The Eliashberg theory of superconductivity is based on a dynamical electron-phonon interaction as opposed to a static interaction present in BCS theory. The standard derivation of Eliashberg theory is based on an equation of motion approach, which incorporates certain approximations such as Migdal's approximation for the pairing vertex. In this paper we provide a functional-integral-based derivation of Eliashberg theory and we also consider its Gaussian-fluctuation extension. The functional approach enables a self-consistent method of computing the mean-field equations, which arise as saddle-point conditions, and here we observe that the conventional Eliashberg self energy and pairing function both appear as Hubbard-Stratonovich transformations. An important consequence of this fact is that it provides a systematic derivation of the Cooper and density-channel interactions in the Gaussian fluctuation response. We also investigate the strong-coupling fluctuation diamagnetic susceptibility near the critical temperature.
The temperature and frequency dependences of the optical conductivity of Co and Ni-doped BaFe2As2 are analyzed and the electron-boson spectral density a2F(w) extracted using Eliashberg's formalism. The characteristic energy of a large peak in the spectrum around 10 meV coincide with the resonance peak in the spin excitation spectra, giving compelling evidence that in iron-based superconductors spin fluctuations strongly couple to the charge carriers and mediate superconductivity. In addition the spectrum is found to evolve with temperature towards a less structured background at higher energie…
We present a novel method for embedding spin and charge fluctuations in an anisotropic, multi-band and full-bandwidth Eliashberg treatment of superconductivity. Our analytical framework, based on the random phase approximation, allows for a selfconsistent calculation of material specific characteristics in the interacting, and more specifically, the superconducting state. We apply this approach to bulk FeSe as representative for the iron-based superconductors and successfully solve for the superconducting transition temperature $T_c$, the gap symmetry and the gap magnitude. We obtain $T_c \app…
In this paper we propose a model to reproduce superconductive and normal properties of the iron pnictide LiFeAs in the framework of the four-band spm wave Eliashberg theory. A confirmation of the multiband nature of the system rises from the experimental measurements of the superconductive gaps and resistivity as function of temperature. We found that the most plausible mechanism is the antiferromagnetic spin fluctuation and the estimated values of the total antiferromagnetic spin fluctuation coupling constant in the superconductive and normal state are lambda{tot}=2.00 and lambda{tot,tr}=0.77…
We study charge ordered solutions for fermions on a square lattice interacting with dynamic antiferromagnetic fluctuations. Our approach is based on real space Eliashberg equations which are solved self-consistently. We first show that the antiferromagnetic fluctuations can induce arc features in the spectral functions, as spectral weight is suppressed at the hot spots; however, no real pseudogap is generated. At low temperature spontaneous charge order with a $d$-form factor can be stabilized for certain parameters. As long as the interacting Fermi surfaces possesses hot spots, the ordering w…
High-Tc pairing mechanism absent of hole-pockets in heavily electron-doped FeSe is one of the key unsolved problems in Fe-based superconductors. Here, this problem is attacked by focusing on the higher-order many-body effects neglected in conventional Migdal-Eliashberg formalism. We uncover two significant many-body effects for high-Tc superconductivity: (i) Due to the "vertex correction", the dressed multiorbital Coulomb interaction acquires prominent orbital dependence for low-energy electrons. The dressed Coulomb interaction not only induces the orbital fluctuations, but also magnifies the …
We investigated the characteristics of the spin fluctuations mediated superconductivity employing the Eliashberg formalism. The effective interaction between electrons was modeled in terms of the spin susceptibility measured by the inelastic neutron scattering experiments on single crystal La2-xSrxCuO4 superconductors. The diagonal self-energy and off-diagonal self-energy were calculated by solving the coupled Eliashberg equation self-consistently for chosen spin susceptibility and tight-binding dispersion of electrons. The full momentum and frequency dependence of the self-energy is presented…
We deduce the normal state angle-resolved single-particle self-energy $Σ(θ, ω)$ and the Eliashberg function (i.e., the product of the fluctuation spectrum and its coupling to fermions) $α^2 F(θ,ω)$ for the high temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ from the ultra high resolution laser angle-resolved photoemission spectroscopy (ARPES). The self-energy $Σ(θ, ω)$ at energy $ω$ along several cuts normal to the Fermi surface at the tilt angles $θ$ with respect to the nodal direction in a slightly underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ were extracted by fitting the ARPES momentu…
The pairing mechanism and gap structure in Ba122 pnictides have been hotly discussed for long time as one of the central issues in Fe-based superconductors. Here, we attack this problem by taking account of the vertex corrections (VCs) for the Coulomb interaction $U$ ($U$-VCs), which are totally dropped in conventional Migdal-Eliashberg formalism. The $U$-VC in the charge susceptibility induces strong orbital fluctuations, and the $U$-VC also enlarges the orbital-fluctuation-mediated attractive interaction. By analyzing the effective multiorbital Hubbard model for Ba122 pnictides, we find that…
Fast Real-Axis Eliashberg Calculations: Full-bandwidth solutions beyond the constant density of states approximation
Research Laboratory of Electronics, Massachusetts Institute of Technology（麻省理工学院电子研究实验室）；Institute of Theoretical and Computational Physics, Graz University of Technology（格拉茨工业大学理论与计算物理研究所）
We present a necessary condition for odd-frequency (odd-f) superconductivity (SC) to occur in a large class of materials described by Eliashberg theory. We use this condition to prove a no-go theorem ruling out the occurrence of odd-f SC in standard one-band superconductors with pairing interactions mediated by phonon exchange. We also present a corresponding no-go theorem for superconductors with interactions mediated by spin-fluctuations. Our results explain why odd-f SC is rare in conventional materials, and they open up the possibility for a search for materials with interactions designed …
The paradigmatic Migdal-Eliashberg theory of the electron-phonon problem is central to the understanding of superconductivity in conventional metals. This powerful framework is justified by the smallness of the Debye frequency relative to the Fermi energy, and allows an enormous simplification of the full many-body problem. However, superconductivity is found also in many families of strongly-correlated materials, in which there is no {\it a priori} justification for the applicability of Eliashberg theory. In these systems, superconductivity emerges out of an anomalous metallic state, calling …
Superconductivity is abundant near quantum-critical points, where fluctuations suppress the formation of Fermi liquid quasiparticles and the BCS theory no longer applies. Two very distinct approaches have been developed to address this issue: quantum-critical Eliashberg theory and holographic superconductivity. The former includes a strongly retarded pairing interaction of ill-defined fermions, the latter is rooted in the duality of quantum field theory and gravity theory. We demonstrate that both are different perspectives of the same theory. We derive holographic superconductivity in form of…
We theoretically investigate the electric transport in the pseudogap state of High-Tc cuprates. Starting from the repulsive Hubbard model, we perform the microscopic calculation to describe the pseudogap phenomena which are induced by the superconducting fluctuations. The single particle Green function, spin susceptibility and superconducting fluctuations are self-consistently determined by the SC-FLEX+T-matrix approximation. The longitudinal and transverse conductivities are calculated by using the Eliashberg and Kohno-Yamada formalism. The effects of the spin fluctuations and superconducting…
We investigate the effects of electron-phonon coupling, as well as of spin and charge fluctuations on the superconducting state in a single layer of FeSe on SrTiO$_3$ substrate. These three bosonic mediators of Cooper pairing are treated on equal footing in a multichannel, full-bandwidth, multiband, and anisotropic Eliashberg theory of the interacting state. Our self-consistent calculations show that an s-wave symmetry of the superconducting gap is compatible only with a complete absence of spin fluctuations. When spin fluctuations are present, the sign-changing nodeless d-wave pairing symmetr…
We analyze the ultra high resolution laser angle resolved photo-emission spectroscopy (ARPES) intensity from the slightly underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ in the superconductive (SC) state. The momentum distribution curves (MDC) were fitted at each energy $\w$ employing the SC Green's function along several cuts perpendicular to the Fermi surface with the tilt angle $θ$ with respect to the nodal cut. The clear observation of particle-hole mixing was utilized such that the complex self-energy as a function of $ω$ is directly obtained from the fitting. The obtained angle resolved self-en…
Even- and odd-frequency superconductivity coexist due to broken time-reversal symmetry under magnetic field. In order to describe this mixing, we extend the linearized Eliashberg equation for the spin and charge fluctuation mechanism in strongly correlated electron systems. We apply this extended Eliashberg equation to the odd-frequency superconductivity on a quasi-one-dimensional isosceles triangular lattice under in-plane magnetic field and examine the effect of the even-frequency component.
In the Eliashberg integral equations for d-wave superconductivity, two different functions $(α^2 F)_n(ω, θ)$ and $(α^2 F)_{p,d}(ω)$ determine, respectively, the "normal" and the "pairing" self-energies. We present a quantitative analysis of the high-resolution laser based ARPES data on the compound Bi-2212 to deduce the function$(α^2 F)_n(ω, θ)$. Besides its detailed $ω$ dependence, we find the remarkable result that this function is nearly independent of $θ$ between the ($π,π$)-direction and 25 degrees from it. Assuming that the same fluctuations determine both the normal and the pairing self…
We study the superconducting instability mediated by spin fluctuations in the Eliashberg theory for a minimal two-band model of iron-based superconductors. While antiferromagnetic spin fluctuations can drive superconductivity (SC) as is well established, we find that spin fluctuations necessarily contain a contribution to suppress SC even though SC can eventually occur at lower temperatures. This self-restraint effect stems from a general feature of the spin-fluctuation mechanism, namely the repulsive pairing interaction, which leads to phase frustration of the pairing gap and consequently the…
We study the relation between the spin fluctuation and superconductivity in an heavily hole doped end material KFe$_2$As$_2$. We construct a five orbital model by approximately unfolding the Brillouin zone of the three dimensional ten orbital model obtained from first principles calculation. By applying the random phase approximation, we obtain the spin susceptibility and solve the linearized Eliashberg equation. The incommensurate spin fluctuation observed experimentally is understood as originating from interband interactions, where the multiorbital nature of the band structure results in an…
Although correlated electronic-structure calculations explain very well the normal state of Sr$_2$RuO$_4$, its superconducting symmetry is still unknown. Here we construct the spin and charge fluctuation pairing interactions based on its correlated normal state. Correlations significantly reduce ferromagnetic in favor of antiferromagnetic fluctuations and increase inter-orbital pairing. From the normal-state Eliashberg equations, we find spin-singlet $d$-wave pairing close to magnetic instabilities. Away from these instabilities, where charge fluctuations increase, we find two time-reversal sy…
The isotope effects for Fe based superconductors are considered by including the phonon and magnetic fluctuations within the two band Eliashberg theory. We show that the recently observed inverse isotope effects of Fe, $α_{Fe} \approx -0.18 \pm 0.03 $,\cite{Shirage0903.3515} as well as the large positive isotope exponent ($α\approx 0.35$) can naturally arise for the magnetically induced sign revered s-wave pairing state within reasonable parameter range. Either experimental report can not be discarded from the present analysis based on the parameter values they require. The inverse and positiv…
We consider normal state properties, the pairing instability temperature, and the structure of the pairing gap in electron-doped cuprates. We assume that the pairing is mediated by collective spin excitations, with antiferromagnetism emerging with the appearance of hot spots. We use a low-energy spin-fermion model and Eliashberg theory up to two-loop order. We justify ignoring vertex corrections by extending the model to N >>1 fermionic flavors, with 1/N playing the role of a small Eliashberg parameter. We argue, however, that it is still necessary to solve coupled integral equations for the f…
We study the five-orbital Hubbard-Holstein model for iron pnictides with small electron-phonon interaction due to Fe-ion Einstein oscillators. Using the fluctuation-exchange (FLEX) approximation, orbital fluctuations evolve inversely proportional to the temperature, and therefore the resistivity shows linear or convex T-dependence for wide range of temperatures. We also analyze the Eliashberg gap equation, and show that s-wave superconducting state without sign reversal (s_{++}-wave state) emerges when the orbital fluctuations dominate the spin fluctuations. When both fluctuations are comparab…
We investigate the superconducting (SC) state in two-dimensional extended Hubbard model at a quarter-filling with the on-site repulsive interaction ($U$) and the nearest-neighbor one ($V$), which lead to the spin fluctuation and the charge fluctuation, respectively. The effect of the charge fluctuation on the onset of the SC state is calculated by estimating the pairing interaction within the random phase approximation. Solving the linearized Eliashberg equation with the anomalous self-energy as the function of both momentum and frequency, we examined the singlet SC state on the plane of $U$ a…
We study the five-orbital Hubbard model including the charge quadrupole interaction for iron pnictides. Using the fluctuation-exchange (FLEX) approximation, orbital fluctuations evolve inversely proportional to the temperature, and therefore the resistivity shows linear or convex T-dependence for wide range of temperatures. We also analyze the Eliashberg gap equation, and show that s-wave superconducting state without sign reversal (s++ -wave state) emerges when the orbital fluctuations dominate over the spin fluctuations. When both fluctuations are comparable, their competition gives rise to …
We present an advanced method to study spin fluctuations in superconductors quantitatively, and entirely from first principles. This method can be generally applied to materials where electron-phonon coupling and spin fluctuations coexist. We employ it here to examine the recently synthesized superconductor iron tetraboride (FeB$_4$) with experimental $T_{\mathrm{c}}\sim 2.4$ K [H. Gou \textit{et al.}, Phys. Rev. Lett. \textbf{111}, 157002 (2013)]. We prove that FeB$_4$ is particularly prone to ferromagnetic spin fluctuations due to the presence of iron, resulting in a large Stoner interaction…
We investigate the electronic state and the superconductivity in the 5-orbital Hubbard model for iron pnictides by using the dynamical mean-field theory in conjunction with the Eliashberg equation. The renormalization factor exhibits significant orbital dependence resulting in the large change in the band dispersion as observed in recent ARPES experiments. The critical interactions towards the magnetic, orbital and superconducting instabilities are suppressed as compared with those from the random phase approximation (RPA) due to local correlation effects. Remarkably, the s++-pairing phase due…
We investigate interplay between magnetic fluctuations and superconductivity in the effective five-band Hubbard model for iron-oxypnictide superconductors on the basis of the fluctuation-exchange approximation. As for the normal-state properties, we find the pseudogap behavior in the NMR relaxation rate and the spectral weight in the electron-doped region, while we cannot find such behavior in the hole-doped region. The pseudogap behavior originates from the band structure effect, that is, existence of high density of states just below the Fermi level. Solving the superconducting Eliashberg eq…
Despite of many experimental and theoretical efforts, the pairing symmetry of superconductivity in Sr$_2$RuO$_4$ remains undecided. The accidentally degenerate $d_{x^2-y^2}+ig$ is consistent with most current experiments and seems to be one of the most probable candidates, but we still lack a satisfactory theoretical mechanism for its appearance. Here we construct a phenomenological model combining realistic electronic band structures and all symmetry-allowed multipole fluctuations as potential pairing glues, and make a systematic survey of major pairing states within the Eliashberg framework.…
We study superconducting instability from orbital nematic fluctuations in a minimal model consisting of the $d_{xz}$ and $d_{yz}$ orbitals, and choose model parameters which capture the typical Fermi surface geometry observed in iron-based superconductors. We solve the Eliashberg equations down to low temperatures with keeping the renormalization function and a full momentum dependence of the pairing gap. When superconductivity occurs in the tetragonal phase, we find that the pairing gap exhibits a weak momentum dependence over the Fermi surfaces. The superconducting instability occurs also in…
Possibility of unconventional pairing due to Coulomb interaction in iron-pnictide superconductors is studied by applying a perturbative approach to realistic 2- and 5-band Hubbard models. The linearized Eliashberg equation is solved by expanding the effective pairing interaction perturbatively up to third order in the on-site Coulomb integrals. The numerical results for the 5-band model suggest that the eigenvalues of the Eliashberg equation are sufficiently large to explain the actual high Tc for realistic values of Coulomb interaction and the most probable pairing state is spin-singlet s-wav…
Microscopic theory of a high Tc cuprate BiSCO based on main pairing channel of electrons in CuO planes due to 40meV lateral vibrations of the apical oxygen atoms in adjacent the SrO ionic insulator layer is proposed. The separation between the vibrating charged atoms and the 2D electron gas creates the forward scattering peak leading in turn to the d-wave pairing within Eliashberg formalism. The phonon mode naturally explain the kink in dispersion relation observed by ARPES and the and effect of the O16 to O18 isotope substitution in the normal state. To describe the pseudogap physics a single…
We study the effect of combining spin fluctuations and forward scattering electron-phonon ({\eph}) coupling on the superconductivity in the FeSe/SrTiO$_3$ system modeled by a phenomenological two-band Hubbard model with long-range {\eph} interactions. We treat the electron and phonon degrees of freedom on an equal footing using a \emph{fully} self-consistent FLEX plus Migdal-Eliashberg calculation, which includes a self-consistent determination of the spin fluctuation spectrum. Based on FeSe monolayers, we focus on the case where one of the bands lies below the Fermi level (i.e. incipient), an…
The self-energy, spectral functions and susceptibilities of 2D systems with strong ferromagnetic fluctuations are considered within the quasistatic approach. The self-energy at low temperatures T has a non-Fermi liquid form in the energy window w<Delta near the Fermi level, where Delta is the ground-state spin splitting for magnetically ordered ground state, and Delta ~ T^(1/2) ln^(1/2)(vF/T) in the quantum critical regime (vF is the Fermi velocity). Spectral functions have a two-peak structure at finite T above the magnetically ordered ground state, which implies quasi-splitting of the Fermi …
In various multiorbital systems, the emergence of the orbital fluctuations and its role on the pairing mechanism attract increasing attention. To archive deep understanding on these issues, we perform the functional-renormalization-group (fRG) study for the two-orbital Hubbard model. The vertex corrections for the electron-boson coupling ($U$-VC), which are dropped in the Migdal-Eliashberg gap equation, are obtained by solving the RG equation. We reveal that the dressed electron-boson coupling for the charge-channel, $U_{eff}^c$, becomes much larger than the bare Coulomb interaction, $U^0$, du…
I examine electron-phonon mediated superconductivity in the intermediate coupling and phonon frequency regime of the quasi-2D Holstein model. I use an extended Migdal-Eliashberg theory which includes vertex corrections and spatial fluctuations. I find a d-wave superconducting state that is unique close to half-filling. The order parameter undergoes a transition to s-wave superconductivity on increasing filling. I explain how the inclusion of both vertex corrections and spatial fluctuations is essential for the prediction of a d-wave order parameter. I then discuss the effects of a large Coulom…
Identification of pairing mechanisms leading to the unconventional superconductivity realized in copper-oxide, heavy-fermions, and organic compounds is one of the most challenging issues in condensed-matter physics. Clear evidence for an electron-phonon mechanism in conventional superconductors is seen by the isotope effect on the superconducting transition temperatures $T_{\rm SC}$, since isotopic substitution varies the phonon frequency without affecting the electronic states. In unconventional superconductors, magnetic fluctuations have been proposed to mediate superconductivity, and consid…
We consider the real part of the conductivity, σ_1(ω), arising from classical phase fluctuations in a model for high-T_c superconductors. We show that the frequency integral of that conductivity, \int_0^\infty σ_1 dω, is non-zero below the superconducting transition temperature $T_c$, provided there is some quenched disorder in the system. Furthermore, for a fixed amount of quenched disorder, this integral at low temperatures is proportional to the zero-temperature superfluid density, in agreement with experiment. We calculate σ_1(ω) explicitly for a model of overdamped phase fluctuations.
Electron-phonon interaction plays an important role in metals and can lead to superconductivity and other instabilities. Previous theoretical studies on superconductivity are largely based on the Migdal-Eliashberg theory, which neglects all the vertex corrections to electron-phonon coupling and breaks down in many unconventional superconductors. Here, we go beyond the Migdal-Eliashberg approximation and develop a nonperturbative Dyson-Schwinger equation approach to deal with the superconducting transition. Remarkably, we take into account all the vertex corrections by solving two coupled Ward-…
We reformulate the Eliashberg's equation for the superconducting transition within the quasi-particle description, which takes account of the heavy electron mass. We discuss the superconductivity of CeIr_{x}Co_{1-x}In_5 by using such a renormalized formula. On the basis of an effective two-dimensional Hubbard model which represents the most heavy quasi-two dimensional Fermi surface with f-character of CeIr_{x}Co_{1-x}In_5, both normal and anomalous self-energies are calculated up to third order with respect to the renormalized on-site repulsion U between quasi-particles. The superconducting tr…
By using a Path Integral formalism, we study the fluctuating fields in the order parameter for a 3D uniform Fermi gas at T<Tc. The BCS-BEC crossover condition is included into the analysis in the framework of the usual s-wave scattering approximation. The general expressions for the temperature dependence of the partition function and the quadratic average in the fluctuations fields are explicitly established in terms of the BCS-Gor'kov correlation propagators.
In Part I we discuss accumulating experimental evidence related to the structure and origin of the bosonic spectral function in high-temperature superconducting (HTSC) cuprates at and near optimal doping. Some global properties of the spectral function, such as number and positions of peaks, are extracted by combining optics, neutron scattering, ARPES and tunnelling measurements. These methods give convincing evidence for strong electron-phonon interaction (EPI) with the coupling constant between 1-3 in cuprates near optimal doping. Here we clarify how these results are in favor of the Eliashb…
We study the combined effect of thermal and quantum fluctuations in a zero dimensional superconductor. By using path integral techniques, we obtain novel expressions for the partition function and the superconducting order parameter which include both types of fluctuations. Our results are valid for any temperature and to leading order in δ/Δ_{0} where δis the mean level spacing and Δ_{0} is the bulk energy gap. We avoid divergences at low temperatures, previously reported in the literature, by identifying and treating non-perturbatively a low-energy collective mode. In the low and high temper…
The opening of a critical-fluctuation induced pseudogap (or precursor pseudogap) in the one-particle spectral weight of the half-filled two-dimensional Hubbard model is discussed. This pseudogap, appearing in our Monte Carlo simulations, may be obtained from many-body techniques that use Green functions and vertex corrections that are at the same level of approximation. Self-consistent theories of the Eliashberg type (such as the Fluctuation Exchange Approximation) use renormalized Green functions and bare vertices in a context where there is no Migdal theorem. They do not find the pseudogap, …
Using the fluctuation-exchange (FLEX) approximation we study an effective five-band Hubbard model for iron-pnictide superconductors obtained from the first-principles band structure. We preclude deformations of the Fermi surface due to electronic correlations by introducing a static potential, which mimics the effect of charge relaxation. Evaluating the Eliashberg equation for various dopings and interaction parameters, we find that superconductivity can sustain higher hole than electron doping. Analyzing the symmetry of the superconducting order parameter we observe clear differences between …
We present a standard field theoretical derivation of the dynamic density and spin linear response functions of a dilute superfluid Fermi gas in the BCS-BEC crossover in both three and two dimensions. The derivation of the response functions is based on the elegant functional path integral approach which allows us to calculate the density-density and spin-spin correlation functions by introducing the external sources for the density and the spin density. Since the generating functional cannot be evaluated exactly, we consider two gapless approximations which ensure a gapless collective mode (G…
