Anomalous Green Functions Research Articles

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.

Migdal-Eliashberg theory as a classical spin chain

We formulate the Migdal-Eliashberg theory of electron-phonon interactions in terms of classical spins by mapping the free energy to a Heisenberg spin chain in a Zeeman magnetic field. Spin components are energy-integrated normal and anomalous Green's functions and sites of the chain are fermionic Matsubara frequencies. The Zeeman field grows linearly with the spin coordinate and competes with ferromagnetic spin-spin interaction that falls off as the square of the inverse distance. The spin-chain representation makes a range of previously unknown properties plain to see. In particular, infinitely many new solutions of the Eliashberg equations both in the normal and superconducting states emerge at strong coupling. These saddle points of the free-energy functional correspond to spin flips. We argue that they are also fixed points of kinetic equations and play an essential role in far from equilibrium dynamics of strongly coupled superconductors. Up to an overall phase, the frequency-dependent gap function that minimizes the free energy must be non-negative. There are strong parallels between our Eliashberg spins and Anderson pseudospins, though the two sets of spins never coincide.

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.

Efficient numerical method for evaluating normal and anomalous time-domain equilibrium Green's functions in inhomogeneous systems

In this work we develop EPOCH: Equilibrium Propagator by Orthogonal polynomial CHain, a computationally efficient method to calculate the time-dependent equilibrium Green's functions, including the anomalous Green's functions of superconductors, to capture the time-evolution in large inhomogeneous systems. The EPOCH method generalizes the Chebyshev wave-packet propagation method from quantum chemistry and efficiently incorporates the Fermi-Dirac statistics that is needed for equilibrium quantum condensed matter systems. The computational cost of EPOCH scales only linearly in the system degrees of freedom, generating an extremely efficient algorithm also for very large systems. We demonstrate the power of the EPOCH method by calculating the time-evolution of an excitation near a superconductor-normal metal interface in two and three dimensions, capturing transmission as well as normal and Andreev reflections.

Superconducting phases and the second Josephson harmonic in tunnel junctions between diffusive superconductors

We consider a planar SIS-type Josephson junction between diffusive superconductors (S) through an insulating tunnel interface (I). We construct fully self-consistent perturbation theory with respect to the interface conductance. As a result, we find correction to the first Josephson harmonic and calculate the second Josephson harmonic. At arbitrary temperatures, we correct previous results for the nonsinusoidal current-phase relation in Josephson tunnel junctions, which were obtained with the help of conjectured form of solution. Our perturbation theory also describes the difference between the phases of the order parameter and of the anomalous Green functions.

Electronic spectrum and superconductivity in the extended [formula omitted]–[formula omitted]–[formula omitted] model

A consistent microscopic theory of superconductivity for strongly correlated electronic systems is presented within the extended t–J–V model where the intersite Coulomb repulsion and the electron-phonon interaction are taken into account. The exact Dyson equation for the normal and anomalous (pair) Green functions is derived for the projected (Hubbard) electronic operators. The equation is solved in the self-consistent Born approximation for the self-energy. We obtain the d-wave pairing with high-Tc induced by the strong kinematical interaction of the order of the kinetic energy ∼t of electrons with spin fluctuations which is much larger than the exchange interaction J. The Coulomb repulsion and the electron-phonon interaction give small contributions for the d-wave pairing. These results support the spin-fluctuation mechanism of high-temperature superconductivity in cuprates previously proposed in phenomenological models.

Edge-induced pairing states in a Josephson junction through a spin-polarized quantum anomalous Hall insulator

Despite the robustness of the chiral edge modes of quantum Hall systems against the superconducting proximity effect, Cooper pairs can penetrate into the chiral edge channels and carry the Josephson current in an appropriate setup. In our work, the Josephson junction of a spin-polarized quantum anomalous Hall insulator (QAHI) with a Chern number $\nu=1$ connecting conventional superconductors is studied from the perspective of pairing symmetry consistent with the chiral edge mode. Induced pairing states are equal-spin triplet, a combination of the even- and odd-frequency components, nonlocally extended, and have a finite momentum $2k_F$. The signature of the equal-spin triplet pairings is confirmed via the dependence on the interface-magnetization direction, and that of the finite-momentum pairing states via the spatial profile of the anomalous Green's function. In the presence of disorder, the robustness of the chiral edge mode leads to high sensitivity of the critical current and the equilibrium phase difference to disorder configurations, which is resulting from the interference of current-carrying channels. The numerical calculations on a lattice model are also examined by a simplified analytical model.

Odd-parity pairing correlations in a d -wave superconductor

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-frequency pairing correlation, chirality of surface bound states at the zero energy, and the anomalous proximity effect. The obtained results enable us to design a superconductor which causes the strong anomalous proximity effect.

Rotational symmetry breaking and partial Majorana corner states in a heterostructure based on high- Tc superconductors

Applying a microscopic model, we study theoretically the quasiparticle excitation of a twodimensional topological insulator (TI) being in proximity to a high-Tc superconductor. In the momentum space, the proximity induced pairing term in the TI layer includes both the singlet channel and triplet channel, leading to the C4 rotational symmetry breaking of the energy bands and the quasiparticle spectra. For a cylinder geometry, the zero energy edge states may appear but they are localized at the upper boundary. For the finite-size system with open boundaries, the zero energy states at the upper boundary disappear and the Majorana bound states emerge at the two lower corners. All of the results can be understood well through exploring the pairing order parameter and the anomalous Green's function.

Josephson effect in two-band superconductors

We study theoretically the Josephson effect between two time-reversal two-band superconductors, where we assume the equal-time spin-singlet $s$-wave pair potential in each conduction band. %as well as the band asymmetry and the band hybridization in the normal state. The superconducting phase at the first band $\varphi_1$ and that at the second band $\varphi_2$ characterize a two-band superconducting state. We consider a Josephson junction where an insulating barrier separates two such two-band superconductors. By applying the tunnel Hamiltonian description, the Josephson current is calculated in terms of the anomalous Green's function on either side of the junction. We find that the Josephson current consists of three components which depend on three types of phase differences across the junction: the phase difference at the first band $\delta\varphi_1$, the phase difference at the second band $\delta\varphi_2$, and the difference at the center-of-mass phase $\delta(\varphi_1+\varphi_2)/2$. A Cooper pairs generated by the band hybridization carries the last current component. In some cases, the current-phase relationship deviates from the sinusoidal function as a result of time-reversal symmetry breaking down.

Evolution of pair correlation symmetries and supercurrent reversal in tilted Weyl semimetals

We study the effective symmetry profiles of superconducting pair correlations and the flow of charge supercurrent in ballistic Weyl semimetal systems with a tilted dispersion relation. Utilizing a microscopic method in the ballistic regime and starting from both opposite-pseudospin and equal-pseudospin phonon-mediated opposite-spin electron-electron couplings, we calculate the anomalous Green's function to study various superconducting pair correlations that Weyl semimetal systems may develop. The momentum-space profile reveals that by properly manipulating the parameters of Weyl semimetal systems, including the tilting parameter, the effective symmetry class of even-parity s-wave (odd-parity p-wave) superconducting correlations can be converted into a d-wave (f-wave) symmetry class that consists of equal-pseudospin and opposite-pseudospin channels. We also find that the supercurrent in a ballistic Weyl Josephson junction can be made to vanish or switch directions, depending on the tilt of the Weyl cones, in addition to the relevant parameters characterizing the Weyl semimetal and junction. We show that inversion symmetry breaking terms introduce transitions that result in the appearance of self-biased current at zero difference between the macroscopic phases of the superconducting segments, creating a phi0 Josephson state. Weyl semimetal systems are shown to offer several experimentally tunable parameters to control the induction of higher harmonics into the current phase relations.

A Renormalization-Group Study of Interacting Bose–Einstein Condensates: II. Anomalous Dimension η for \(d \lesssim 4\) at Finite Temperatures

We study the anomalous dimension $\eta$ of homogeneous interacting single-component Bose-Einstein condensates at finite temperatures for $d\lesssim 4$ dimensions. This $\eta$ is defined in terms of the one-particle density matrix $\rho({\bf r})\equiv \langle \hat\psi^\dagger({\bf r}_1)\hat\psi({\bf r}_1+{\bf r})\rangle$ through its asymptotic behavior $\rho({\bf r})\rightarrow N_{\bf 0}/V+C r^{-d+2-\eta}$ for $r\rightarrow \infty$, where $N_{\bf 0}/V$ is the condensate density and $C$ is a constant. It is shown that the anomalous dimension is given by $\eta=0.181\epsilon^2$ to the leading order in $\epsilon\equiv d-4$. The change of the prefactor $0.181$ from the value $0.02$ at the transition point of the ${\rm O}(2)$ symmetric $\phi^4$ model is attributed to the emergence of three-point vertices and the anomalous Green's function when $N_{\bf 0}$ acquires a finite value.

Superconductivity of strongly correlated electrons on the honeycomb lattice

A microscopic theory of the electronic spectrum and of superconductivity within the t-J model on the honeycomb lattice is developed. We derive the equations for the normal and anomalous Green functions in terms of the Hubbard operators by applying the projection technique. Superconducting pairing of d + id′-type mediated by the antiferromagnetic exchange is found. The superconducting Tc as a function of hole doping exhibits a two-peak structure related to the van Hove singularities of the density of states for the two-band t-J model. At half-filling and for large enough values of the exchange coupling, gapless superconductivity may occur. For small doping the coexistence of antiferromagnetic order and superconductivity is suggested. It is shown that the s-wave pairing is prohibited, since it violates the constraint of no-double-occupancy.Graphical abstract

Inverse proximity effect in s -wave and d -wave superconductors coupled to topological insulators

We study the inverse proximity effect in a bilayer consisting of a thin $s$- or $d$-wave superconductor (S) and a topological insulator (TI). Integrating out the topological fermions of the TI, we find that spin-orbit coupling is induced in the S, which leads to spin-triplet $p$-wave ($f$-wave) correlations in the anomalous Green's function for an $s$-wave ($d$-wave) superconductor. Solving the self-consistency equation for the superconducting order parameter, we find that the inverse proximity effect can be strong for parameters for which the Fermi momenta of the S and TI coincide. The suppression of the gap is approximately proportional to $e^{-1/\lambda}$, where $\lambda$ is the dimensionless superconducting coupling constant. This is consistent with the fact that a higher $\lambda$ gives a more robust superconducting state. For an $s$-wave S, the interval of TI chemical potentials for which the suppression of the gap is strong is centered at $\mu_{TI} = \pm\sqrt{2mv_F^2\mu}$, and increases quadratically with the hopping parameter $t$. Since the S chemical potential $\mu$ typically is high for conventional superconductors, the inverse proximity effect is negligible except for $t$ above a critical value. For sufficiently low $t$, however, the inverse proximity effect is negligible, in agreement with what has thus far been assumed in most works studying the proximity effect in S-TI structures. In superconductors with low Fermi energies, such as high-$T_c$ cuprates with $d$-wave symmetry, we again find a suppression of the order parameter. However, since $\mu$ is much smaller in this case, a strong inverse proximity effect can occur at $\mu_{TI}=0$ for much lower values of $t$. Moreover, the onset of a strong inverse proximity effect is preceded by an increase in the order parameter, allowing the gap to be tuned by several orders of magnitude by small variations in $\mu_{TI}$.

Electronic spectrum and superconductivity in the t-J model on the honeycomb lattice

A microscopic theory of electronic spectrum and superconductivity within the $t$-$J$ model on the honeycomb lattice is formulated. The Dyson equation for the normal and anomalous Green functions for the two-band model in terms of the Hubbard operators is derived by applying the Mori-type projection technique. The self-energy is evaluated in the self-consistent Born approximation for electron scattering on spin and charge fluctuations induced by the kinematical interaction for the Hubbard operators. Superconducting pairing mediated by the antiferromagnetic exchange and spin fluctuations is discussed.

Theoretical Study of Magnetism Induced by Proximity Effect in a Ferromagnetic Josephson Junction with a Normal Metal

We theoretically study the magnetism induced by the proximity effect in the normal metal of ferromagnetic Josephson junction composed of two $s$-wave superconductors separated by ferromagnetic metal/normal metal/ferromagnetic metal junction (${S}/{F}/{N}/{F}/{S}$ junction). We calculate the magnetization in the $N$ by solving the Eilenberger equation. We show that the magnetization arises in the ${N}$ when the product of anomalous Green's functions of the spin-triplet even-frequency odd-parity Cooper pair and spin-singlet odd-frequency odd-parity Cooper pair in the ${N}$ has a finite value. The induced magnetization $M(d,\theta)$ can be decomposed into two parts, $M(d,\theta)=M^{\rm I}(d)+M^{\rm II}(d,\theta)$, where $d$ is the thickness of $N$ and $\theta$ is superconducting phase difference between two ${S}$s. Therefore, $\theta$ dependence of $M(d,\theta)$ allows us to control the amplitude of magnetization by changing $\theta$. The variation of $M(d,\theta)$ with $\theta$ is indeed the good evidence of the magnetization induced by the proximity effect, since some methods of magnetization measurement pick up total magnetization in the ${S}/{F}/{N}/{F}/{S}$ junction.

Excitonic Insulator State of the Extended Falicov–Kimball Model in the Cluster Dynamical Impurity Approximation

We comparatively study the excitonic insulator state in the extended Falicov-Kimball model (EFKM, a spinless two-band model) on the two-dimensional square lattice using the variational cluster approximation (VCA) and the cluster dynamical impurity approximation (CDIA). In the latter, the particle-bath sites are included in the reference cluster to take into account the particle-number fluctuations in the correlation sites. We thus calculate the particle-number distribution, order parameter, ground-state phase diagram, anomalous Green's function, and pair coherence length, thereby demonstrating the usefulness of the CDIA in the discussion of the excitonic condensation in the EFKM.

Majorana STM as a perfect detector of odd-frequency superconductivity

We propose a novel scanning tunneling microscope (STM) device in which the tunneling tip is formed by a Majorana bound state (MBS). This peculiar bound state exists at the boundary of a one-dimensional topological superconductor. Since the MBS has to be effectively spinless and local, we argue that it is the smallest unit that shows itself odd-frequency superconducting pairing. Odd-frequency superconductivity is characterized by an anomalous Green function which is an odd function of the time arguments of the two electrons forming the Cooper pair. Interestingly, our Majorana STM can be used as the perfect detector of odd-frequency superconductivity. The reason is that a supercurrent between the Majorana STM and any other superconductor can only flow if the latter system exhibits itself odd-frequency pairing. To illustrate our general idea, we consider the tunneling problem of the Majorana STM coupled to a quantum dot in vicinity to a conventional superconductor. In such a (superconducting) quantum dot, the effective pairing can be tuned from even- to odd-frequency behavior if an external magnetic field is applied to it.

Optical properties of Bogoliubov quasiparticles

We calculated the optical conductivity $\sigma(T,\Omega)$ of a gas of Bogoliubov quasiparticles (BQP) from their Green's function and the Kubo formula. We compare with corresponding normal state (N) and superconducting state (SC) results. The superconducting case includes the dynamic response of the condensate through additional contributions to the Kubo formula involving the Gor'kov anomalous Green's function. The differences in the optical scattering rate are largest just above the optical gap and become progressively smaller as the photon energy is increased or the temperature is raised. Our results are compared with those obtained using a recently advocated phenomenological procedure for eliminating the effect of the condensate [1]. The $\delta$-function contribution at zero photon energy, proportional to the superfluid density, is dropped in the real part of the conductivity $[\sigma_1(T,\Omega)]$ and its Kramers-Kronig transform is subtracted from the imaginary part $\sigma_2(T,\Omega)$. This results in deviations from our BQP and superconducting state optical scattering rates even in the region where these have merged and are, in addition, close to the normal state result.

Odd-frequency superconductivity in driven systems

We show that Berezinskii's classification of the symmetries of Cooper pair amplitudes holds for driven systems even in the absence of translation invariance. We then consider a model Hamiltonian for a superconductor coupled to an external driving potential and, treating the driving potential as a perturbation, we investigate the corrections to the anomalous Green's function, density of states, and spectral function. We find that in the presence of an external drive the anomalous Green's function develops terms that are odd in frequency and that the same mechanism responsible for these odd-frequency terms generates additional features in the density of states and spectral function.