Integrability and Applications of the Exactly-Solvable Haldane-Shastry One-Dimensional Quantum Spin Chain

TL;DR

This paper explores the integrability, eigenfunctions, and physical properties of the exactly solvable Haldane-Shastry spin chain, highlighting its unique features, symmetry algebra, and potential relevance to superconductivity.

Contribution

It provides a detailed analysis of the Haldane-Shastry model's eigenfunctions, symmetry structure, and dynamical properties, extending understanding beyond traditional models.

Findings

Eigenfunctions constructed and generated by Yangian symmetry

Explicit constants of motion identified for the model

Connection established between spinon propagator and off-diagonal long-range order

Abstract

Recently, the one dimensional model of $N$ spins with $S=\frac{1}{2}$ on a circle, interacting with an exchange that falls off with the inverse square of the separation: $H_{\rm ISE} =\sum_{i\neq j} \frac{1}{[\frac{N}{\pi} \sin(\frac{i-j}{N}\pi)]^2}(\svec_i\cdot \svec_j-\frac{1}{4})$, or ISE-model, has received ample attention. Its special features include: relatively simple eigenfunctions, non-interacting elementary excitations that obey semionic statistics (spinons), and a large ``quantum group'' symmetry algebra called the Yangian. This model is fully integrable, albeit in a slightly different sense than the more traditional nearest neighbor exchange (NNE) Heisenberg chain. This thesis comes in 4 chapters. Chapter 1 introduces the model and presents the construction of a subset of the eigenfunctions. The other eigenfunctions are shown to be generated by the action of the Yangian symmetry algebra of $H_{\rm ISE}$. Chapter 2 presents a method to construct the set of constants of the motion of the ISE-model. The ISE-model is tractable enough to obtain its zero-magnetic-field dynamical structure factors. Chapter 3 attempts to extend this to a non-zero magnetic field, where, due to the presence of spinons in the groundstate, more complicated excitations contribute: small numbers of magnons and spinons. discuss the relation to the more complicated NNE-model. Finally chapter 4 illustrates how a recently conjectured new form of Off Diagonal Long Range Order in antiferromagnetic spin chains can be reinterpreted as a spinon propagator in the ISE-model, and verified numerically. We briefly comment on its relevance to stabilizing superconductivity in the layered cuprates.