Topology and Edge Modes in Quantum Critical Chains

Area of Science:

• Condensed Matter Physics
• Topological Phases of Matter
• Quantum Field Theory

Background:

• Symmetry-protected topological phases exhibit unique edge states.
• Understanding critical points between these phases is crucial for classifying topological matter.
• Previous studies often required gapped bulk states to observe edge modes.

Purpose of the Study:

• To demonstrate that topology can protect zero-energy edge modes at critical points between 1D symmetry-protected topological phases.
• To investigate the existence of these edge modes without requiring gapped bulk degrees of freedom.
• To classify critical phases in the noninteracting BDI class.

Main Methods:

• Development of an intuitive picture for edge mode existence in noninteracting spinless fermions with time-reversal symmetry (BDI class).
• Utilizing a topological invariant based on the zeros and poles of a complex function within the unit circle.
• Leveraging conformal field theory (CFT) to label and classify critical phases.

Main Results:

• Identification of exponentially localized, zero-energy edge modes at critical points.
• A full classification of critical phases in the noninteracting BDI class, labeled by central charge (c) and topological invariant (ω).
• Demonstration that numerical simulations show stability of these topological edge modes against interactions and disorder.

Conclusions:

• Topology provides a mechanism to protect edge modes at critical points, even in gapless systems.
• The study establishes a robust classification of critical phases in the BDI class.
• The findings suggest potential for robust topological phenomena in realistic, interacting, and disordered systems.