Odd-frequency Superconductivity Research Articles

Odd-frequency superfluidity from a particle-number-conserving perspective

We investigate odd-in-time—or —pairing of fermions in equilibrium systems within the particle-number-conserving framework of Penrose, Onsager, and Yang, where superfluid order is defined by macroscopic eigenvalues of reduced density matrices. We show that odd-frequency pair correlations are synonymous with even fermion-exchange symmetry in a time-dependent correlation function that generalises the two-body reduced density matrix. Macroscopic even-under-fermion-exchange pairing is found to emerge from conventional Penrose-Onsager-Yang condensation in two-body or higher-order reduced density matrices through the symmetry-mixing properties of the Hamiltonian. We identify and characterize a matrix responsible for producing macroscopic even fermion-exchange correlations that coexist with a conventional Cooper-pair condensate, while a matrix is shown to be responsible for creating macroscopic even fermion-exchange correlations from hidden orders such as a multiparticle condensate. The transformer scenario is illustrated using the spin-balanced s-wave superfluid with Zeeman splitting as an example. The generator scenario is demonstrated by the composite-boson condensate arising for itinerant electrons coupled to magnetic excitations. Structural analysis of the transformer and generator matrices is shown to provide general conditions for odd-frequency pairing order to arise in a given system. Our formalism facilitates a fully general derivation of the Meissner effect for odd-frequency superconductors that holds also beyond the regime of validity for mean-field theory. Published by the American Physical Society 2024

Singlet Even-Frequency and Triplet Odd-Frequency Superconductivity in the Two-Band Hubbard Model Based on the Dynamical Mean-Field Theory

Singlet Even-Frequency and Triplet Odd-Frequency Superconductivity in the Two-Band Hubbard Model Based on the Dynamical Mean-Field Theory

Impurity and hybridization effects on the symmetry classification and magnetic response function of a two-band superconductor with interband pairing order

Abstract The effects of hybridization and impurity (magnetic and nonmagnetic) potentials on the pairing symmetries and magnetic response of a two-band superconductor with an equal-time s-wave interband pairing order parameter in the framework of Green’s function technique are investigated theoretically. First, the effects of spin-independent and spin-dependent hybridization on the generation of even- or odd-frequency Cooper pairs which determines the symmetry classification and the response of the superconductor are studied. Next, the impurity effect on creating different symmetry classes and the kernel response function of a two-band superconductor are discussed. By separating the contributions of even- and odd-frequency pairing to the Meissner kernel, it is shown that the competition between these two terms determines the total Meissner effect of the superconductor. For a two-band spin-singlet superconductor, nonmagnetic impurity scatterings do not change transition temperature according to Anderson’s theorem, while both intra- and interband magnetic impurity scattering cause superconducting transition temperature suppression with the rate following the Abrikosov–Gor’kov theory. For spin-triplet pairing, interband magnetic scattering has no impact on pair breaking, whereas intraband magnetic scattering acts as a pair breaker and suppresses the transition temperature in the Born limit. In this case, the odd-frequency superconducting pairs can be induced in the simultaneous presence of both intra- and interband magnetic impurities. Thus, by controlling the concentration of magnetic impurities, it is possible to engineer triplet-pairing odd-frequency superconductors with a total diamagnetic Meissner response which stabilizes the superconducting state. This technique opens up an avenue for designing stable odd-frequency superconductors.

Appearance of odd-frequency superconductivity in a relativistic scenario

Appearance of odd-frequency superconductivity in a relativistic scenario

Triplet odd-frequency superconductivity in hybrid superconductor–ferromagnet structures

We present an overview of the contributions made by Konstantin B. Efetov and the authors to the field of triplet odd-frequency superconductivity in hybrid ferromagnet–superconductor structures (Bergeret et al., 2005) and the repercussions they had on experimental and theoretical research. Konstantin B. Efetov passed away in August 2021. We hope the present short review gives a faithful testimony of an important part of his scientific legacy.

Effective quasiclassical models for odd-frequency superconductivity: Energy-inversion symmetry, preserved spectral weight, and Meissner response

The odd-frequency superconducting state appears generally in hybrid structures consisting of conventional superconductors and other materials, and features electrons that form temporally nonlocal Cooper pairs. The quasiclassical theory of superconductivity has been extensively used to model such systems, finding in many cases excellent agreement with experimental measurements. Therefore, it is of interest to study effective models of odd-frequency superconductivity to predict new phenomena associated with this form of unconventional pairing. We establish necessary criteria that the quasiclassical Green's functions in odd-frequency superconducting systems in the dirty limit must satisfy in order to be physically reasonable, including conservation of spectral weight. We show that it is possible to write down effective models, which satisfy all the above-mentioned criteria, but which still predict different behavior when it comes to the density of states and the magnetic response of the superconductor. For instance, an effective model for the odd-frequency anomalous Green's function that gives a conserved spectral weight can yield either a peaked or gapped density of states at the Fermi energy, and exhibit conventional, zero, or unconventional Meissner response. This finding demonstrates the importance of carefully considering the properties of effective models describing odd-frequency superconductivity in order to obtain physically reasonable results.

Odd frequency pairing in a quantum critical metal

We analyze a possibility for odd-frequency pairing near a quantum critical point(QCP) in a metal. We consider a model with dynamical pairing interaction $V(\Omega_n)\sim 1/|\Omega_n|^\gamma$ (the $\gamma$-model). This interaction gives rise to a non-Fermi liquid in the normal state and is attractive for pairing. The two trends compete with each other. We search for odd-frequency solutions for the pairing gap $\Delta (\omega_m)= -\Delta(\omega_m)$. We show that for $\gamma <1$, odd-frequency superconductivity looses the competition with a non-Fermi liquid and does not develop. We show that the pairing does develop in the extended model in which interaction in the pairing channel is larger than the one in the particle-hole channel. For $\gamma >1$, we argue that the original model is at the boundary towards odd-frequency pairing and analyze in detail how superconductivity is triggered by a small external perturbation. In addition, we show that for $\gamma >2$, the system gets frozen at the critical point towards pairing in a finite range in the parameter space. This give rise to highly unconventional phase diagrams with flat regions.

Direct detection of odd-frequency superconductivity via time- and angle-resolved photoelectron fluctuation spectroscopy

We propose a measurement scheme to directly detect odd-frequency superconductivity via time- and angle-resolved photoelectron fluctuation spectroscopy. The scheme includes two consecutive, non-overlapping probe pulses applied to a superconducting sample. The photoemitted electrons are collected in a momentum-resolved fashion. Correlations between signals with opposite momenta are analyzed. Remarkably, these correlations are directly proportional to the absolute square of the time-ordered anomalous Green's function of the superconductor. This setup allows for the direct detection of the "hidden order parameter" of odd-frequency pairing. We illustrate this general scheme by analyzing the signal for the prototypical case of a two-band superconductor.

Induced odd-frequency superconducting state in vertex-corrected Eliashberg theory

We show that vertex corrections to Migdal's theorem in general induce an odd-frequency spin-triplet superconducting order parameter, which coexists with its more commonly known even-frequency spin-singlet counterpart. Fully self-consistent vertex-corrected Eliashberg theory calculations for a two dimensional cuprate model, isotropically coupled to an Einstein phonon, confirm that both superconducting gaps are finite over a wide range of temperatures. The subordinate $d$-wave odd-frequency superconducting gap is found to be one order of magnitude smaller than the primary even-frequency $d$-wave gap. Our study provides a direct proof of concept for a previously unknown generation mechanism of odd-frequency superconductivity as well as for the generic coexistence of both superconducting states in bulk materials.

Spin injection and spin relaxation in odd-frequency superconductors

The spin transport inside an odd-frequency spin-triplet superconductor differs from that of a conventional superconductor due to its distinct symmetry properties. We study spin transport inside an emergent odd-frequency superconductor by replacing the spin-singlet gap matrix in the Usadel equation with a matrix representing spin-triplet pairing that is odd under inversion of energy. We show that the peculiar nature of the density of states allows for an even larger spin injection than in the normal-state. Moreover, when the odd-frequency pairing inherits its temperature dependence from a conventional superconductor through the proximity effect, the density of states can transition from gapless to gapped as the temperature decreases. At the transition point, the spin accumulation inside the odd-frequency superconductor is peaked and larger than in the normal-state. While the spin-flip scattering time is known to decrease below the superconducting transition temperature in conventional superconductors, we find that the same is true for the spin-orbit scattering time in odd-frequency superconductors. This renormalization is particularly large for energies close to the gap edge, if such a gap is present.

Odd-frequency pair density wave correlations in underdoped cuprates

Pair density waves, identified by Cooper pairs with finite center-of-mass momentum, have recently been observed in copper oxide based high T c superconductors (cuprates). A charge density modulation or wave is also ubiquitously found in underdoped cuprates. Within a general mean-field one-band model we show that the coexistence of charge density waves (CDWs) and uniform superconductivity in d-wave superconductors like cuprates, generates an odd-frequency spin-singlet pair density wave, in addition to the even-frequency counterparts. The strength of the induced odd-frequency pair density wave depends on the modulation wave vector of the CDW, with the odd-frequency pair density waves even becoming comparable to the even-frequency ones in parts of the Brillouin zone. We show that a change in the modulation wave vector of the CDW from bi-axial to uni-axial, can enhance the odd-frequency component of the pair density waves. Such a coexistence of superconductivity and uni-axial CDW has already been experimentally verified at high magnetic fields in underdoped cuprates. We further discuss the possibility of an odd-frequency spin-triplet pair density wave generated in the coexistence regime of superconductivity and spin density waves, applicable to the iron-based superconductors. Our work thus presents a route to bulk odd-frequency superconductivity in high T c superconductors.

Odd-frequency superconductivity in dilute magnetic superconductors

We show that dilute magnetic impurities in a conventional superconductor give origin to an odd-frequency component of superconductivity, manifesting itself in Yu-Shiba-Rusinov bands forming within the bulk superconducting gap. Our results are obtained in a general model solved within the dynamical mean field theory. By exploiting a disorder analysis and the limit to a single impurity, we are able to provide general expressions that can be used to extract explicitly the odd-frequency superconducting function from scanning tunneling measurements.

Proximity-Induced Odd-Frequency Superconductivity in a Topological Insulator.

At an interface between a topological insulator (TI) and a conventional superconductor (SC), superconductivity has been predicted to change dramatically and exhibit novel correlations. In particular, the induced superconductivity by an s-wave SC in a TI can develop an order parameter with a p-wave component. Here we present experimental evidence for an unexpected proximity-induced novel superconducting state in a thin layer of the prototypical TI, Bi_{2}Se_{3} proximity coupled to Nb. From depth-resolved magnetic field measurements below the superconducting transition temperature of Nb, we observe a local enhancement of the magnetic field in Bi_{2}Se_{3} that exceeds the externally applied field, thus supporting the existence of an intrinsic paramagnetic Meissner effect arising from an odd-frequency superconducting state. Our experimental results are complemented by theoretical calculations supporting the appearance of such a component at the interface which extends into the TI. This state is topologically distinct from the conventional Bardeen-Cooper-Schrieffer state it originates from. To the best of our knowledge, these findings present a first observation of bulk odd-frequency superconductivity in a TI. We thus reaffirm the potential of the TI-SC interface as a versatile platform to produce novel superconducting states.

Large Josephson current in Weyl nodal loop semimetals due to odd-frequency superconductivity

Weyl nodal loop semimetals (WNLs) host a closed nodal line loop Fermi surface in the bulk, protected zero-energy flat band, or drumhead, surface states, and strong spin-polarization. The large density of states of the drumhead states makes WNL semimetals exceedingly prone to electronic ordering. At the same time, the spin-polarization naively prevents conventional superconductivity due to its spin-singlet nature. Here we show the complete opposite: WNLs are extremely promising materials for superconducting Josephson junctions, entirely due to odd-frequency superconductivity. By sandwiching a WNL between two conventional superconductors we theoretically demonstrate the presence of very large Josephson currents, even up to orders of magnitude larger than for normal metals. The large currents are generated both by an efficient transformation of spin-singlet pairs into odd-frequency spin-triplet pairing by the Weyl dispersion and the drumhead states ensuring exceptionally proximity effect. As a result, WNL Josephson junctions offer unique possibilities for detecting and exploring odd-frequency superconductivity.

Suppression of odd-frequency pairing by phase disorder in a nanowire coupled to Majorana zero modes

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 variability in both its magnitude and phase. Using this, we demonstrate that, upon averaging over phase-disorder, the induced superconducting, including odd-frequency, correlations in the spin-polarized wire are significantly suppressed. We perform both a rigorous analytical evaluation of the disorder-averaged $T$-matrix in the wire, as well as numerical calculations based on a tight-binding model, and find that the anomalous, i.e. superconducting, part of the $T$-matrix is highly suppressed with phase disorder. We also demonstrate that this suppression is concurrent with the filling of the single-particle excitation gap by smearing the near-zero frequency peaks, due to formation of bound states that satisfy phase-matching conditions between spatially separated Majorana zero modes. Our results convey important constraints on the parameter control needed in practical realizations of Majorana zero mode structures and suggest that the achievement of a bulk 1D odd-$\omega$ superconductivity from MZMs demand full control of the system parameters.

Odd-frequency superconductivity near a magnetic impurity in a conventional superconductor

Superconductor-ferromagnetic heterostructures have been suggested as one of the most promising alternatives of realizing odd-frequency superconductivity. In this work we consider the limit of shrinking the ferromagnetic region to the limit of a single impurity embedded in a conventional superconductor, which gives raise to localized Yu-Shiba-Rusinov (YSR) bound states with energies inside the superconducting gap. We demonstrate that all the sufficient ingredients for generating odd-frequency pairing are present at the vicinity of these impurities. We investigate the appearance of all possible pair amplitudes in accordance with the Berezinskii $SP^{\ast}OT^{\ast} = -1$ rule, being the symmetry under the exchange of spin, spatial, orbital (in our case $O=+1$) and time index, respectively. We study the spatial and frequency dependence of of the possible pairing amplitudes, analyzing their evolution with impurity strength and identifying a reciprocity between different symmetries related through impurity scattering. We show that the odd-frequency spin-triplet pairing amplitude dominates at the critical impurity strength, where the YSR states merge at the middle of the gap, while the even components cancel out close to the impurity. We also show that the spin-polarized local density of states exhibits the same spatial and frequency behavior as the odd-$\omega$ spin-triplet component at the critical impurity strength.

Influence of electron–phonon coupling strength on signatures of even and odd-frequency superconductivity

The recently discovered APt3P (A=Sr,Ca,La) family of superconductors offers a platform to study frequency dependent superconducting phenomena as the electron–phonon coupling varies from weak to strong. Here we perform ab initio Eliashberg theory calculations to investigate two such phenomena, the occurrence of dip-hump structures in the tunneling spectra and the magnetic field induced coexistence of even and odd frequency superconductivity in these compounds. By calculating the superfluid density, we make materials specific predictions for the occurrence of the paramagnetic Meissner effect as a hallmark of odd frequency pairing. Our results provide a link between two seemingly uncorrelated aspects of even and odd frequency superconductivity and provide theoretical guidance for the future experimental identification of bulk odd frequency superconductivity in this materials’ family.

Odd-frequency superconductivity

This article reviews odd-frequency (odd-w) pairing with focus on superconducting systems. Since Berezinskii introduced the concept of odd frequency order in 1974 it has been viewed as an exotic and rarely occurring in nature. Here, we present a view that the Berezinskii state is in fact a ubiquitous superconducting order that is both non-local and odd in time. It appears under quite general circumstances including in bulk materials, heterostructures and dynamically driven superconducting states. It is therefore important to understand the nature of odd-w pairing. We present the properties of odd-w pairing in bulk materials, including possible microscopic mechanisms, discuss definitions of the odd-w superconducting order parameter, and the unusual Meissner response of odd-frequency superconductors. Next, we present how odd-w pairing is generated in hybrid structures of nearly any sort and focus on its relation to Andreev bound states, spin polarized Cooper pairs, and Majorana states. We overview how odd-w pairing can be applied to non-superconducting systems such as ultracold Fermi gases, Bose-Einstein condensates, and chiral spin-nematics. Due to the growing importance of dynamic orders in quantum systems we also discuss the emergent view that the odd-w state is an example of phase coherent dynamic order. We summarize the recent progress made in understanding the emergence of odd-w states in driven superconducting systems. A more general view of odd-w superconductivity suggests an interesting approach to this state as a realization of the hidden order with inherently dynamic correlations. We overview progress made in this rapidly evolving field and illustrate the ubiquity of the odd-w states and potential for future discoveries of these states in variety of settings. We sum up the general rules/design principles to induce odd-w components using the SPOT rule.

Spectroscopic and optical response of odd-frequency superconductors

The optical response of superconductors with odd-frequency Berezinskii pairing is studied. By using a simple model with a parabolic dispersion law and a non-magnetic disorder, the spectral function, the electron density of states, and the optical conductivity are calculated for a few gap ansatzes. The spectral function and the electron density of states clearly reveal the gap for the Berezinskii pairing for the sufficiently strong frequency dependence of the order parameters. It is found that, similarly to the conventional BCS pairing, the odd-frequency gaps induce peaks in the real part of the conductivity, which, however, are sharper than in the BCS case. The magnitude and position of these peaks are determined by the frequency profile of the gap. The imaginary part of the optical conductivity for the Berezinskii pairing demonstrates sharp cusps that are absent in the case of the BCS superconductors. The corresponding results suggest that the Berezinskii pairing might allow for the optical transparency windows related to the onsets of the attenuation peaks in the real part of the conductivity. Thus, the study of the optical response not only provides an alternative way to probe the odd-frequency gaps but can reveal also additional features of the dynamic superconducting pairing.

Odd-frequency superconductivity induced by nonmagnetic impurities

A growing body of literature suggests that odd-frequency superconducting pair amplitudes can be generated in normal metal-superconductor junctions. The emergence of odd-frequency pairing in these systems is often attributed to the breaking of translation invariance. In this work, we study the pair symmetry of a one-dimensional $s$-wave superconductor in the presence of a single non-magnetic impurity and demonstrate that translation symmetry breaking is not sufficient for inducing odd-frequency pairing. We consider three kinds of impurities: a local perturbation of the chemical potential, an impurity possessing a quantum energy level, and a local perturbation of the superconducting gap. Surprisingly, we find local perturbations of the chemical potential do not induce any odd-frequency pairing, despite the fact that they break translation invariance. Moreover, although odd-frequency can be induced by both the quantum impurity and the perturbation of the gap, we find these odd-frequency amplitudes emerge from entirely different kinds of scattering processes. The quantum impurity generates odd-frequency pairs by allowing one of the quasiparticles belonging to an equal-time Cooper pair to tunnel onto the impurity state and then back to the superconductor, giving rise to odd-frequency amplitudes with a temporal broadening inversely proportional to the energy level of the impurity. In contrast to this, the perturbation of the gap leads to odd-frequency pairing by "gluing-together" normal state quasiparticles from different points in space and time, leading to odd-frequency amplitudes which are very localized in the time domain.