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Odd-frequency Cooper pairs with chiral symmetry emerging at the edges oftopological superconductors are a useful physical quantity for characterizingthe topological properties of these materials. In this work, we show that theodd-frequency Cooper pair amplitudes can be expressed by a winding numberextended to a nonzero frequency, which is called a `spectral bulk-boundarycorrespondence,' 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 $\sim1/z$ (where $z$ is a complex frequency) that reflects the presence of atopological surface Andreev bound state, whereas the amplitudes in the secondcategory have the regular form $\sim z$ and are regarded as non-topological. Wediscuss the topological phase transition by using the coefficient in the lattercategory, which undergoes a power-law divergence at the topological phasetransition point and is used to indicate the distance to the critical point.These concepts are established based on several concrete models, including aRashba nanowire system that is promising for realizing Majorana fermions.

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