Odd-frequency superconducting pairing in Kitaev-based junctions

Abstract

We investigate odd-frequency superconducting correlations in normal-superconductor (NS) and short superconductor-normal-superconductor (SNS) junctions with the S region described by the Kitaev model of spinless fermions in one dimension. We demonstrate that, in both the trivial and topological phases, Andreev reflection is responsible for the coexistence of even-and odd-frequency pair amplitudes at interfaces, while normal reflections solely contribute to odd-frequency pairing. At NS interfaces we find that the odd-frequency pair amplitude exhibits large, but finite, values in the topological phase at low frequencies. This enhancement is due to the emergence of a Majorana zero mode at the interface, but notably there is no divergence and a finite odd-frequency pair amplitude also exists outside the topological phase. We also show that the local density of states and local oddfrequency pairing can be characterized solely by Andreev reflections deep in the topological phase. Moreover, in the topological phase of short SNS junctions, we find that both even-and odd-frequency amplitudes capture the emergence of topological Andreev bound states. For a superconducting phase difference 0 < φ < π the odd-frequency magnitude exhibits a linear frequency (∼ |ω|) dependence at low-frequencies, while at φ = π it develops a resonance peak (∼ 1/|ω|) due to the protected Majorana zero modes.

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].…”

“…On the other hand, the low-frequency odd-ω components develop a huge increase near the edges, from where they decay, but do not vanish, towards the bulk of the system in an exponentially oscillatory fashion [(b)]. We attribute this enhancement to the emergence of MZMs in the topological phase, in a similar way as in other topological systems [3,5,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. Moreover, in order to identify the emergence of odd-ω amplitudes in the bulk as well as compare with the results presented in the previous section, we present in the inset of Fig.…”

“…Moreover, under the presence of a spin field in such junctions, e.g., from a magnetic field or spin-orbit coupling, spins get mixed, which then gives rise to even more exotic spin-triplet odd-ω correlations [2][3][4][5]. This has allowed for an understanding of a number of exotic phenomena: long-range proximity effect [2,15] and paramagnetic Meissner effect [16][17][18][19][20] in superconductor-ferromagnet junctions, anomalous proximity effect in spin-triplet superconductor junctions [21][22][23][24][25][26], Majorana zero modes [3,5,[27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], and surface impedance in topological superconductors [43]. Nowadays, induced odd-ω pairing in junctions is well established even experimentally [44,45], which reflects the vast activity made towards the understanding of this induced effect.…”

“…Since these nanowires represent one of the most investigated platforms for one-dimensional topological superconductivity [57][58][59], the reported staggered-induced topological phase is within experimental reach and its fundamental understanding is, therefore, important. Generally, this can be achieved by a proper investigation of the emergent superconducting pair correlations, which has not been carried out so far; odd-ω correlations are expected to play an important role due to the topological nature of this exotic staggered phase, as has been shown for other topological systems [3,5,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. As a consequence, a general exploration of systems with intrinsic staggered properties is timely, can reveal the emergence of exotic bulk superconducting properties, and add fundamental understanding to these effects.…”

“…This system has been shown to exhibit a one-dimensional topological superconducting phase with Majorana zero modes (MZMs) [63,64]. MZMs appear exponentially localized at the ends and their pair amplitudes have been shown to exhibit odd-ω symmetry [3,5,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. MZMs have attracted an enormous amount of attention due to their potential use for building topological protected qubits [57][58][59]65].…”