Universal broadening of zero modes: A general framework and identification.

Area of Science:

• Quantum mechanics
• Condensed matter physics
• Random matrix theory

Background:

• Perturbed Hermitian operators often exhibit zero modes, which can be topological or system-specific.
• Understanding the behavior of these zero modes under perturbation is crucial for various physical systems.

Purpose of the Study:

• To analyze the behavior of the smallest eigenvalues of perturbed Hermitian operators with zero modes.
• To unify and extend existing results in chiral random matrix theory and effective field theory.
• To identify experimental indicators for distinguishing zero modes.

Main Methods:

• Leading-order analysis of small generic perturbations.
• Application of concepts from chiral random matrix theory and effective field theory.
• Utilizing the Altland-Zirnbauer classification of symmetric spaces.

Main Results:

• The smallest eigenvalues broaden to a Gaussian random matrix ensemble of size ν×ν, where ν is the number of zero modes.
• The broadened zero modes decouple from bulk eigenvalues.
• The scaling of zero modes with volume differs from bulk eigenvalues, providing an experimental signature.

Conclusions:

• The study unifies and extends results in random matrix theory and effective field theory.
• The proposed experimental indicator can help identify topological zero modes.
• The findings are applicable across all 10 symmetric spaces in the Altland-Zirnbauer classification.