calogero-Sutherland Model and Multicomponent Generalization

• SU(n) t-J Model

• SU(n) Spin Chain

• Twisted Boundary Condition

• Harmonic Fluid Description

• Conclusion

• Proof of Theorem 3.1

• Proof of Theorem 3.2

The following sections are included:

• Introduction

• Yangian Algebra

• Squeezed Strings in the ISE model

• Construction of Motifs

• Thermodynamics

• Conclusion

The following sections are included:

• Introduction

• Supersymmetric t-J Model

• Elementary Excitations

• Compact Supports of Various Correlation Functions

• Interpolation of Anti-Holon Charge from NNE to ISE Model

• Conclusion

• Normalization constant of the wavefunction

• Phase shift function

The following sections are included:

• Introduction

• Eigenstates of Calogero-Sutherland Model

• Jack Symmetric Polynomials
  • Introduction to notations
  • General properties of Jack polynomials

• Fractional statistics
  • Exchange Statistics
  • Exclusion Statistics

• Dynamical density-density correlation function

• One-Particle Green's Function

• Low-energy effective theory and Coupled Fractional Quantum Hall Edge States

• Finite-size Scaling and correlation exponents

• Lattice Models

• Conclusion

• How to take the thermodynamic limit

• Asymptotic Expansion
  • Dynamical density-density correlation function
  • One-particle Green's function

In this book some elementary and some novel features of two families of exactly solvable models are studied. It has been observed that the strings, which are introduced originally for the NNE Bethe-ansatz-solvable Heisenberg spin chain, can be used to represent the eigenstates of the one-band Hubbard model. Furthermore, exact and complete eigenstates of the finite size ISE spin chain are also represented by the strings. The strings are also known to be present in the t-J models. It appears that the string structure is quite general and is probably a generic feature of one-dimensional integrable many-body systems and it is connected with the underlying dynamically generated quantum symmetry. The strings with length longer than one probably represent the internal quantum structures of solitonic excitations of the strongly interacting 1D systems and, therefore, may be related to non-Abelian fractional statistics which are presumably described by the quantum Yang-Baxter equations.

The ISE models have very rich and elegant structures and seem to represent idealized 1D quantum fluids supporting fractional statistics. By explicitly calculating the correlation functions and examining the excitation contents I am convinced that this class of model systems represents an emerging fully solvable paradigm that is a full generalization of the ideal Fermi gas. (This finite energy generalization needs to be contrasted with the Luttinger liquid which is an universal T → 0 1D fluid.) Future works should include the effects of residual interactions or gauge fluctuations on these ideal systems and examine their stabilities and instabilities.

The following sections are included:

• Bibliography

• Index