Anomalous Green Functions Research Articles

Maximum entropy analytic continuation for spectral functions with nonpositive spectral weight

Information about the pairing mechanism for superconductivity is contained in the spectral weight for the anomalous (Gorkov) Green function. In the most general case, this spectral weight can change sign on the positive real axis or even be complex in the presence of broken time-reversal symmetry. This renders impossible the direct analytic continuation with Maximum Entropy methods of numerical results obtained for the anomalous Green function either in Matsubara frequency or in imaginary time. Here we show that one can define auxiliary Green functions that allow one to extract the anomalous spectral weight in the most general multi-orbital spin-singlet or triplet cases with no particular symmetry. As a simple example, we treat the case of lead in Eliashberg theory: Maximum Entropy analytic continuation with auxiliary Green functions (MaxEntAux) allows us to recover spectral functions that change sign on the positive real axis. The approach can be extended to transport quantities.

Probing Phase Coherence Via Density of States for Strongly Correlated Excitons

We present the calculation of the coherent spectral functions and density of states (DOS) for excitonic systems in the frame of the three dimensional extended Falicov-Kimball model. By using gauge-invariant U(1) transformation to the usual fermions, we represent the electron operator as a fermion attached to the U(1) phase-flux tube. The emergent bosonic gauge field, related to the phase variables is crucial for the Bose-Einstein condensation (BEC) of excitons. Employing the path-integral formalism, we manipulate the bosonic and fermionic degrees of freedom to obtain the effective actions related to fermionic and bosonic sectors. Considering the normal and anomalous excitonic Green functions, we calculate the spectral functions, which have the forms of convolutions in the reciprocal space between bosonic and fermionic counterparts. For the fermionic incoherent part of the DOS we have found the strong evidence of the hybridization-gap in DOS spectra. Furthermore, considering Bogoliubov coherence mechanism, we calculate the coherent DOS spectra. For the coherent normal fermionic DOS, there is no hybridization-gap found in the system due to strong coherence effets and phase stiffness. The similar behavior is observed also for the condensate part of the anomalous excitonic DOS spectra. We show that for small values of the Coulomb interaction, fermionic DOS exhibits a Bardeen-Cooper-Schriffer (BCS) -like double-peak structure. In the BEC region of the BCS-BEC crossover, the double-peak structure disappears totally for both: coherent and incoherent DOS spectra. We discuss also, temperature dependence of DOS functions.

Unconventional proximity-induced superconductivity in bilayer systems

We study the proximity-induced superconducting state in a general bilayer -- conventional $s$-wave superconductor hybrid structure. For the bilayer we include a general parabolic dispersion, Rashba spin-orbit coupling, and finite layer tunneling as well as the possibility to apply a bias potential and a magnetic Zeeman field, in order to address experimentally relevant bilayer systems, ranging from topological insulator thin films to generic double quantum well systems. By extracting the proximity-induced anomalous Green's function in the bilayer we show on a very rich structure for the superconducting pairing, including different spin states and odd-frequency pairing. Equal-spin spin-triplet $p_x\pm ip_y$-wave pairing is induced in both layers in the presence of a finite spin-orbit coupling and opposite-spin spin-triplet $s$-wave pairing with odd-frequency dependence appears for an applied magnetic Zeeman field. Finite interlayer pairing is also generally present in the bilayer. The interlayer pairing can be either even or odd in the layer index, with a complete reciprocity between parity in frequency and in layer index. We also find that a bilayer offers the possibility of sign reversal of the superconducting order parameters, both between the two layers and between multiple Fermi surfaces.

Phase transition of a heavy-fermion superconductor in a high magnetic field: Entanglement analysis

When the magnetic field is only acting on the spin of electrons, a transition from a normal to a modulated superconducting state or Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state may occur at low temperatures. A FFLO superconducting state, which accompanies an order parameter that oscillates spatially, may be stabilized by a high applied magnetic field or a molecular field. Quantum multipartite entanglement is a new procedure for investigating quantum phase transitions. In this article, we deal with the phase transition of the FFLO state of CeCoIn5 to a normal state by obtaining quantum multipartite entanglement of the system. For this purpose, using normal and anomalous Green functions and the density matrix, we obtain concurrence, as a measure of bipartite entanglement. Then, the order parameter and the magnetic -field dependence of multipartite entanglement in momentum space is calculated. The phase transition is determined, and the behavior of the system based on order parameter is discussed. Furthermore, the phase transitions of both the Bardeen-Cooper-Schrieffer (BCS) and FFLO states to the normal state are compared.

Exact-diagonalization study of exciton condensation in electron bilayers

We report on small-cluster exact-diagonalization calculations which prove the formation of electron-hole pairs (excitons) as a prerequisite for spontaneous interlayer phase coherence in double-layer systems described by the extended Falicov-Kimball model. Evaluating the anomalous Green's function and momentum distribution function of the pairs, and thereby analyzing the dependence of the exciton binding energy, condensation amplitude, and coherence length on the Coulomb interaction strength, we demonstrate a crossover between a BCS-like electron-hole pairing transition and a Bose-Einstein condensation of tightly bound preformed excitons. We furthermore show that a mass imbalance between electrons and holes tends to suppress the condensation of excitons.

Phonon Origin of High T c in Superconducting Cuprates

Eliashberg theory (ET), generalized for the account of the peculiar properties of the finite zone width electron-phonon (EP) systems with the non-constant electron density of states, the electron-hole nonequivalence, chemical potential renormalization with doping and frequency, and electron correlations in the vertex function, is used for the study of T c in cuprates. The phonon contribution to the nodal anomalous electron Green function (GF) in cuprates is considered. The pairing on the full width of the electron zone was taken into account, not just on the Fermi surface. It is found that the finite zone width phenomenon in the newly derived Eliashberg equations for the finite zone width EP system together with the abrupt fall of the density of states above the Fermi surface are the crucial factors for the appearance of the high temperature superconductivity phenomenon. It is shown that near the optimal doping in the hole-doped cuprates high T c value is reproduced with the EP interaction constant obtained from tunnel experiments.

A BCS-BEC crossover in the extended Falicov-Kimball model: Variational cluster approach

Motivated by recent experimental and theoretical studies of excitonic insulators, we use the variational cluster approximation to calculate the single-particle Green's function and anomalous Green's function of the Falicov-Kimball model extended by including a finite valence bandwidth. We thus determine the single-particle excitation gap due to the bound-state formation of an electron and a hole. We also evaluate the order parameter of the excitonic insulating state indicating quantum coherence of condensed excitons. We thereby discuss the excitonic insulator that typifies either a BCS condensate of electron-hole pairs (weak-coupling regime) or a Bose-Einstein condensate of preformed excitons (strong-coupling regime). A BCS-BEC crossover of the model thus manifests itself as a function of the Coulombic coupling strength.

Variational cluster approach to the extended Falicov-Kimball model: A BCS-BEC crossover in the Excitonic insulators

We study the Coulomb-interaction induced spontaneous symmetry breaking of the excitonic insulator state in the two-dimensional extended Falicov-Kimball model. Using the variational cluster approximation, we evaluate the order parameter, single-particle excitation gap, momentum distribution functions, and anomalous Green's function as a function of U at zero temperature. We find that in the weak-to-intermediate couplinb regime, the Fermi surface is well-defined and the calculated results can be understood in the close correspondence with the BCS theory, whereas in the strong-coupling regime, the Fermi surface is ill-defined and the results are consistent with the picture of BEC.

Josephson and proximity effects on the surface of a topological insulator

We investigate Josephson and proximity effects on the surface of a topological insulator on which superconductors and a ferromagnet are deposited. The superconducting regions are described by the conventional BCS Hamiltonian, rather than the superconducting Dirac Hamiltonian. Junction interfaces are assumed to be dirty. We obtain analytical expressions of the Josephson current and the proximity-induced anomalous Green's function on the topological insulator. The dependence of the Josephson effect on the junction length, the temperature, the chemical potential, and the magnetization is discussed. It is also shown that the proximity-induced pairing on the surface of a topological insulator includes even- and odd-frequency triplet pairings as well as a conventional $s$-wave pairing.

Excitonic insulator state in the two-orbital Hubbard model: Variational cluster approach

The variational cluster approximation is used to study the spontaneous symmetry breaking of the excitonic insulator (EI) state in the two-orbital Hubbard model defined on the two-dimensional square lattice. Evaluating the order parameter of the EI state, staggered magnetization, and number of particles on each orbital, we obtain the ground-state phase diagram of the model in a wide parameter space of the intra and interorbital Coulomb interactions. We also calculate the single-particle and anomalous Green's functions. We show that the normal metallic phase is unstable to the formation of the EI state and the EI phase appears in a wide parameter region between the band-insulator and Mott-insulator phases.

Superconducting phase transistor in diffusive four-terminal ferromagnetic Josephson junctions

We study diffusive magnetic Josephson junctions with four superconducting terminals in the weak proximity limit where the leads are arranged in cross form. Employing the linearized Keldysh-Usadel technique, the anomalous Green's function and Josephson current are analytically obtained based on a quasiclassical theory using the Fourier series method. The derived results may be reduced to non-magnetic junctions by setting the exchange field equal to zero. We find that increments of the magnetic barrier thickness may cause a reversal of the supercurrent direction flowing into some of the leads, whereas the direction of current-flow remains invariant at the others. The reversal direction can be switched by tuning the perpendicular superconducting phases. In the non-magnetic case, we find that the supercurrent flowing between the leads in one direction can be tuned by changing the superconducting phase difference in the perpendicular direction. These findings suggest the possibility of constructing a nano-scale superconducting phase transistor whose core element consists of the proposed four-terminal Josephson junction with rich switching aspects.

Influence of spin-dependent quasiparticle distribution on the Josephson current through a ferromagnetic weak link

The Josephson current flowing through weak links containing ferromagnetic elements is studied theoretically under the condition that the quasiparticle distribution over energy states in the interlayer is spin dependent. It is shown that the interplay between the spin-dependent quasiparticle distribution and the triplet superconducting correlations induced by the proximity effect between the superconducting leads and ferromagnetic elements of the interlayer leads to the appearence of an additional contribution to the Josephson current. This additional contribution ${j}_{t}$ can be extracted from the full Josephson current in experiment. The features of the additional supercurrent ${j}_{t}$, which are of main physical interest are the following: i) We propose the experimental setup, where the contributions given by the short-range triplet component (SRTC) and long-range triplet component (LRTC) of superconducting correlations in the interlayer can be measured separately. It can be realized on the basis of a S/N/F/N/S junction, where the interlayer is composed of two normal metal regions with a spiral ferromagnet layer sandwiched between them. For the case of tunnel junctions, the measurement of ${j}_{t}$ in such a system can provide direct information about the energy-resolved anomalous Green's function components describing SRTC and LRTC. ii) In some cases the exchange field-suppressed supercurrent can be not only recovered but also enhanced with respect to its value for a nonmagnetic junction with the same interface resistances by the presence of a spin-dependent quasiparticle distribution. This effect is demonstrated for the S/N/S junction with magnetic S/N interfaces. In addition, it is also found that under the considered conditions the dependence of the Josephson current on temperature can be nontrivial: At first, the current rises upon the temperature increasing and only after that starts to decline.

Variational cluster approach for strongly correlated lattice bosons in the superfluid phase

We extend the variational cluster approach to deal with strongly correlated lattice bosons in the superfluid phase. To this end, we reformulate the approach within a pseudoparticle formalism, whereby cluster excitations are described by particlelike excitations. The approximation amounts to solving a multicomponent noninteracting bosonic system by means of a multimode Bogoliubov approximation. A source-and-drain term is introduced in order to break U(1) symmetry at the cluster level. We provide an expression for the grand potential, the single-particle normal and anomalous Green's functions, the condensate density, and other static quantities. As a first nontrivial application of the method we choose the two-dimensional Bose-Hubbard model and evaluate results in both the Mott and the superfluid phases. Our results show an excellent agreement with quantum Monte Carlo calculations.

Andreev reflection at half-metal/superconductor interfaces with nonuniform magnetization

Andreev reflection at the interface between a half-metallic ferromagnet and a spin-singlet superconductor is possible only if it is accompanied by a spin flip. Here we calculate the Andreev reflection amplitudes for the case that the spin flip originates from a spatially non-uniform magnetization direction in the half metal. We calculate both the microscopic Andreev reflection amplitude for a single reflection event and an effective Andreev reflection amplitude describing the effect of multiple Andreev reflections in a ballistic thin film geometry. It is shown that the angle and energy dependence of the Andreev reflection amplitude strongly depends on the orientation of the gradient of the magnetization with respect to the interface. Establishing a connection between the scattering approach employed here and earlier work that employs the quasiclassical formalism, we connect the symmetry properties of the Andreev reflection amplitudes to the symmetry properties of the anomalous Green function in the half metal.

Spin-polarized Josephson current in superconductor/ferromagnet/superconductor junctions with inhomogeneous magnetization

We study numerically the properties of spin transport and charge transport in a current-biased nanoscale diffusive superconductor/ferromagnet/superconductor junction when the magnetization texture is nonuniform. Specifically, we incorporate the presence of a Bloch/Neel domain walls and conical ferromagnetism, including the role of spin-active interfaces. The superconducting leads are assumed to be of the conventional $s$-wave type. In particular, we investigate how the $0\text{\ensuremath{-}}\ensuremath{\pi}$ transition is influenced by the inhomogeneous magnetization texture and focus on the particular case where the charge current vanishes while the spin current is nonzero. In the case of a Bloch/Neel domain wall, the spin current can be seen only for one component of the spin polarization, whereas in the case of conical ferromagnetism the spin current has the three components. This is in contrast to a scenario with a homogeneous exchange field, where the spin current vanishes completely. We explain all of these results in terms of the interplay between the triplet anomalous Green's function induced in the ferromagnetic region and the local direction of the magnetization vector in the ferromagnet. Interestingly, we find that the spin current exhibits discontinuous jumps at the $0\text{\ensuremath{-}}\ensuremath{\pi}$ transition points of the critical charge current. This is seen both in the presence of a domain wall and for conical ferromagnetism. We explain this result in terms of the different symmetry obeyed by the current-phase relation when comparing the charge and spin currents. Specifically, we find that whereas the charge current obeys the well-known relation ${I}_{c}(\ensuremath{\phi})=\ensuremath{-}{I}_{c}(\ensuremath{-}\ensuremath{\phi})$, the spin current satisfies ${I}_{s}(\ensuremath{\phi})={I}_{s}(\ensuremath{-}\ensuremath{\phi})$, where $\ensuremath{\phi}$ is the superconducting phase difference.

Spin fluctuation processes in the problem of s-wave superconducting phase of the heavy-fermion intermetallides

In the framework of the periodic Anderson model in the limit of strong electron correlations a theory of s-wave superconductivity (this phase has been recently discovered in the skutterudite LaFe4P12) has been developed. The theory is based on kinematic mechanism, provided that the spin fluctuation processes (SFP) are taken into account.To obtain the self-consistent equations for the superconducting phase the diagram technique for Hubbard operators has been used. The exact representations of normal and anomalous Green's functions in terms of normal and anomalous components of mass and strength operators have been obtained. A set of integral equations for superconducting phase was deduced with the assumption that SFP have been accounted for. In the solution of these equations normal and anomalous components of the mass and strength operators have been calculated in the one-loop approximation. Computation has revealed that SFP result in substantial decreasing of critical temperature Tc and shifting the concentration critical point of the superconducting phase transition.

AC Josephson effect in the long voltage-biased SINIS junction

Theory of non-stationary coherent effects is developed for superconductor-normal-superconductor (SNS) structures with relatively strong normal scattering on S/N interfaces (interface resistance is large compared to intrinsic resistance of N metal). Analitical expressions are found for the time-dependent anomalous Green functions induced in the N region under the fixed-voltage-bias. The amplitude of the current oscillations is determined in non-equilibrium conditions. Non-stationary correction to the distribution function is calculated in high-temperature limit and found to be slowly decreasing with the temperature, leading to the dominance of the second-harmonic term in the Josepshon current at high temperatures and low voltage.

Proximity effect in ferromagnet/superconductor hybrids: From diffusive to ballistic motion

We present an analytical study of the proximity effect in ferromagnet/superconductor (F/S) heterostructures, allowing for an arbitrary magnetic exchange energy as well as arbitrary impurity and spin-flip scattering rates within a quasiclassical approach. While previous studies mainly have focused on the clean or dirty limits, our results grant access to the regime of intermediate impurity concentrations, thus allowing us to probe the crossover from the clean to the dirty limit. We find that in the crossover regime, all possible symmetry correlations of the proximity-induced anomalous Green's function are induced in the ferromagnet. We also point out that the local density of states oscillates spatially not only for an F/S bilayer but also for a normal/superconductor (N/S) bilayer in the diffusive limit, a fact which appears to have gone unnoticed in the literature. Within the weak proximity-effect regime, we present compact analytical expressions valid for arbitrary exchange fields and impurity-scattering rates for (i) the local density of states in an F/S bilayer, (ii) the Josephson current in an S/F/S junction, and (iii) the critical temperature in an F/S/F multilayer. For all cases, we study in particular the crossover regime between diffusive and ballistic motion. Our results may be useful for analyzing experimental data in cases when the dirty limit is not fully reached, thus invalidating the use of the Usadel equation.

Electronic spectrum and superconductivity in Hubbard model

A microscopic theory of electronic spectrum and superconducting pairing within the Hubbard model is formulated. The Dyson equation for the normal and anomalous Green functions in terms of the Hubbard operators is derived by applying the Mori-type projection technique. The self-energy is evaluated in the noncrossing approximation for electron scattering on spin and charge uctuations induced by kinematic interaction for Hubbard operators. Numerical results for electron dispersion in the strong correlation limit are presented. Superconducting pairing mediated by antiferromagnetic exchange and spin uctuations is discussed.

Role of interface transparency and spin-dependent scattering in diffusive ferromagnet/superconductor heterostructures

We present a numerical study of the density of states in a ferromagnet/superconductor junction and the Josephson current in a superconductor/ferromagnet/superconductor junction in the diffusive limit by solving the Usadel equation with Nazarov's boundary conditions. Our calculations are valid for an arbitrary interface transparency and an arbitrary spin-dependent scattering rate, which allows us to explore the entire proximity-effect regime. We first investigate how the proximity-induced anomalous Green's function affects the density of states in the ferromagnet for three magnitudes of the exchange field $h$ compared to the superconducting gap $\ensuremath{\Delta}$: (i) $h\ensuremath{\lesssim}\ensuremath{\Delta}$, (ii) $h\ensuremath{\gtrsim}\ensuremath{\Delta}$, and (iii) $h⪢\ensuremath{\Delta}$. In each case, we consider the effect of the barrier transparency and allow for various concentrations of magnetic impurities. We clarify features that may be expected in the various parameter regimes accessible for the ferromagnetic film, with regard to thickness and exchange field. In particular, we address how the zero-energy peak and minigap observed in experiments may be understood in terms of the interplay between the singlet and the triplet anomalous Green's functions and their dependence on the concentration of magnetic impurities. Our results should serve as a useful tool for the quantitative analysis of experimental data. We also investigate the role of the barrier transparency and spin-flip scattering in a superconductor/ferromagnet/superconductor junction. We suggest that such diffusive Josephson junctions with large residual values of the supercurrent at the $0\text{\ensuremath{-}}\ensuremath{\pi}$ transition, where the first harmonic term in the current vanishes, may be used as efficient supercurrent-switching devices. We numerically solve for the Josephson current in such a junction to clarify to what extent this idea may be realized in an experimental setup. It is also found that uniaxial spin-flip scattering has a very different effect on the $0\text{\ensuremath{-}}\ensuremath{\pi}$ transition points depending on whether one considers the width or the temperature dependence of the current. Our theory takes into account vital elements that are necessary to obtain quantitative predictions of the supercurrent in such junctions.