Odd-Frequency Pairs in Chiral Symmetric Systems: Spectral Bulk-Boundary Correspondence and Topological Criticality

TL;DR

This paper introduces a spectral bulk-boundary correspondence linking odd-frequency Cooper pairs to topological invariants in chiral symmetric systems, revealing their role in topological phase transitions and criticality.

Contribution

It establishes a new framework connecting odd-frequency pair amplitudes with a nonzero frequency winding number, providing insights into topological phase transitions in superconductors.

Findings

Spectral bulk-boundary correspondence relates odd-frequency pairs to bulk spectral features.

Classification of odd-frequency pairs into topological and non-topological categories.

Identification of a power-law divergence as an indicator of topological phase transition.

Abstract

Odd-frequency Cooper pairs with chiral symmetry emerging at the edges of topological superconductors are a useful physical quantity for characterizing the topological properties of these materials. In this work, we show that the odd-frequency Cooper pair amplitudes can be expressed by a winding number extended to a nonzero frequency, which is called a `spectral bulk-boundary correspondence,' and can be evaluated from the spectral features of the bulk. The odd-frequency Cooper pair amplitudes are classified into two categories: the amplitudes in the first category have the singular functional form $\sim 1/z$ (where $z$ is a complex frequency) that reflects the presence of a topological surface Andreev bound state, whereas the amplitudes in the second category have the regular form $\sim z$ and are regarded as non-topological. We discuss the topological phase transition by using the coefficient in the latter category, which undergoes a power-law divergence at the topological phase transition point and is used to indicate the distance to the critical point. These concepts are established based on several concrete models, including a Rashba nanowire system that is promising for realizing Majorana fermions.