#### Date of Award

1-1-2021

#### Document Type

Dissertation

#### Degree Name

Ph.D. in Physics

#### First Advisor

Kevin SD Beach

#### Second Advisor

Samuel Lisi

#### Third Advisor

Leo Stein

#### Relational Format

dissertation/thesis

#### Abstract

The Heisenberg Hamiltonian is the prototype model for quantum magnetism. The Hamilto- nian includes exchange couplings between vector spins, which are the intrinsic magnetic moments of localized electrons. The classical energy between any pair of interacting spins can be minimized by aligning them parallel (ferromagnetic case) or antiparallel (antiferromagnetic), depending on the sign of the coupling. But quantum fluctuations play a key role, and a full quantum mechanical treatment of the many-body system is generally required. Finding the exact ground-state energy and wave function of isotropic antiferromagnetic (AFM) systems for an arbitrary spin-S and lattice morphology is still a challenging and hotly debated topic. A questions arises: Which is the best quantum representation (fermion, boson, or other) in which to describe the spin model when the charge component of an electron is quenched at the atomic limit? For example, physicists often formulate a half-odd spin chain in the fermionic basis; while the integer spin chain is more typically treated with bosonic auxiliary particles. In my dissertation, I propose a bond-operator algebra for entangled spin pairs which are constructed using hard-core bosons as a book-keeping trick. I show how this bond operator formalism can be used to express the Hamiltonians for generic quantum spin systems, but especially Affleck, Kennedy, Lieb, and Tasaki (AKLT) and AKLT-like models. The AKLT class of models is an extension of the AFM Heisenberg paradigm, with localized quantum spins on a lattice directly coupled to their neighbors on adjacent sites. While the ground-state energy and wave function of most AFM Heisenberg Hamiltonians are unknown, they are known explicitly in the fixed-point AKLT Hamiltonian for the standard cases of the spin-1 chain, spin-3/2 honeycomb lattice, and spin-2 in a chain and square lattice. We confirm that our bond-operator approach reproduces the original results of Affleck et al. [Phys. Rev. Lett. 59, 799, 1987]. Further- more, we use the bond operator framework to construct an “update projector” which is the main ingredient for valence-bond quantum Monte Carlo sampling. We derive a valence-bond “coherent state” as a superposition of bond creation operators and show that its energy expectation value can be evaluated exactly for a broad class of quantum antiferromagnets.

#### Recommended Citation

DO, HUU TRAN, "A VALENCE-BOND OPERATOR ALGEBRA FOR QUANTUM SPIN MODELS AND ITS APPLICATIONS" (2021). *Electronic Theses and Dissertations*. 2095.

https://egrove.olemiss.edu/etd/2095