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, both of which vanish locally. Interestingly, 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 identify a notable effect of the odd-frequency pairing in the superconductor on local quantities accessible by experiments. Specifically, we find that the odd-frequency pairing plays a direct role in the emergence of certain subgap features in the local density of states, and, importantly, it is responsible for a reduction of the maximum Josephson current between the two nanowires, measurable using Josephson scanning tunneling microscopy. We discuss ways to control the sizes of these effects induced by odd-frequency superconductivity by tuning the parameters describing the nanowires.
We show that the topological Majorana modes in nanowires much longer than the superconducting coherence length are adiabatically connected with discrete zero-energy states generically occurring in short nanowires. We demonstrate that these zero-energy crossings can be tuned by an external magnetic field and are protected by the particle-hole symmetry. We study the evolution of the low-energy spectrum and the splitting oscillations as a function of magnetic field, wire length, and chemical potential, manifestly establishing that the low-energy physics of short wires is related to that occurring…
We consider a system of two one-dimensional nanowires coupled via an $s$-wave superconducting strip, a geometry that is capable of supporting Kramers pairs of Majorana fermions. By performing an exact analytical diagonalization of a tunneling Hamiltonian describing the proximity effect (via a Bogoliubov transformation), we show that the excitation gap of the system varies periodically on the scale of the Fermi wavelength in the limit where the interwire separation is shorter than the superconducting coherence length. Comparing with the excitation gaps in similar geometries containing only dire…
Odd-frequency superconductivity, originally proposed by Berezinskii in 1974, is an exotic phase of matter in which Cooper pairing between electrons is entirely dynamical in nature. The pair potential is an odd function of frequency, leading to a vanishing static superconducting order parameter and exotic types of pairing seemingly inconsistent with Fermi statistics. Motivated by recent experimental progress in the realization of Majorana zero modes in semiconducting nanowires, we show that odd-frequency superconductivity generically appears in a spin-polarized nanowire coupled to Majorana zero…
We first show the realization of exceptional points in a non-Hermitian superconducting system based on a conventional superconductor and then demonstrate that, surprisingly, the system hosts odd-frequency pairing, solely generated by the non-Hermiticity. While there is a coexistence of even- and odd-frequency pairs under general conditions, we find that the even-frequency term vanishes at the exceptional degeneracies, leaving only odd-frequency pairing. This exceptional odd-frequency pairing is directly given by the imaginary part of the eigenvalues at the exceptional points and can be measure…
We theoretically investigate the transport properties of a quasi one dimensional ferromagnet-superconductor junction where the superconductor consists of mixed singlet and triplet pairings. We show that the relative orientation of the stoner field ($\bf{\tilde{h}}$) in the ferromagnetic lead and the $\bf{d}$ vector of the superconductor acts like a on-off switch for the zero bias conductance of the device. In the regime, where triplet pairing amplitude dominates over the singlet counterpart (topological phase), a pair of Majorana zero modes appear at each end of the superconducting part of the…
In this Letter, we first formulate an effective theory, which generally captures long-range proximity effects of a surface system weakly coupled to an s-wave superconductor. The long-range proximity effects include both the emergent long-range pairing and hopping interactions in the surface system. We then model the Rashba spin-orbit-coupled nanowire in proximity with an s-wave superconductor by taking into account the emergent nonlocal effects in the weak-coupling limit. In this limit the induced superconducting pair potential is found much smaller than that of the host superconductor, which …
It is difficult to unambiguously confirm the existence of Majorana zero modes (MZMs) due to the absence of smoking-gun signatures in charge transport measurements. Recent studies suggest that the spin degree of freedom of MZMs may provide an alternative detection method. We study the spin properties of the superconducting state in Majorana nanowires and the associated unconventional Josephson effect with realistic experimental parameters taken from [Zhang {\it et al.}, Nature 556, 74 (2018)]. For a superconducting thin film with in-plane polarized spin-triplet pairing, an out-of-plane electric…
A semiconducting nanowire with strong Rashba coupling and in proximity of a superconductor hosts Majorana edge modes. An array of such nanowires with inter-wire coupling gives an approximate description of a two-dimensional topological superconductor, where depending on the strength of the magnetic field and the chemical potential, a rich phase diagram hosting trivial and different types of non-trivial phases can be achieved. Here, we theoretically consider such a two-dimensional assembly of spin-orbit coupled superconducting nanowires and calculate the collective tunneling conductance between…
We investigate the induced superconducting pair correlations in junctions between a conventional spin-singlet $s$-wave superconductor and a disordered normal metal. Decomposing the pair amplitude based on its symmetries in the time domain, we demonstrate that the odd-time, or equivalently odd-frequency, spin-singlet $p$-wave correlations are both significant in size and entirely robust against random non-magnetic disorder. We find that these odd-frequency correlations can even be generated by disorder. Our results show that anisotropic odd-frequency pairing represent an important fraction of t…
Vertical graphene heterostructures made up of graphene layers separated by BN spacers allow for novel ways of tuning the interactions between electrons. We study the possibility of electron pairing mediated by modified repulsive interactions. Long range intravalley and short range intervalley interactions give rise to different anisotropic phases. We show that a superconducting state with gaps of opposite signs in different valleys, an odd momentum pairing state, can exist above carrier densities of $5-10 \times 10^{13} {\rm cm}^{-2}$. The dependence of the transition temperature on the differ…
We consider time-reversal-symmetric two-channel semiconducting quantum wires proximity coupled to an s-wave superconductor. We analyze the requirements for a nontrivial topological phase and find that necessary conditions are 1) the determinant of the pairing matrix in channel space must be negative, 2) inversion symmetry must be broken, and 3) the two channels must have different spin-orbit couplings. The first condition can be implemented in semiconducting nanowire systems where interactions suppress intra-channel pairing, while the inversion symmetry can be broken by tuning the chemical pot…
We consider normal-superconductor (NS) and superconductor-normal-superconductor (SNS) junctions based on one-dimensional nanowires with Rashba spin-orbit coupling and proximity-induced $s$-wave spin-singlet superconductivity and analytically demonstrate how both even- and odd-frequency and spin-singlet and -triplet superconducting pair correlations are always present. In particular, by using a fully quantum mechanical scattering approach, we show that Andreev reflection induces mixing of spatial parities at interfaces, thus being the unique process which generates odd-frequency pairing; on the…
We theoretically analyze the Andreev bound states and their coupling to external radiation in superconductor-nanowire-superconductor Josephson junctions. We provide an effective Hamiltonian for the junction projected onto the Andreev level subspace and incorporating the effects of nanowire multichannel structure, Rashba spin-orbit coupling, and Zeeman field. Based on this effective model, we investigate the dependence of the Andreev levels and the matrix elements of the current operator on system parameters such as chemical potential, nanowire dimensions, and normal transmission. We show that …
We have investigated the proximity effect in superconductor/ferromagnet junctions in a systematic manner to discuss the relationship between the zero-energy peak (ZEP) of the local density of states (LDOS) and spin-triplet odd-frequency pairing. By exactly solving the nonlinear Usadel equations, we have found that the ZEP is realized in a wide range of geometrical and material parameters in the case of the noncollinear magnetization. This strongly suggests the robustness of the ZEP induced by spin-triplet odd-frequency pairing in such systems. We also found that the crossover from singlet pair…
Odd-frequency superconductivity represents a truly unconventional ordered state which, in contrast to conventional superconductivity, exhibits pair correlations which are odd in relative time and, hence, inherently dynamical. In this review article we provide an overview of recent advances in the study of odd-frequency superconducting correlations in one-dimensional systems. In particular, we focus on recent developments in the study of nanowires with Rashba spin-orbit coupling and metallic edges of two-dimensional topological insulators in proximity to conventional superconductors. These syst…
The novel idea that spin-orbit coupling (SOC) and $s$-wave pairing can lead to induced $p$-wave pairing at strong magnetic limit has stimulated widespread interests in the whole community for the searching of Majorana Fermions (MFs), a self-hermitian particle, in semiconductor-superconductor hybrid structures. However, this system has several inherent limitations that prohibit the realization and identification of MFs with the major advances of semiconductor nanotechnology. We show that these limitations can be resolved by replacing the $s$-wave superconductor with type-II Fulde-Ferrell (FF) s…
We discuss pairing signatures in mesoscopic nanowires with variable attractive pairing interaction. Depending on wire length, density, and interaction strength, these systems realize a simultaneous bulk-to-mesoscopic and BCS-BEC crossover, which we describe in terms of the parity parameter that quantifies the odd-even energy difference and generalizes the bulk Cooper pair binding energy to mesoscopic systems. We show that the parity parameter can be extracted from recent measurements of conductance oscillations in SrTiO$_3$ nanowires by G. Cheng et al. [Nature 521, 196 (2015)], where it marks …
We theoretically predict proximity-induced odd-frequency (odd-$ω$) pairing on the surface of a Weyl nodal loop semimetal, characterized by a nodal loop Fermi surface and drumhead-like surface states (DSSs), attached to conventional spin-singlet $s$-wave superconducting leads. Due to the complete spin polarization of the DSS, odd-$ω$ equal-spin triplet pairing is present, and we show that it gives rise to a finite Josephson current. Placing an additional ferromagnet in the junction can also generate odd-$ω$ mixed-spin triplet pairing, but the pairing and current is not affected if the magnetiza…
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…
Among the different platforms to engineer Majorana fermions in one-dimensional topological superconductors, topological insulator nanowires remain a promising option. Threading an odd number of flux quanta through these wires induces an odd number of surface channels, which can then be gapped with proximity induced pairing. Because of the flux and depending on energetics, the phase of this surface pairing may or may not wind around the wire in the form of a vortex. Here we show that for wires with discrete rotational symmetry, this vortex is necessary to produce a fully gapped topological supe…
Odd frequency (odd-$ω$) electron pair correlations naturally appear at the interface between BCS superconductors and other materials. The detection of odd-$ω$ pairs, which are necessarily non-local in time, is still an open problem. The main reason is that they do not contribute to static measurements described by time-local correlation functions. Therefore, dynamical measurements, which depend on non-local time correlations, are suitable for detecting these pairs. In this work, we study the signatures of odd-$ω$ pairs in the supercurrent noise through a weak link between two superconductors a…
We point out that essentially all multi-band superconductors have an odd-frequency pairing component, as follows from a general symmetry analysis of even- and odd-frequency pairing states. We show that odd-frequency superconducting pairing requires only a finite band hybridization, or scattering, and non-identical intraband order parameters, of which only one band needs to be superconducting. Under these conditions odd-frequency odd-interband pairing is always present. From a symmetry analysis we establish a complete reciprocity between parity in band-index and frequency.
The Su-Schrieffer-Heeger model describes fermions that hop on a one-dimensional chain with staggered hopping amplitude, where the unit cell contains two sites, or two sublattices. In this work we consider the Su-Schrieffer-Heeger model with superconducting pairing and show that the sublattice index acts as an additional quantum number in the classification of Cooper pairs, giving rise to inter- and intra-sublattice odd-frequency pair correlations in the bulk. Interestingly, this system behaves as a two band superconductor where the bulk odd-frequency correlations depend solely on the intrinsic…
In this article we review recent progress in the understanding of multiband superconductivity and its relationship to odd-frequency pairing. We begin our discussion by reviewing the emergence of odd-frequency pairing in a simple two-band model, providing a brief pedagogical overview of the formalism. We then examine several examples of multiband superconducting systems in each case describing, both, the origin of the band degree of freedom and the nature of the odd-frequency pairing. Throughout, we attempt to convey a unified picture of how odd-frequency pairing emerges in these materials and …
A theoretical study is presented on the odd-frequency spin-singlet pairing that arises in nonuniform even-frequency superconductors as a consequence of broken translation symmetry. The effect of the odd-frequency pairing on the superfluid density and the spin susceptibility is analyzed by using the quasiclassical theory of superconductivity. It is shown that (1) the superfluid density is reduced by the formation of the odd-frequency singlet pairs and (2) the odd-frequency pairing increases the spin susceptibility even though its spin symmetry is singlet. The two unusual phenomena are related t…
Quantum dots are gate-defined within InSb nanowires, in proximity to NbTiN superconducting contacts. As the coupling between the dot and the superconductor is increased, the odd-parity occupations become non-discernible (erased) both above and below the induced superconducting gap. Above the gap, conductance in the odd Coulomb valleys increases until the valleys are lifted. Below the gap, Andreev bound states undergo quantum phase transitions to singlet ground states at odd occupancy. We observe that the apparent erasure of odd-parity regimes coincides at low-bias and at high-bias. This observ…
Using a simple theoretical model, we demonstrate the emergence of odd-frequency pair amplitudes in conventional Josephson junctions both in the absence of a voltage (DC effect) and in the presence of a finite voltage (AC effect). In both cases, we find that odd-frequency interlead pairing emerges whenever a Josephson current is expected to flow. Additionally, we show that the interlead spin-susceptibility is directly influenced by the presence of the odd-frequency pair amplitudes. Specifically, we find that the spin-susceptibility is suppressed when the odd-frequency component is the largest. …
We explore the role of proximity-induced odd-frequency pairing in the thermoelectricity of a ferromagnet when coupled to a conventional $s$-wave spin-singlet superconductor through a spin-active interface. By varying both the polarization and its direction in the ferromagnet and the interfacial spin-orbit interaction strength, we analyze the behavior of all proximity-induced pair amplitudes in the ferromagnet and their contributions to the thermoelectric coefficients. Based on our results for the Seebeck coefficient, we predict that odd-frequency spin-triplet Cooper pairs are much efficient th…
Superconducting correlations which are long-ranged in magnetic systems have attracted much attention due to their spin-polarization properties and potential use in spintronic devices. Whereas experiments have demonstrated the slow decay of such correlations, it has proven more difficult to obtain a smoking gun signature of their odd-frequency character which is responsible, e.g. for their gapless behavior. We here demonstrate that the magnetic susceptibility response of a normal metal in contact with a superconducting spin-valve provides precisely this signature, namely in form of an anomalous…
We show that quasiparticle interference (QPI) due to omnipresent weak impurities and probed by Fourier transform scanning tunneling microscopy and spectroscopy acts as a direct experimental probe of bulk odd-frequency superconducting pairing. Taking the example of a conventional $s$-wave superconductor under applied magnetic field, we show that the nature of the QPI peaks can only be characterized by including the odd-frequency pairing correlations generated in this system. In particular, we identify that the defining feature of odd-frequency pairing gives rise to a bias asymmetry in the QPI, …
We study and classify the proximity-induced superconducting pairing in a topological insulator (TI)-superconductor (SC) hybrid structure for SCs with different symmetries. The Dirac surface state gives a coupling between spin-singlet and spin-triplet pairing amplitudes as well as pairing that is odd in frequency for p-wave SCs. We also find that all SCs induce pairing that is odd in both frequency and orbital (band) index, with oddness in frequency and orbital index being completely interchangeable. The different induced pairing amplitudes significantly modifies the density of states in the TI…
We study proximity effect in diffusive ferromagnet/normal-metal/superconductor junction with precessing magnetization of the ferromagnet. We find that the odd-frequency pairing induced in the normal metal is modified by spin pumping from the ferromagnet and hence can be tuned by changing the precessional frequency. At the frequency corresponding to twice the superconducting gap, the odd-frequency pairing is strongly enhanced. We find a crossover from the even- to the odd-frequency superconductivity in the normal metal by tuning the precessional frequency. This gives a clearcut signature of the…
We study the emergence of odd-frequency superconducting pairing in UPt$_3$. Starting from a tight-binding model accounting for the nonsymmorphic crystal symmetry of UPt$_3$ and assuming an order parameter in the $E_{2u}$ representation, we demonstrate that odd-frequency pairing arises very generally, as soon as inter-sublattice hopping or spin-orbit coupling is present. We also show that in the low temperature superconducting $B$ phase, the presence of a chiral order parameter together with spin-orbit coupling, leads to additional odd-frequency pair amplitudes not present in the $A$ or $C$ pha…
We study the unconventional superconducting correlations caused by a single isolated magnetic impurity in a conventional s-wave superconductor. Due to the local breaking of time-reversal symmetry, the impurity induces unconventional superconductivity which is even in both space and spin variables but odd under time inversion. We derive an exact proportionality relation between the even-frequency component of the local electron density of states and the imaginary part of the odd-frequency local pairing function. By applying this relation to scanning tunneling microscopy spectra taken on top of …
A one-dimensional spin-orbit coupled nanowire with proximity-induced pairing from a nearby s-wave superconductor may be in a topological nontrivial state, in which it has a zero energy Majorana bound state at each end. We find that the topological trivial phase may have fermionic end states with an exponentially small energy, if the confinement potential at the wire's ends is smooth. The possible existence of such near-zero energy levels implies that the mere observation of a zero-bias peak in the tunneling conductance is not an exclusive signature of a topological superconducting phase even i…
We present a microscopic theory of fluctuation-mediated pairing mechanism in a two-dimensional extended Hubbard model. In contrast to conventional wisdom, odd-frequency spin-triplet pairing can be stabilized near the spin-density-wave critical point. Favorable conditions for the odd-frequency pairing are the presence of geometrical frustration and off-site Coulomb interaction.
In this paper, we investigate the relation between odd-frequency pairing and proximity effect in non-uniform Kitaev chain systems with a particular interest in the topological critical point. First, we correlate the odd-frequency pairing and Majorana fermion in a semi-infinite Kitaev chain, where we find that the spatial dependence of the odd-frequency pair amplitude coincides with that of the local density of states at low frequencies. Second, we demonstrate that, contrary to the standard view, the odd-frequency pair amplitude spreads into the bulk of a semi-infinite Kitaev chain at the topol…
The effects of spin independent hybridization potential and spin orbit coupling on two band superconductor with equal time s-wave inter band pairing order parameter is investigated theoretically. To study symmetry classes in two band superconductors the Gorkov equations are solved analytically. By defining spin singlet and spin triplet s wave order parameter due to two band degree of freedom the symmetry classes of Cooper pair are studied. For spin singlet case it is shown that spin independent hybridization generates Cooper pair belongs to even frequency spin singlet even momentum even band p…
The Meissner effect is one of the defining properties of superconductivity, with a conventional superconductor completely repelling an external magnetic field. In contrast to this diamagnetic behavior, odd-frequency superconducting pairing has often been seen to produce a paramagnetic Meissner effect, which instead makes the superconductor unstable due to the attraction of magnetic field. In this work we study how both even- and odd-frequency superconducting pairing contributes to the Meissner effect in a generic two-orbital superconductor with a tunable odd-frequency pairing component. By div…
In several members of the pnictide materials, spin density wave order coexists with superconductivity over a range of dopings and temperature. In this paper we show that odd-frequency superconductivity emerges on the edges of pnictides in such a coexistence phase. In particular, the breaking of spin-rotation symmetry by SDW and translation symmetry by the edge can lead to the development of odd-frequency spin-triplet Cooper pairing on edges of superconducting pnictide samples. In this case, the odd-frequency pairing has even parity components, which are immune to disorder. Our results show tha…
Using the quasiclassical Green's function formalism, we study the influence of the odd-frequency spin-triplet superconductivity on the local density of states (LDOS) in a diffusive ferromagnet (DF) attached to a superconductor. Various possible symmetry classes in a superconductor are considered which are consistent with the Pauli's principle: even-frequency spin-singlet even-parity (ESE) state, even-frequency spin-triplet odd-parity (ETO) state, odd-frequency spin-triplet even-parity (OTE) state and odd-frequency spin-singlet odd-parity (OSO) state. For each of these states, the pairing state…
We theoretically study the effects of spin-orbit interactions on symmetry of a Cooper pair in a spin-singlet d-wave superconductor in two-dimension. The pairing symmetry is analyzed in terms of the anomalous Green's function which is obtained by solving the Gor'kov equation analytically. A spin-orbit interaction induces a spin-triplet p-wave pairing correlation in a uniform superconductor. An odd-frequency spin-triplet s-wave pairing correlation appears at a surface of such superconductor as a result of breaking inversion symmetry locally. We also discuss a close relationship among the odd-fre…
We study superconductivity in spin-orbit-coupled systems in the vicinity of inversion symmetry breaking. We find that due to the presence of spin-orbit coupling, fluctuations of the incipient parity-breaking order generate an attractive pairing interaction in an odd-parity pairing channel, which competes with the s-wave pairing. We show that applying a Zeeman field suppresses the s-wave pairing and promotes the odd-parity superconducting state. Our work provides a new mechanism for odd-parity pairing and opens a route to novel topological superconductivity.
Recent experiments indicate that superconductivity in Bi$_2$Se$_3$ intercalated with Cu, Nb or Sr is nematic with rotational symmetry breaking. Motivated by this observation, we present a model study of nematic and chiral superconductivity induced by odd-parity fluctuations. We show that odd-parity fluctuations in the two-component $E_u$ representation of $D_{3d}$ crystal point group can generate attractive interaction in both the even-parity $s$-wave and odd-parity $E_u$ pairing channels, but repulsive interaction in other odd-parity pairing channels. Coulomb repulsion can suppress $s$-wave p…
We study the superconductivity of spin-polarized electrons in centrosymmetric ferromagnetic metals. Due to the spin-polarization and the Fermi statistics of electrons, the superconducting pairing function naturally has odd parity. According to the parity formula proposed by Fu, Berg, and Sato, odd-parity pairing leads to conventional first-order topological superconductivity when a normal metal has an odd number of Fermi surfaces. Here, we derive generalized parity formulae for the topological invariants characterizing higher-order topology of centrosymmetric superconductors. Based on the form…
The local density of states (LDOS) around a vortex core is studied theoretically in terms of the odd-frequency (odd-omega) Cooper pairing induced around the center of the vortex core. We find a robust zero-energy peak in the LDOS at a vortex core in a chiral p-wave superconductor originating from an odd-omega s-wave pair amplitude. We suggest how to discriminate a spin-triplet pairing symmetry and spatial chiral-domain structure by scanning tunneling spectroscopy via odd-omega pair amplitudes inside vortex cores.
We demonstrate that, quite generally, the spin-singlet even-parity (spin-triplet odd-parity) pair potential in a superconductor induces the odd-frequency pairing component with spin-singlet odd-parity (spin-triplet even-parity) near interfaces. The magnitude of the induced odd-frequency component is enhanced in the presence of the midgap Andreev resonant state due to the sign change of the anisotropic pair potential at the interface. The Josephson effect should therefore occur between odd- and even-frequency superconductors, contrary to the standard wisdom. A method to probe the odd-frequency …
Odd-frequency superconductivity is an exotic phase of matter in which Cooper pairing between electrons is entirely dynamical in nature. Majorana zero modes exhibit pure odd-frequency superconducting correlations due to their specific properties. Thus, by tunnel-coupling an array of Majorana zero modes to a spin-polarized wire, it is in principle possible to engineer a bulk one-dimensional odd-frequency spinless $s$-wave superconductor. We here point out that each tunnel coupling element, being dependent on a large number of material-specific parameters, is generically complex with sample varia…
We investigate how hybridization (single-quasiparticle scattering) between two superconducting bands induces odd-frequency superconductivity in a multiband superconductor. An explicit derivation of the odd-frequency pairing correlation and its full frequency dependence is given. We find that the density of states is modified, from the sum of the two BCS spectra, at higher energies by additional hybridization gaps with strong coherence peaks when odd-frequency pairing is present.
We investigate the influence of critical spin fluctuations on electromagnetic responses in the odd-frequency superconductivity. It is shown that the Meissner kernel of the odd-frequency superconductivity is strongly reduced by the critical spin fluctuation or the massless spin wave mode in the antiferromagnetic phase. These results imply that the superfluid density is reduced, and the London penetration depth is lengthened for the odd-frequency pairing. It is also shown that the zero field Flude-Ferrell-Larkin-Ovchinnikov pairing is spontaneously realized both for even- and odd-frequency in th…
