Topological Phase Transition and Z_{2} Index for S=1 Quantum Spin Chains

Area of Science:

• Condensed Matter Physics
• Quantum Mechanics
• Statistical Mechanics

Background:

• S=1 quantum spin systems are crucial for understanding exotic phases of matter.
• Short-ranged Hamiltonians with specific symmetries are key to exploring novel quantum phenomena.
• The Affleck-Kennedy-Lieb-Tasaki (AKLT) model serves as a foundational example in topological quantum matter.

Purpose of the Study:

• To rigorously prove the existence of a topological phase transition in S=1 quantum spin systems.
• To establish that the AKLT model resides in a nontrivial symmetry-protected topological phase.
• To introduce a Z2 index for characterizing gapped unique ground states.

Main Methods:

• Definition of a Z2 index for gapped unique ground states.
• Proof of the Z2 index's invariance under smooth deformation.
• Utilizing the Z2 index to demonstrate a topological phase transition.

Main Results:

• A rigorous proof for a topological phase transition between the AKLT model and trivial models.
• Demonstration that this transition is undetectable by conventional order parameters.
• Establishment of the AKLT model as a nontrivial symmetry-protected topological phase.

Conclusions:

• The Z2 index provides a robust tool for identifying and characterizing topological phases.
• Symmetry-protected topological phases represent a distinct and important class of quantum matter.
• This work rigorously confirms the topological nature of the AKLT model, advancing the field of quantum information and condensed matter theory.