[Paper Review] Integrability and Applications of the Exactly-Solvable Haldane-Shastry One-Dimensional Quantum Spin Chain

J. C. Talstra|arXiv (Cornell University)|Sep 30, 1995

Physics of Superconductivity and Magnetism1 references3 citations

TL;DR

This PhD thesis investigates the integrable Haldane-Shastry spin chain with inverse-square exchange interactions, demonstrating its exact solvability via Yangian symmetry and constructing eigenfunctions through this algebra. It derives dynamical structure factors at zero field and identifies spinon excitations, while also linking a novel form of off-diagonal long-range order to spinon propagation, suggesting relevance to high-temperature superconductivity in cuprates.

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: $H_{ m ISE} =\sum_{i eq j} \frac{1}{[\frac{N}π \sin(\frac{i-j}{N}π)]^2}(\svec_i\cdot \svec_j-\frac{1}{4})$, 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 $H_{ m ISE}$. 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.

Motivation & Objective

• To establish the exact solvability of the Haldane-Shastry spin chain with inverse-square exchange interactions.
• To analyze the role of Yangian symmetry in generating the full set of eigenfunctions.
• To compute the dynamical structure factors in the absence of a magnetic field.
• To extend the analysis to non-zero magnetic fields, accounting for magnon and spinon contributions.
• To reinterpret a recently conjectured form of off-diagonal long-range order as a spinon propagator and verify it numerically.

Proposed method

• Constructs a subset of eigenfunctions using the model's specific form and symmetry.
• Uses the Yangian symmetry algebra to generate the remaining eigenfunctions of the Hamiltonian.
• Derives a complete set of conserved quantities (constants of motion) for the ISE-model.
• Computes the zero-field dynamical structure factors using the exact eigenstates and their matrix elements.
• Applies perturbative and algebraic techniques to analyze the spectrum under non-zero magnetic fields.
• Reinterprets the conjectured off-diagonal long-range order as a spinon propagator and performs numerical verification in the ISE-model framework.

Experimental results

Research questions

• RQ1 How does the Yangian symmetry algebra fully generate the eigenstates of the Haldane-Shastry Hamiltonian?
• RQ2 What are the dynamical structure factors of the ISE-model in the absence of an external magnetic field?
• RQ3 How do magnon and spinon excitations contribute to the spectrum when a magnetic field is applied?
• RQ4 Can the recently conjectured form of off-diagonal long-range order be interpreted as a spinon propagator in the ISE-model?
• RQ5 What is the potential relevance of spinon-mediated off-diagonal long-range order to superconductivity in layered cuprates?

Key findings

• The full set of eigenfunctions of the Haldane-Shastry model is generated by the action of the Yangian symmetry algebra.
• The model's dynamical structure factors are analytically tractable and computed exactly at zero magnetic field.
• In the presence of a magnetic field, the low-energy spectrum includes both magnons and spinons, with spinons contributing to the ground state.
• The conjectured form of off-diagonal long-range order is reinterpreted as a spinon propagator and verified numerically within the ISE-model.
• The spinon propagator exhibits long-range coherence, suggesting a mechanism relevant to stabilizing superconductivity in layered cuprates.