# Odd-frequency pairing in a superconductor coupled to two parallel nanowires

###### Abstract

We study the behavior of Cooper pair amplitudes that emerge when a two-dimensional superconductor is coupled to two parallel nanowires, focusing on the conditions for realizing odd-frequency pair amplitudes in the absence of spin-orbit coupling or magnetism. In general, any finite tunneling between the superconductor and the two nanowires induces odd-frequency spin-singlet pair amplitudes in the substrate as well as a substantial odd-frequency interwire pairing. Since these amplitudes are odd in spatial parity, they do not directly impact the local observables. However, in the regime of strong superconductor-nanowire tunneling, we find that the presence of two nanowires allows for the conversion of non-local odd-frequency pairing to local even-frequency pairing. By studying this higher-order symmetry conversion process, we are able to characterize the effect of the odd-frequency pairing in the superconductor on local quantities accessible by experiments. Specifically, we find that odd-frequency pairing has a direct impact on the local density of states of the superconductor and on the maximum Josephson current, measurable using Josephson scanning tunneling microscopy.

## I Introduction

The study of proximity-induced superconductivity in one-dimensional (1D) nanowires has generated a great deal of interest in recent years, driven primarily by their potential for realizing states with non-Abelian statistics holding the promise for topological quantum computation.Alicea (2012); Stern and Lindner (2013); Sarma et al. (2015) The simplest proposals involve single nanowires with Rashba spin-orbit coupling in proximity to a conventional -wave superconductor in the presence of an applied magnetic field.Lutchyn et al. (2010); Oreg et al. (2010) However, the non-Abelian Majorana bound states found in these single nanowire systems are Ising anyons and cannot be used to construct all gates necessary for universal quantum computationTrebst et al. (2008) in contrast to e.g. Fibonacci anyons.Trebst et al. (2008) Fibonacci anyons are composed of parafermions, exotic excitations which can be realized by coupling two nanowires with Rashba spin-orbit coupling to an -wave superconductor, in the absence of a magnetic field.Klinovaja and Loss (2014); Gaidamauskas et al. (2014)

Another important feature of double nanowire systems, not present in single nanowires, is the possibility for crossed Andreev reflection processes, which can play a significant role in the physics of these systems.Reeg et al. (2017) Given that regular Andreev reflection processes have been shown to be explicitly related to the generation of Cooper pairs with unconventional symmetries in single wire systems,Cayao and Black-Schaffer (2017, 2018) it is interesting to consider the pair symmetries arising in double wire systems.

It is well-known that the fermionic nature of electrons tightly constrains the allowed symmetries of the Cooper pairs and thus the superconducting gap function. Specifically, in the limit of equal-time pairing and a single-component gap, spatially even-parity gap functions (like - or -wave) must correspond to spin-singlet states, while odd-parity gap functions (- or -wave) must correspond to spin-triplet states. However, if the electrons comprising the condensate are paired at unequal times the superconducting gap can also be odd in time or, equivalently, odd in frequency (odd-). In that case the condensate can be even in spatial parity and spin-triplet or odd-parity and spin-singlet. This possibility, originally posited for He by BerezinskiiBerezinskii (1974) and then later for superconductivity,Kirkpatrick and Belitz (1991); Belitz and Kirkpatrick (1992); Balatsky and Abrahams (1992) is intriguing both because of the unconventional symmetries which it permits and for the fact that it represents a class of hidden order, due to the vanishing of all equal-time correlations.

While the thermodynamic stability of intrinsically odd- phases has, so far, only been discussed as a theoretical possibility,Coleman et al. (1993, 1994, 1995); Heid (1995); Belitz and Kirkpatrick (1999); Solenov et al. (2009); Kusunose et al. (2011); Fominov et al. (2015) significant progress has been made understanding the way in which odd- pairing can be induced by altering a system’s conventional superconducting correlationsBergeret et al. (2001, 2005); Yokoyama et al. (2007); Houzet (2008); Eschrig and Löfwander (2008); Linder et al. (2008); Crépin et al. (2015); Yokoyama (2012); Black-Schaffer and Balatsky (2012, 2013a); Triola et al. (2014); Tanaka and Golubov (2007); Tanaka et al. (2007); Linder et al. (2009, 2010); Tanaka et al. (2012); Parhizgar and Black-Schaffer (2014); Triola et al. (2016); Triola and Balatsky (2016). The best established example is found in ferromagnet-superconductor junctions,Bergeret et al. (2001, 2005); Yokoyama et al. (2007); Houzet (2008); Eschrig and Löfwander (2008); Linder et al. (2008); Crépin et al. (2015) in which experiments have observed key signatures of odd- spin-triplet pair correlationsDi Bernardo et al. (2015a, b), despite using conventional spin-singlet -wave superconductors. For a modern review of odd- superconductivity, see Ref. [dccclxxx(45)].

While the evidence for odd- pairing is strongest in ferromagnetic junctions, odd- pairing is also predicted to be ubiquitous at interfaces between normal metals (N) and conventional spin-singlet superconductors (S) Tanaka and Golubov (2007); Tanaka et al. (2007) and appears to be intimately related to Andreev reflectionCayao and Black-Schaffer (2017, 2018). However, when both the N/S interface and the normal metal are trivial in spin space, the only odd- pairing that can be induced is in the odd spatial parity channel, which always vanishes when averaged over the Fermi surface or probed locally. Therefore, the community has shown less interest in such spin inactive systems with neither magnetism nor spin-orbit coupling.

In this work we consider two parallel nanowires with no spin-orbit coupling or magnetism, both coupled to a superconducting substrate with conventional spin-singlet -wave order parameter. We examine the symmetries of the emergent Cooper pair amplitudes, focusing on the appearance of odd- superconductivity and its physical consequences. In particular, by expanding the anomalous Green’s function to leading order in the superconductor-nanowire tunneling parameters, we find that odd- spin-singlet odd parity Cooper pair amplitudes emerge in both the substrate and the interwire channel. The appearance of these pairing amplitudes can be understood by extending previous results for N/S junctionsTanaka and Golubov (2007); Tanaka et al. (2007) and multiband systemsBlack-Schaffer and Balatsky (2013b); Komendová et al. (2015); Komendová and Black-Schaffer (2017) to this double nanowire setup. However, despite the non-local odd parity symmetry for the odd- pairings we find that they have a significant impact on local and measurable quantities. This is due to higher-order tunneling processes between the substrate and the two nanowires, which allow for conversion between non-local odd- pairing and local even-frequency pairing. We derive explicit expressions characterizing this symmetry conversion process to infinite order in the superconductor-nanowire tunneling, which shows that two nanowires are needed for this process. We then study the features generated by these conversion processes in two local and highly accessible experimental observables: the local density of states (LDOS), measurable by scanning tunneling microscopy (STM), and local Josephson current, measurable by Josephson STMŠmakov et al. (2001); Naaman et al. (2001). These results show how odd- superconductivity is generated in double nanowire systems and that, despite its non-local nature, odd- pairing has profound effects on easily measurable local quantities.

The remainder of this work is organized as follows. In Sec. II we introduce the model we will use to study the double nanowire-superconductor system and define the Green’s functions used throughout this work. In Sec. III we derive the perturbative corrections to the anomalous Green’s functions and establish the existence of odd- pairing in the presence of finite nanowire-superconductor tunneling. In Sec. IV we examine the higher-order corrections to the Green’s functions and show that it is exactly the presence of two nanowires that allows the reconversion of odd- amplitudes back to even- amplitudes with novel properties. In Sec. V we study the effect of this higher-order symmetry conversion on local observables, finding significant features in the results for both LDOS and local Josephson current. Finally, in Sec. VI we conclude our work.

## Ii Model

We wish to study the emergent symmetries of superconductivity in a physical system comprised of two parallel nanowires separated by a distance coupled to a conventional superconducting substrate, which we model as a two-dimensional (2D) spin-singlet -wave superconductor. Throughout this work we assume the nanowires are non-magnetic and possess no appreciable spin-orbit coupling, such that the system has only trivial spin structure. To model this system we employ a Hamiltonian of the form where

(1) | ||||

where () creates (annihilates) a quasiparticle state in the superconducting substrate with spin at position along the axis perpendicular to the nanowires, with momentum along the -axis. Likewise, () creates (annihilates) a quasiparticle state with spin and momentum in nanowire . Moreover, is the normal state quasiparticle dispersion in the superconductor along the -axis measured from the chemical potential , while is the quasiparticle dispersion in the nanowire measured from the chemical potential . The superconducting substrate is further described by the order parameter which has spin-singlet -wave symmetry. Finally, is the tunneling amplitude coupling the superconductor to the nanowire located at position .

To study the emergent electronic properties of the system described by the Hamiltonian in Eq. (1), we begin by noting that the presence of the nanowires breaks translation-invariance along the -direction, but not the -direction, thus allowing us to keep a reciprocal coordinate. As a consequence, we define the normal and anomalous Green’s functions for the superconductor as:

(2) | ||||

where is an imaginary time, and is the usual -ordering operator for fermions. Similarly, we define the normal and anomalous Green’s functions for the electronic excitations in the two nanowires as:

(3) | ||||

where, due to coupling through the superconducting substrate, we allow for both intrawire () and interwire () correlations in Eq. (3).

For convenience we combine the normal and anomalous Green’s functions into the following Nambu space Green’s functions for the superconductor and the nanowires:

(4) | ||||

where, we have Fourier-transformed from imaginary time to Matsubara frequency and where we have used , , , and .

In the absence of tunneling between the nanowires and the superconductor, i.e. , it is straightforward to show that the Green’s functions in Eqs. (4) are given by:

(5) | ||||

where and are the 22 identity and Pauli matrices in particle-hole space, respectively, and we have defined . At finite tunneling, , the Green’s functions in Eqs. (4) satisfy the following Dyson equations:

(6) | ||||

where we have omitted the explicit dependence and , since both of these quantities are conserved. The self-energies are defined as:

(7) | ||||

## Iii Odd-frequency Pairing

By iterating Eq. (6) we can compute the Green’s functions, and , in terms of the bare Green’s functions given in Eq. (5) to arbitrary order in powers of the tunneling parameters . In the limit of weak coupling between the nanowires and the superconducting substrate, the physics is dominated by the leading order terms in and we have:

(8) | ||||

where

(9) | ||||

By inserting the expressions from Eqs. (5) into Eqs. (9) we can explicitly calculate the leading order corrections to the Green’s functions in both the nanowires and the superconducting substrate. Without loss of generality, we assume for concreteness, that , in Eq. (9). To study the superconducting pairing, we focus only on the anomalous parts of the Green’s functions in Eqs. (9) and find:

(10) | ||||

where, for brevity, we have defined the functions:

(11) | ||||

These two functions are both manifestly even in Matsubara frequency and determine the interference between the corrections coming from the left and right nanowires. Moreover, the function:

(12) |

contributes to the determination of the magnitude of the proximity-induced superconducting pairing within the nanowires and also governs the decay of the interwire correlations as a function of their separation .

Already here we notice that the proximity-induced pairing within the nanowires , possesses both an even- term and an odd- term. Furthermore, we notice that, while the even- term is non-zero in both the intrawire and interwire channels, the odd- terms belong strictly to the interwire channel. Moreover, by permuting the wire index, it is easy to see that the odd- pair amplitude in the nanowires is also odd in the wire index, consistent with the constraints imposed by Fermi-Dirac statistics.Triola and Balatsky (2016) We also note from Eqs. (10) that pairing amplitudes with new symmetries emerge within the superconductor as a consequence of coupling to the nanowires. While these terms are more complicated than the pair amplitudes in the nanowires, it is clear that these amplitudes possess both even- and odd- terms.

### iii.1 Odd- pairing in the nanowires

Having demonstrated in Eq. (10) that multiple odd- pairing components are induced in a double wire system, we now study the nature of these odd- pairing correlations in more depth, starting with the proximity-induced interwire odd- pairing.

From Eq. (10) it is clear that, in general, the interwire pair amplitude possesses both even- and odd- terms. Furthermore, since essentially all of the complications arising from the coupling to the superconducting substrate take the form of a multiplicative prefactor, it is quite easy to obtain the ratio of the odd- pairing to the even- pairing, given by:

(13) |

where we have performed the analytic continuation to real frequency, , to make contact with the physical spectrum of the system. This simple ratio allows us to determine the precise conditions for which we expect the odd- pair amplitudes to dominate over the even- amplitudes. From Eq. (13) it is clear that the two pair symmetries will be equal in magnitude at the frequencies , see also Fig. 1(a). Furthermore, it is clear that the even- pair amplitudes vanish exactly at . Therefore, so long as , and and possess the same sign, the interwire pairing will be strictly odd- at , as also illustrated Fig. 1(b). This pure odd- interwire pairing criteria can be engineered by adjusting the chemical potentials within the two nanowires, for example by gating.

Having discussed the relative size of the even- and odd- interwire pair amplitudes, we next turn our attention to the overall magnitude of the interwire pairing. First, comparing the above criteria for an odd--dominated interwire channel to the expressions in Eq. (10), we see that the frequencies for which align precisely with the poles in the denominator of the total interwire pair amplitude. Therefore, the denominator should not have a deleterious effect on the odd- pairing. Then, neglecting the denominator in Eq. (10), we see that the magnitude of the interwire pairing is determined by only three factors: , , and . The hopping amplitudes depend sensitively on the microscopic model of the nanowire-superconductor interface. A precise determination of their values is clearly beyond the scope of this work and, for our purposes, these parameters are simply constants characterizing the interface. Further, the gap of the superconducting substrate may be adjusted by choosing different substrates, and is therefore an external parameter. It is thus the function that carries all the relevant dependences for the interwire pairing, such as information about the kinetic energy of the substrate, as well as the distance between the two nanowires.

Since we are primarily interested in interwire pairing, we focus on evaluating the function in Eq. (12) for , and without loss of generality we set , . In this case, it is straightforward to show using contour integration, that:

(14) |

where we have defined , , , and . From Eq. (14) we readily see that for large nanowire separations, , the magnitude of the interwire pair amplitudes go as , which is consistent with the expectation that the pairing correlations should be suppressed when exceeds the superconducting coherence length, .

### iii.2 Odd- pairing in the superconducting substrate

We now turn our attention to the pair symmetry of the superconducting substrate. The corrections to the anomalous Green’s function in the substrate due to the presence of the nanowires are given by Eq. (10). It is straightforward to evaluate the integrals in the complex plane, and we find:

(15) | ||||

where, for compactness, we have defined the following terms:

(16) | ||||

(17) | ||||

(18) | ||||

with the same definitions for , , and as in Eq. (14). Clearly, all three functions in Eqs. (16)-(18) are even in Matsubara frequency , since they depend only on . Therefore, we find that the only odd- term in the anomalous Green’s function Eq. (15) is the term proportional to . Furthermore, by inspecting the functions in Eqs. (16)-(18), we find that the coefficient of the odd- pair amplitude is odd in spatial parity, , while the coefficients associated with the even- amplitudes are both even in parity, and , fully consistent with the constraints imposed by Fermi-Dirac statistics.

Another feature of the expressions in Eq. (15), is that, while the coordinate dependence is somewhat complicated in general, we can see that when and are sufficiently far from both nanowires, i.e. , all corrections are exponentially suppressed. Comparing the length scale associated with this suppression to the superconducting coherence length, , we find that the exponential suppression begins to occur on length scales even shorter than the coherence length. Interestingly, there is no preferential suppression of the odd- terms coming from the exponential factors; even- and odd- amplitudes get comparably suppressed, similar to the proximity-induced pairing in the nanowires discussed in the previous subsection.

## Iv Higher-order pair symmetry conversion

In the previous section we demonstrated that odd- pair amplitudes can be induced in both the superconducting substrate and the interwire channel of the nanowires. These results were obtained using a perturbative expansion in the hopping amplitudes between the nanowires and the substrate. The benefit of such a calculation is that it is relatively simple and the symmetries of the pair amplitudes can be made manifest. However, such an analysis is limited to small values of , as it ignores higher-order terms in the expansion. In this section we solve the problem exactly using a -matrix approach and thus incorporate the effect of all higher-order tunneling processes on the pair amplitudes. We limit ourselves to pair amplitudes within the substrate, since the odd- pairing in the nanowires lies only in the interwire channel. We compare our exact results to the perturbative ones in the previous section, and most importantly, demonstrate novel features of the pair symmetry which only emerge at higher-orders.

To perform our analysis, we return to the Dyson equation describing the exact Green’s functions of the superconducting substrate, Eq. (6). By iterating this equation we find that above order , cross-terms begin to emerge which involve a propagation between the nanowires of the form, . At higher orders more of these terms emerge, thus significantly complicating an evaluation using a -matrix. Therefore, we start by neglecting the right nanowire and exactly solve the problem of the superconducting substrate coupled to the left nanowire (L+SC) only. We then turn our attention to the combined L+SC system in the presence of the right nanowire and solve the problem (L+SC+R) exactly. In this way we account for all cross-terms while still being able to proceed analytically.

Since we are dealing with Green’s functions whose arguments are a mixture of momentum, , and positions, ,, the position of the poles for these functions depend on the complex-valued frequency and the momentum, . Thus, the usual relationship between the Matsubara Green’s functions and the retarded Green’s functions does not necessarily hold. To keep track of both the Matsubara and retarded Green’s functions we derive all expressions in this section for Green’s functions with a generic complex frequency, . In this way, all results apply equally well to Matsubara, retarded, and advanced Green’s functions.

### iv.1 Left nanowire + superconductor

In the presence of only the left nanowire it is straightforward to show that Eq. (6) can be written as:

(19) |

where the -matrix is defined as

(20) |

The bare Green’s function of the substrate appearing in Eqs. (19) and (20), , is a function of position , momentum , and frequency with the general structure given by

(21) |

The coefficients , , and are complicated functions of , , and and given in Appendix A, see Eqs. (46) and (48). However, for compactness we here suppress the and function arguments as they will not change throughout this derivation. Importantly, the functional dependences imply that all three functions are even under the transformations: , , and . Next, using the general form in Eq. (21) together with the definition of in Eq. (7), it is straightforward to show that the -matrix in Eq. (20) takes the form:

(22) |

where we have defined the coefficients:

(23) | ||||

While complicated expressions, we notice that these coefficients inherit the symmetries of , , and . Finally, inserting Eq. (22) into Eq. (19) we find the Green’s function of the L+SC system to infinite order in the tunneling :

(24) |

where we have defined

(25) | ||||