We develop the symmetry classification of superconducting gap functions in electron bands that do not transform under the crystal point group operations like the pure spin-1/2 states. The Bloch state bases in twofold degenerate bands with spin-orbit coupling are defined across the Brillouin zone in the way which satisfies the symmetry and continuity requirements. These bases are used to construct general multiband pairing Hamiltonians in centrosymmetric crystals. Focusing on single-band pairing, four exceptional cases are identified in which the triplet gap function does not transform under the point group operations as a pseudovector, with a significant impact on the nodal structure.
Hund's coupling in multiorbital systems allows for the possibility of even-parity orbital-antisymmetric spin-triplet pairing, which can be stabilized by spin-orbit coupling (SOC). While this pairing expressed in the orbital basis is uniform and spin-triplet, it appears in the band basis as a pseudospin-singlet, with the momentum dependence determined by the SOC and the underlying triplet character remaining in the form of interband pairing active away from the Fermi energy. Here, we examine the role of momentum-dependent SOC in generating nontrivial pairing symmetries, as well as the hidden tr…
The lasting puzzle of the superconducting order parameter of Sr$_2$RuO$_4$ calls for theoretical studies that include seldom-considered effects. Here we include spin-orbit coupling effects on the electronic structure and then solve the linearized Eliashberg equation in a pseudospin basis, including the possibility that spin and charge fluctuations induce frequency-dependent superconducting order parameters. We find that spin-orbit coupling mixes even and odd contributions in orbital, spin and frequency spaces and that leading inter-pseudospin symmetries, B$_{1g}^+$ and A$_{2g}^-$, have intra-o…
We study the spin susceptibility of the orbital-singlet pairings, including the spin-triplet/orbital-singlet/$s$-wave $E_g$ representation proposed by Suh \textit{et al}., [H.\ G.\ Suh \textit{et al}., Phys.\ Rev.\ Research \textbf{2}, 032023 (2020)], for a three-orbital model of superconducting Sr$_2$RuO$_4$ in three dimensions. For the pseudospin-singlet states represented in the band basis, the spin susceptibility decreases when reducing the temperature, irrespective of the direction of the applied magnetic fields, even if they are spin-triplet/orbital-singlet pairings in the spin-orbital s…
We study possible superconductivity in a carrier-doped iridium oxide insulator Sr$_{2}$IrO$_{4}$ based on an effective $t_{2g}$ three-orbital Hubbard model on the square lattice with a large spin-orbit coupling (SOC). Numerically solving the linearized Eliashberg equation for the superconducting (SC) gap function with the random phase approximation, we systematically examine both singlet and triplet SC gap functions with possible pairing symmetries and the parameter dependence of the superconductivity. For the realistic SOC $λ$ and Hund's coupling $J/U$ relevant to Sr$_{2}$IrO$_{4}$, namely, f…
The angular momentum of an electron is characterized well by pseudospin with $J=3/2$ in the presence of strong spin-orbit interactions. We study theoretically the Josephson effect of superconductors in which such two $J=3/2$ electrons form a Cooper pair. Within even-parity symmetry class, pseudospin-quintet pairing states with $J=2$ can exist as well as pseudospin-singlet state with $J=0$. We focus especially on the Josephson selection rule among these even-parity superconductors. We find that the selection rule between quintet states is severer than that between spin-triplet states formed by …
The superconducting order parameter of strontium ruthenate is the center of a lasting puzzle calling for theoretical studies that include the seldom-considered effects of spin-orbit coupling and the frequency-dependence of the order parameters. Here we generalize the frequency-dependent theory of superconductivity mediated by spin and charge fluctuations to include spin-orbit coupling in multi-orbital systems and we characterize the superconducting states using the spin-parity-orbital-time $SPOT$ quantum numbers, group theory, and phase distributions in the complex plane. We derive a pseudospi…
We introduce a minimal model for 2D cuprates with the on-site Hilbert space reduced to only three effective valence centers CuO$_4^{7-,6-,5-}$ (nominally Cu$^{1+,2+,3+}$) and make use of the S=1 pseudospin formalism. Despite its seeming simplicity the model is believed to capture the salient features both of the hole- and electron-doped cuprates. The pseudospin formalism elucidates an unique fermion-boson duality of the doped cuprates, does provide an unified standpoint for classification of the "myriad"\, of electronic phases in cuprates and the evolution of the CuO$_2$ planes under a nonisov…
We point out that essentially all multi-band superconductors have an odd-frequency pairing component, as follows from a general symmetry analysis of even- and odd-frequency pairing states. We show that odd-frequency superconducting pairing requires only a finite band hybridization, or scattering, and non-identical intraband order parameters, of which only one band needs to be superconducting. Under these conditions odd-frequency odd-interband pairing is always present. From a symmetry analysis we establish a complete reciprocity between parity in band-index and frequency.
We present a theory of superconductivity in twisted bilayer graphene in which attraction is generated between electrons on the same honeycomb sublattice when the system is close to a sublattice polarization instability. The resulting Cooper pairs are spin-polarized valley-singlets. Because the sublattice polarizability is mainly contributed by interband fluctuations, superconductivity occurs over a wide range of filling fraction. It is suppressed by i) applying a sublattice polarizing field (generated by an aligned BN substrate) or ii) changing moiré band filling to favor valley polarization. …
Motivated by recent efforts to achieve cold fermions pairing near a Feshbach resonance, we consider the dynamics of formation of the Bardeen-Cooper-Schrieffer (BCS) state. At times shorter than the quasiparticle energy relaxation time, after the interaction is turned on, the dynamics of the system is nondissipative. We show that this collective nonlinear evolution of the BCS-Bogoliubov amplitudes (u,v) along with the pairing function, is an integrable dynamical problem, and obtain a family of exact solutions in the form of single solitons and soliton trains. We interpret the collective oscilla…
Using the {\it ab initio} anisotropic Eliashberg theory including Coulomb interactions, we investigate the electron-phonon interaction and the pairing mechanism in the recently-reported superconducting Ca-intercalated bilayer graphene. We find that C$_6$CaC$_6$ can support phonon-mediated superconductivity with a critical temperature $T_{\rm c}=$6.8-8.1~K, in good agreement with experimental data. Our calculations indicate that the low-energy Ca$_{xy}$ vibrations are critical to the pairing, and that it should be possible to resolve two distinct superconducting gaps on the electron and hole Fe…
We consider the necessary conditions of superconducting pairing at repulsive interaction between particles composing a pair with large total momentum: (1) the existence of, at least, one negative eigenvalue of the repulsion potential and (2) mirror nesting of the Fermi contour. Under these conditions, we represent the solution of the self-consistency equation continuously depending on the momentum of the relative motion of the pair. The corresponding superconducting order parameter changes its sign on a line crossing the Fermi contour inside the domain of definition of the relative motion pair…
We analyze various possible superconducting pairing states and their relative stabilities in lightly doped graphene. We show that, when inter-sublattice electron-electron attractive interaction dominates and Fermi level is close to Dirac points, the system will favor intra-valley spin-triplet $p+\mathrm{i}p$ pairing state. Based on the novel pairing state, we further propose a scheme for doing topological quantum computation in graphene by engineering local strain fields and external magnetic fields.
Finite-momentum superconducting states due to odd-frequency Cooper pairing correlations
Graduate School of Science, Hokkaido University（北海道大学大学院理学研究科）；RIKEN Center for Emergent Matter Science（理化学研究所新兴物质科学中心）；Department of Applied Physics, Hokkaido University（北海道大学应用物理系）
Some of the iron pnictides show coexisting superconductivity and spin-density-wave order. We study the superconducting pairing instability in the spin-density-wave phase. Assuming that the pairing interaction is due to spin fluctuations, we calculate the effective pairing interactions in the singlet and triplet channels by summing the bubble and ladder diagrams taking the reconstructed band structure into account. The leading pairing instabilities and the corresponding superconducting gap structures are then obtained from the superconducting gap equation. We illustrate this approach for a mini…
Starting from a two-orbital model and based on the random phase approximation, spin excitations in the superconducting state of the newly discovered BiS$_2$ superconductors with three possible pairing symmetries are studied theoretically. We show that spin response is uniquely determined by the pairing symmetry. Possible spin resonance excitations might occur for the d-wave symmetry at an incommensurate momentum about $(0.7π,0.7π)$. For the p-wave symmetry the transverse spin excitation near $(0,0)$ is enhanced. For the s-wave pairing symmetry there is no spin resonance signature. These distin…
Single-particle spectra are calculated in the superconducting state for a fermionic system with an attractive interaction, as functions of temperature and coupling strength from weak to strong. The fermionic system is described by a single-particle self-energy that includes pairing-fluctuation effects in the superconducting state. The theory reduces to the ordinary BCS approximation in weak coupling and to the Bogoliubov approximation for the composite bosons in strong coupling. Several features of the single-particle spectral function are shown to compare favorably with experimental data for …
We show that quasiparticle interference (QPI) due to omnipresent weak impurities and probed by Fourier transform scanning tunneling microscopy and spectroscopy acts as a direct experimental probe of bulk odd-frequency superconducting pairing. Taking the example of a conventional $s$-wave superconductor under applied magnetic field, we show that the nature of the QPI peaks can only be characterized by including the odd-frequency pairing correlations generated in this system. In particular, we identify that the defining feature of odd-frequency pairing gives rise to a bias asymmetry in the QPI, …
We present the derivation of an ab-initio and parameter free effective electron-electron interaction that goes beyond the screened RPA and accounts for superconducting pairing driven by spin-fluctuations. The construction is based on many body perturbation theory and relies on the approximation of the exchange-correlation part of the electronic self-energy within time dependent density functional theory. This effective interaction is included in an exchange correlation kernel for superconducting density functional theory, in order to achieve a completely parameter free superconducting gap equa…
We investigate an unusual symmetry of Fe-based superconductors (FeSCs) and find novel superconducting pairing structures. FeSCs have a minimal translational unit cell composed of two Fe atoms due to the staggered positions of anions with respect to the Fe plane. We study the physical consequences of the additional glide symmetry that further reduces the unit cell to have only one Fe atoms. In the regular momentum space, it not only leads to a particular orbital parity separated spectral function but also dictates orbital parity distinct pairing structures. Furthermore, it produces accompanying…
We investigate the combined effect of Hund's and spin-orbit (SO) coupling on superconductivity in multi-orbital systems. Hund's interaction leads to orbital-singlet spin-triplet superconductivity, where the Cooper pair wave function is antisymmetric under the exchange of two orbitals. We identify three d-vectors describing even-parity orbital-singlet spin-triplet pairings among t2g-orbitals, and find that the three d-vectors are mutually orthogonal to each other. SO coupling further assists pair formation, pins the orientation of the d-vector triad, and induces spin-singlet pairings with a rel…
A layered superconductor with a full pairing energy gap can be driven into a nodal superconducting (SC) state by inter-layer pairing when the SC state becomes more quasi-3D. We propose that this mechanism is responsible for the observed nodal behavior in a class of iron-based SCs. We show that the intra- and inter-layer pairings generally compete and the gap nodes develop on one of the {\em hole} Fermi surface pockets as they become larger in the iron-pnictides. Our results provide a natural explanation of the c-axis gap modulations and gap nodes observed by angle resolved photoemission spectr…
We have introduced a Coulomb correlated normal state to study the superconducting pairing with a Fermi liquid like normal phase background in a 3 dimensional system. The role of the Coulomb correlation has been actively incorporated by means of Gutzwiller projection scheme, in the presence of phonon mediated attractive electron-electron interaction throughout the entire regime of its applicability. We variationally modulate the Coulomb correlation strength from the weak to the strong regime in our calculational procedure. The main highlight of our results is the appearance of a '2-gap-like' st…
In order to study the pairing symmetry in non-centrosymmetric superconductors, we solve the linearized Eliashberg's equation on the two-dimensional extended Hubbard model in the presence of the Rashba-type spin-orbit coupling (RSOC) within the random phase approximation. In the presence of the RSOC, three types of pairing symmetries appear in the phase diagram with respect to the on-site Coulomb repulsion U and off-site one V. Each of pairing symmetries is admixture of spin-singlet and -triplet ones. On the basis of analytical study, it is found that the admixture of spin-singlet and -triplet …
It was recently shown that a chiral topological phase emerges from the coupling of two twisted monolayers of superconducting Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ for certain twist angles. In this work, we reveal the behavior of such twisted superconducting bilayers with $d_{x^2-y^2}$ pairing symmetry in presence of applied magnetic field. Specifically, we show that the emergent vortex matter can serve as smoking gun for detection of topological superconductivity in such bilayers. Moreover, we report two distinct skyrmionic states that characterize the chiral topological phase, and provide full accoun…
In several members of the pnictide materials, spin density wave order coexists with superconductivity over a range of dopings and temperature. In this paper we show that odd-frequency superconductivity emerges on the edges of pnictides in such a coexistence phase. In particular, the breaking of spin-rotation symmetry by SDW and translation symmetry by the edge can lead to the development of odd-frequency spin-triplet Cooper pairing on edges of superconducting pnictide samples. In this case, the odd-frequency pairing has even parity components, which are immune to disorder. Our results show tha…
We show that the phase information contained in coherent phonon oscillations generated by a laser pulse in a multi- orbital superconductor can be used as an experimental tool to probe super- conductivity and pairing symmetries. The phase difference between the normal and superconducting states is proportional to the superconducting order parameter just below the superconducting transition temperature, Tc. It also exhibits different behaviors for superconducting states with different pairing symmetries. In particular, if there is an orbital-dependent internal sign change state, the phase differ…
It is shown that an account for the space separatedness of the two-orbital subsystem of the oxygen holes and the subsystem of the localized spins of copper ions in high-T$_c$ cuprate superconductors leads to the stability of the superconducting $d_{x^2-y^2}$-wave pairing towards the strong Coulomb repulsion between holes located at the nearest oxygen ions. This effect is due to the fact that the Coulomb potential slips out of the equation for the Cooper pairing in the $d_{x^2-y^2}$-wave channel owing to the properties of symmetry.
Electron pairing by dispersive phonons in altermagnets: re-entrant superconductivity and continuous transition to finite momentum superconducting state
研究交错磁体中色散声子介导的电子配对，发现热声子散射抑制交错磁性，导致超导电性随温度重入，并在低温稳定 FFLO 有限动量超导态，无需 d 波配对。
It is shown that an extra magnetization is induced by an onset of the equal-spin-pairing of spin triplet superconductivity if the energy dependence of the density of states of quasiparticles exists in the normal state. It turns out that the effect is observable in Sr_2RuO_4 due to the existence of van Hove singularity in the density of states near the Fermi level, explaining the extra contribution in the Knight shift reported by Ishida et al. It is also quite non-trivial that this effect exists even without external magnetic field, which implies that the time reversal symmetry is spontaneously…
The momentum distribution of the energy gap opening at the Fermi level of superconductors is a direct fingerprint of the pairing mechanism. While the phase diagram of the iron-based superconductors promotes antiferromagnetic fluctuations as a natural candidate for electron pairing, the precise origin of the interaction is highly debated. We used angle-resolved photoemission spectroscopy to reveal directly the momentum distribution of the superconducting gap in FeTe1-xSex, which has the simplest structure of all iron-based superconductors. We found isotropic superconducting gaps on all Fermi su…
We study photoinduced dynamics triggered by an inhomogeneity due to competition between charge density waves (CDWs) and superconductivity. As a simple example, we consider the superconducting (SC) interface between two CDW domains with opposite signs. The real-time dynamics are calculated within the time-dependent Hartree--Fock--Bogoliubov framework, where the order parameter dynamics and the nonequilibrium quasiparticle distribution functions are studied. We also calculate the various dynamical response functions within a generalized random phase approximation. Through comparisons between the…
We theoretically consider superconducting proximity effect, using the Bogoliubov-de Gennes theory, in heterostructure sandwich-type geometries involving a normal s-wave superconductor and a non-superconducting material with the proximity effect being driven by Cooper pairs tunneling from the superconducting slab to the non-superconducting slab. Applications of the superconducting proximity effect may rely on an induced spectral gap or induced pairing correlations without any spectral gap. We clarify that in a non-superconducting material the induced spectral gap and pairing correlations are in…
Cooper's formalism for fermionic pairing has been revisited considering upto 3rd neighbour hopping terms, firstly with a Fermi liquid like background on a square lattice keeping in mind the overdopped cuprates. Then the whole scheme is repeated with a Marginal Fermi liquid- like background, taking into account the self energy correction of the mobile electrons to include a more realitic density of states in the calculation. Detailed comparison of our theoretical results with those from experiments strongly supports the Marginal Fermi liquid- like character of the normal phase with exciton medi…
We discuss general implications of the local spin-triplet pairing among fermions induced by local ferromagnetic exchange, example of which is the Hund's rule coupling. The quasiparticle energy and their wave function are determined for the three principal phases with the gap, which is momentum independent. We utilize the Bogolyubov-Nambu-De Gennes approach, which in the case of triplet pairing in the two-band case leads to the four-components wave function. Both gapless modes and those with an isotropic gap appear in the quasiparticle spectrum. A striking analogy with the Dirac equation is bri…
Starting from a model Hamiltonian for the normal state of the topological insulator Bi2Se3, we construct a pseudospin basis for the single-particle wavefunctions. Considering weak superconducting pairing near the Fermi surface, we express the recently proposed superconducting order parameters for Cu doped Bi2Se3 in this basis. For the odd parity states, the d-vectors specifying the order parameter can have unusual momentum dependence for certain parameter regimes. Some peculiar results in the literature for surface states are discussed in light of the forms of these d(k)'s. Properties of the e…
To investigate the non-Fermi liquid behavior we consider the extended two-dimensional t-J model which includes additional hopping t''. In the regime t,J << t'' we were able to solve the model analytically. It has a very rich phase diagram including antiferromagnetic (AF) insulator and AF strange metal with different kinds of pseudospin-singlet superconducting pairings (p,d,g-waves). We also demonstrate a collective triplet excitation with energy below the superconducting gap
We probe the local quasiparticles density-of-states in micron-sized SmFeAsO$_{1-x}$F$_{x}$ single-crystals by means of Scanning Tunnelling Spectroscopy. Spectral features resemble those of cuprates, particularly a dip-hump-like structure developed at energies larger than the gap that can be ascribed to the coupling of quasiparticles to a collective mode, quite likely a resonant spin mode. The energy of the collective mode revealed in our study decreases when the pairing strength increases. Our findings support spin-fluctuation-mediated pairing in pnictides.
We study the proximity-induced superconducting correlations in a normal metal connected to a superconductor when the interface between them is spin-active and the normal metal is ballistic or diffusive. Remarkably, for any interface spin polarization there is a critical interface resistance, above which the conventional even-frequency proximity component vanishes completely at the chemical potential, while the odd-frequency component remains finite. We propose a way to unambiguously observe the odd-frequency component.
We investigate characteristic features in the spin response of doped three-dimensional topological insulators with odd-parity unequal-spin superconducting pairing, which are predicted to have gapless Majorana surface modes. These Majorana modes contribute to the local spin susceptibility, giving rise to a characteristic temperature behavior of the Knight shift and the spin-lattice relaxation time in magnetic resonance experiments. Because of their different decay lengths, the Majorana modes can be observed and clearly distinguished from the Dirac modes of the topological insulator by local pro…
We show the intrinsic inconsistency of the conventional phonon mediated theory of superconductivity in relation to the observed properties of Rb$_3$C$_{60}$. The recent, highly accurate measurement of the carbon isotope coefficient $α_{\rm C}=0.21$, together with the high value of $T_c$ (30 K) and the very small Fermi energy $E_{\rm F}$ (0.25 eV), unavoidably implies the opening of nonadiabatic channels in the superconducting pairing. We estimate these effects and show that they are actually the key elements for the high value of $T_c$ in these materials compared to the very low values of grap…
Fresnel single aperture diffraction (FSAD) is proposed as a phase-sensitive probe for pairing symmetry and Fermi surface of a superconductor. We consider electrons injected, through a small aperture, into a thin superconducting (SC) layer. It is shown that in case of SC gap symmetry $Δ(-k_x,\mathbf{k}_\parallel)=Δ(k_x,\mathbf{k}_\parallel)$ with $k_x$ and $\mathbf{k}_\parallel$ respectively the normal and parallel component of electron Fermi wavevector, quasiparticle FSAD pattern developed at the image plane is zeroth-order minimum if $k_x x=nπ$ ($n$ is an integer and $x$ is SC layer thickness…
We report the results of a zero-field muon spin relaxation (ZF-muSR) study of superconducting Ba1-xKxFe2As2 (0.5 < x < 0.9) in search of weak spontaneous internal magnetic fields associated with proposed time-reversal-symmetry breaking mixed pairing states. The measurements were performed on polycrystalline samples, which do not exhibit the mesoscopic phase separation previously observed in single crystals of Ba1-xKxFe2As2. No evidence of spontaneous internal magnetic fields is found in any of the samples at temperatures down to T ~ 0.02 K.
Understanding the nature of Cooper pairs is essential to describe the properties of superconductors. The original proposal of Bardeen, Cooper, and Schrieffer (BCS) was based on electrons pairing with same energy and zero center-of-mass momentum. With the advent of new superconductors, different forms of pairing have been discussed. In particular, Cooper pairs with finite center-of-mass momentum have received large interest. Along with such finite-momentum pairs, pairing of electrons at different energies is also central to understanding some superconductors. Here, we investigate the interplay …
We revisit the Aslamazov-Larkin theory of paraconductivity in two dimensions, to distinguish its universal features from the specific features of nearly-free paired fermions. We show that both the numerical prefactor and the temperature dependence of the experimental paraconductivity in underdoped La_{2-x}Sr_xCuO_4 are only compatible with pairing of nearly-free fermionic quasiparticles. This conclusion is strengthened by the analysis of paraconductivity data in the presence of a finite magnetic field, from which we extract a rather low value of the critical field H_{c2}(T=0).
Unconventional superconductivity is usually associated with symmetry breaking in the system. Here we consider a simple setup consisting of a solid state material with conduction electrons and an applied surface acoustic wave (SAW), that can break time and spatial translation symmetries. We study the symmetries of the possible SAW-induced order parameters, showing that even-frequency spin-triplet odd-parity order parameter can occur. We suggest different methods of how to engineer the symmetries of the order parameters using SAWs and the applications of such setups.
We have investigated the doping dependence of the penetration depth vs. temperature in electron doped Pr$_{2-x}$Ce$_x$CuO$_{4-δ}$ using a model which assumes the uniform coexistence of (mean-field) antiferromagnetism and superconductivity. Despite the presence of a $d_{x^2-y^2}$ pairing gap in the underlying spectrum, we find nodeless behavior of the low-$T$ penetration depth in underdoped case, in accord with experimental results. As doping increases, a linear-in-$T$ behavior of the penetration depth, characteristic of d-wave pairing, emerges as the lower magnetic band crosses the Fermi level…
We generalize the $η$-pairing theory in Hubbard models to the ones with spin-orbit coupling (SOC) and obtain the conditions under which the $η$-pairing operator is an eigenoperator of the Hamiltonian. The $η$ pairing thus reveals an exact $SU(2)$ pseudospin symmetry in our spin-orbit coupled Hubbard model, even though the $SU(2)$ spin symmetry is explicitly broken by the SOC. In particular, these exact results can be applied to a variety of Hubbard models with SOC on either bipartite or non-bipartite lattices, whose noninteracting limit can be a Dirac semimetal, a Weyl semimetal, a nodal-line …
