Quantum Spin Chains

Specialeforsvar: Emil Aagaard Clausen

Titel: Quantum Spin Chains
Haldane’s Conjecture and Symmetry-Protected Topological Phases

Abstract: In this thesis, we explore the properties of quantum spin chains. The motivation for our studies stems from Haldane’s surprising 1983 conjecture, which states that the low-energy properties of the antiferromagnetic Heisenberg (AFH) chain differ qualitatively depending on whether the particles that constitute it have quantum spin S ∈ {1, 2, . . . } or S ∈ {1 2 , 3 2 . . . }. In particular, Haldane argued that the ground state is unique and gapped if and only if S ∈ {1, 2, . . . }. For S = 1, this conjectured ground state is the first theoretical example of a unique, gapped ground state in a non-trivial symmetry-protected topological (SPT) phase. Today, this SPT phase is known as the Haldane phase, and models like the Affleck-Kennedy-Lieb-Tasaki (AKLT) chain, which is rigorously known to belong to this phase, are used to study its exotic features. Since the late 1980s, the Haldane phase has been studied intensively, with physicists introducing various indices to characterize it. Between 2019 and 2021, Ogata found indices that are invariant within any SPT phase of quantum spin chains with unique, gapped ground states protected by on-site symmetries, time-reversal symmetry, and reflection symmetry. This remarkable achievement finally proved the existence and stability of the Haldane phase.
By studying the operator algebraic formulation of quantum spin chains, the AFH chain, the AKLT chain, and SPT indices, we aim to present some of the core principles and results in this fascinating field of study. To support our work, we have developed software that utilizes symmetries to diagonalize Hamiltonians on finite quantum spin chains. This software is written in the Rust
programming language and is available at [2].

Vejleder: Jan Philip Solovej
Censor:   Poul Hjorth, DTU