Odd-frequency pairs in chiral symmetric systems: Spectral bulk-boundary correspondence and topological criticality

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 ∼ 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 ∼ 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. arXiv:1809.05687v2 [cond-mat.supr-con]

Bulk odd-frequency pairing in the superconducting Su-Schrieffer-Heeger model

“…This is similar to what has been reported in multiband superconductors [46][47][48], double quantum dots [50,51], and double nanowires [35,52]. Moreover, when the system is of finite length, we find that, at the edges, the low-frequency odd-ω components in the topological phase develop larger values than the even-ω terms due to MZMs [3,5,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], an effect we explicitly associate with the topological bulk invariant through the so-called spectral edge boundary correspondence [66,67].…”

“…In the topological phase in finite length systems we showed that the low-frequency odd-frequency amplitudes are enhanced at the edges of the system due to the presence of Majorana zero modes. Furthermore, we demonstrated that the spectral bulk-boundary correspondence reveals the relation between intrasublattice odd-frequency correlations and winding number, thus extending previous studies [66,67] to systems with intrinsic staggering, such as the SC SSH model. Our results highlight odd-frequency pairing as a bulk phenomenon that goes beyond the induced effect in junctions and does not require any external actor, such as interfaces in junctions, but instead, this bulk effect solely relies on the intrinsic staggered nature of the system.…”

“…Then, following Refs. [66,67], we numerically find that, at small ω, the previous expression can be written as…”

“…Further understanding of the relation between odd-ω correlations and the topological phase can be obtained from a spectral bulk-boundary correspondence (SBBC) [66,67]. The SBBC tells us that some components of the anomalous Green's function are related to an extended version of the winding number defined in the bulk for chiral symmetric systems.…”

“…The relation between odd-ω pair amplitudes and the topological number for a chiral symmetric system is given by the spectral bulk-boundary correspondence [66,67]. For chiral symmetric systems, following the spectral bulk-boundary correspondence, we can write [66,67]…”