Based on a recently developed framework, we conduct classifications of time-reversal symmetric topological superconductors with conventional pairing symmetries. Our real-space approach clarifies the nature of boundary modes in nontrivial phases. The key difference from the calculations for topological crystalline insulators originates from the appearance of vortex zero modes on the interface of several two-dimensional topological superconductors. We find that our classification is complete in the $K$-theory sense for all rod groups, all layer groups, and 159 out of 230 space groups. Our results shed new light on superconductors with conventional pairing as candidates for topological superconductors.
The induction of an s-wave component in a d-wave superconductor is considered. Near the {110}-oriented edges of such a sample, the induced s-wave order parameter together with d-wave component forms a complex combination d+e^{iϕ} s, which breaks the time reversal symmetry (BTRS) of the pairing state. As a result, the spontaneous current is created. We numerically study the current distribution and the formation of the spontaneous flux induced by the current. We show that the spontaneous flux formed from a number of defect lines with {110} orientation has a measurable strength. This result may …
We investigate the competition between the extended $s_{\pm}$-wave and $d_{x^2-y^2}$-wave pairing order parameters in the iron-based superconductors. Because of the frustrating pairing interactions among the electron and the hole fermi pockets, a time reversal symmetry breaking $s+id$ pairing state could be favored. We analyze this pairing state within the Ginzburg-Landau theory, and explore the experimental consequences. In such a state, spatial inhomogeneity induces supercurrent near a non-magnetic impurity and the corners of a square sample. The resonance mode between the $s_{\pm}$ and $d_{…
Recent observations of two nodeless gaps in superconducting CeCu$_2$Si$_2$ have raised intensive debates as to its exact gap structure of either sign-reversal ($s^{+-}$) or sign-preserving ($s^{++}$) pairing. Here we investigate the quasiparticle interference (QPI) using realistic Fermi surface topology for both weak and strong interband impurity scatterings. Our calculations of the QPI and integrated antisymmetrized local density of states reveal qualitative distinctions between $s^{+-}$ and $s^{++}$ pairing states, which include the intragap impurity resonance and a significant energy-depend…
Noncentrosymmetric superconductors have sparked significant research interests due to their exciting properties, such as the admixture of spin-singlet and spin-triplet Cooper pairs. Here we report $μ$SR and thermodynamic measurements on the noncentrosymmetric superconductor La7Rh3 which indicate a fully established superconducting gap and spontaneous time-reversal-symmetry breaking at the onset of superconductivity. We show that our results pose severe constraints on any microscopic theory of superconductivity in the system. A symmetry analysis identifies ground states compatible with time-rev…
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.
We investigate pairing states realized at the (001) interface of a spin-triplet superconductor Sr$_2$RuO$_4$ on the basis of microscopic calculations. Because of a Rashba-type spin-orbit interaction induced at the interface, strong parity-mixing of Cooper pairs between a spin-singlet state and a spin-triplet state occurs in this system. There are also strong inter-band pair correlations between the spin-orbit split bands, in spite of the considerably large spin-orbit splitting. This is due to frustration between the spin-orbit interaction and pairing interactions. In this pairing state, time-r…
The traditional symmetries that protect superconductivity are time-reversal and inversion. Here, we examine the minimal symmetries protecting superconductivity in two dimensions and find that time-reversal symmetry and inversion symmetry are not required, and having a combination of either symmetry with a mirror operation on the basal plane is sufficient. We classify superconducting states stabilized by these two symmetries, when time-reversal and inversion symmetries are not present, and provide realistic minimal models as examples. Interestingly, several experimentally realized systems, such…
The multipocket Fermi surfaces of iron-based superconductors promote pairing states with both s_{+-}-wave and d_{x^2-y^2}-wave symmetry. We argue that the competition between these two order parameters could lead to a time-reversal-symmetry breaking state with s+id-pairing symmetry in the iron-based superconductors, and propose serveral scenarios in which this phase may be found. To understand the emergence of such a pairing state on a more rigorous footing, we start from a microscopic 5-orbital description representative for the pnictides. Using a combined approach of functional renormalizati…
We propose a modified Josephson corner-junction experiment which can test whether the order parameter in the iron pnictides changes sign between the electron and hole pockets of the Fermi surface.
Motivated by recent experimental progress, we study the effect of mechanical deformations on the superconducting pairing symmetries in monolayer black phosphorus (MBP). Starting with phonon-mediated intervalley spin-singlet electron-electron pairing and making use of realistic band parameters obtained through first-principles calculations, we show that the application of weak mechanical strain in the plane of MBP can change the effective $s$-wave and $p$-wave symmetry of the superconducting correlations into effective $d$-wave and $f$-wave symmetries, respectively. This prediction of a change …
We study time-reversal-invariant topological superconductivity in topological insulator (TI) thin films including both intra- and inter-surface pairing. We find a nontrivial topology for multiple different configurations. For intra-surface pairing a $π$-phase difference between the intra-surface pairing states is required. We show that in this case the resulting topological phase is highly tunable by both an applied electric field and varied chemical potential. For spin-singlet inter-surface pairing, a sign-changing tunnel coupling present in many TI thin films is needed, and again, the topolo…
A detailed analysis of the symmetric tetracrystal geometry used in phase-sensitive pairing symmetry experiments on high Tc superconductors is carried out for both bulk and surface time-reversal symmetry-breaking states, such as the d+id' and d+is states. The results depend critically on the substrate geometry. In the general case, for the bulk d+id' (or d+is) state, the measured flux quantization should in general not be too different from that obtained in the pure d-wave case, provided |d'| << |d| (or |s| << |d|). However, in one particular high symmetry geometry, the d+id' state gives result…
The single helical Fermi surface on the surface state of three-dimensional topological insulator Bi2Se3 is constrained by the time-reversal invariant bulk topology to possess a spin-singlet superconducting pairing symmetry. In fact, the Cu-doped, and pressure-tuned superconducting Bi2Se3 show no evidence of the time reversal symmetry breaking. We report on the detection of the time reversal symmetry (TRS) breaking in the topological superconductor Sr0.1Bi2Se3 , probed by zero-field (ZF) μSR measurements. The TRS breaking provides strong evidence for the existence of spin-triplet pairing state.…
We study the pairing of fermions by an interaction consisting of a Hubbard repulsion, mimicking a screened Coulomb potential, and a dynamical phonon-mediated attraction. For such interaction, the gap equation allows even- and odd-frequency solutions $Δ_e$ and $Δ_o$. We show that odd-frequency pairing does not develop within the Eliashberg approximation due to over-critical pair-breaking from the self-energy. When vertex corrections are included, the pairing interaction gets stronger, and $Δ_o$ can develop. We argue that even in this case keeping the self-energy is still a must as it cancels ou…
Pairing symmetries of iron-based superconductors are investigated systematically in a five-orbital model within the different regions of interaction parameters by functional renormalization group(FRG). Even for a fixed Fermi surface with both hole and electron pockets, it is found that depending on interaction parameters, a variety of pairing symmetries, including two types of $d$-wave and two types of $s$-wave pairing symmetries, can emerge. Only the $d_{x^2-y^2}$ and the $s\pm$ waves are robustly supported by the nearest-neighbor(NN) intra-orbital $J_1$ and the next-nearest-neighbor(NNN) int…
The low energy band structure of the FeAs based superconductors is fitted by a tight binding model with two Fe ions per unit cell and two degenerate orbitals per Fe ion. Based on this, superconductivity with extended s-wave pairing symmetry of the form $\cos k_x +\cos k_y$ is examined. The local density of states near an impurity is also investigated by using T-matrix approach. For the nonmagnetic scattering potential, we found that there exist two major resonances inside the gap. The height of the resonance peaks depends on the strength of the impurity potential. These in-gap resonances are o…
For the newly discovered ironbased superconductors we propose a one band model for conduction band with first and secondnearestneighbor electron hoppings with two types of d wave and three types of s wave pairing symmetries for the superconductivity. We have calculated the Greens functions from which the temperature dependent superconducting (SC) gap and tunneling conductance are calculated and computed numerically. The evolution of the SC gap and the tunneling conductance spectra are investigated for different model parameters of the system.
We calculate the tunneling conductance spectra of a ferromagnetic metal/insulator/superconductor using the Blonder-Tinkham-Klapwijk (BTK) formulation. Two possible states for the superconductor are considered with the time reversal symmetry ($\cal{T}$) broken, i.e., $d_{x^2-y^2}+is$, or $d_{x^2-y^2}+id_{xy}$. In both cases the tunneling conductance within the gap is suppressed with the increase of the exchange interaction due to the suppression of the Andreev reflection. In the $(d_{x^2-y^2}+is)$-wave case the peaks that exist when the ferromagnet is a normal metal in the amplitude of the s-wa…
The symmetries of the order parameter (OP) responsible for pairing and the OP responsible for phase coherence in hole-doped cuprates are discussed. We analyze angle-resolved tunneling, ARPES, Raman scattering and torque measurements performed on hole-doped cuprates. We conclude that, most likely, the OP responsible for phase coherence has the d-wave symmetry whereas the OP responsible for pairing has an anisotropic s-wave symmetry.
Geometrical frustration leads to novel quantum phenomena such as the spin-liquid phase in triangular and Kagomé lattices. Intra-band and inter-band Fermi surface (FS) nesting can drive unique superconducting (SC) ground states with $d$-wave and $s^{\pm}$ pairing symmetries, respectively, according to the criterion that the SC gap changes sign across the nesting wavevector. For an odd number of FSs, when multiple inter-band nesting is of comparable strength, the sign-reversal criterion between different FS sheets can leads to frustration, which promotes novel SC order parameters. Here we report…
We consider a generic two-dimensional system of fermionic particles with attractive interactions and no disorder. If time-reversal symmetry is absent, it is possible to obtain incompressible insulating states in addition to the superfluid at zero temperature. The superfluid-insulator phase transition is found to be second order in type-II systems using a perturbative analysis of Cooper pairing instability in quantum Hall states of unpaired fermions. We obtain the pairing phase diagram as a function of chemical potential (density) and temperature. However, a more careful analysis presented here…
In a recent Letter [PRL 103, 186402 (2009)], the energy band structure of Fe-As-based superconductors is fitted with a tight-binding model with two Fe ions per unit cell and two degenerate xz and yz orbitals per Fe ion. The author claims that the proposed model, which differs markedly from a model previously used by other authors for the same two orbitals in the same compounds, possesses the symmetry required to describe the Fe-As planes in iron-pnictide superconductors. In this comment we argue that this is not the case.
In this work, we study the magnetic-impurity resonance states in superconducting phase of twisted bilayer graphene for different pairing symmetries. Using the two-orbital model proposed by Yuan and Fu in [Phys. Rev. B 98, 045103 (2018)], we find that when the impurity is located at one site of the emergent honeycomb lattice, the spacial distributions of the resonance states will break both the threefold and twofold rotation symmetries of $D_3$ group for pairing symmetries which belong to the irreducible representations of this point group. When the magnetic impurity is located at the center of…
We propose a class of topological superconductivity in which the pairing order is $\mathbb{Z}_2$ topologically obstructed in a three-dimensional time-reversal invariant system. When two Fermi surfaces are related by time-reversal and mirror symmetries, such as those in a $\mathbb{Z}_2$ Dirac semimetal, the inter-Fermi-surface pairing in the weak-coupling regime inherits the band topological obstruction. As a result, the pairing order cannot be well-defined over the entire Fermi surface and forms a time-reversal invariant generalization of U($1$) monopole harmonic pairing. A tight-binding model…
A time reversal invariant (TRI) topological superconductor has a full pairing gap in the bulk and topologically protected gapless states on the surface or at the edge. In this paper, we show that in the weak pairing limit, the topological quantum number of a TRI superconductor can be completely determined by the Fermi surface properties, and is independent of the electronic structure away from the Fermi surface. In three dimensions (3D), the integer topological quantum number in a TRI superconductor is determined by the sign of the pairing order parameter and the first Chern number of the Berr…
The symmetries and origins of the order parameters (OPs) for pairing and long-range phase coherence in hole-doped cuprates are discussed. By analyzing tunneling, inelastic neutron scattering and torque measurements we conclude that the Cooper pairs in cuprates consist of spinons, and the pairing OP has an anisotropic s-wave symmetry, whereas the OP for long-range phase coherence has the magnetic origin due to spin fluctuations and, has the d-wave symmetry. All conclusions are based exclusively on experimental facts.
We discuss possible pairing symmetries in the hexagonal pnictide superconductor SrPtAs. The local lack of inversion symmetry of the two distinct conducting layers in the unit cell results in a special spin-orbit coupling with a staggered structure. We classify the pairing symmetry by the global crystal point group D_3d, and suggest some candidates for the stable state using a tight-binding model with an in-plane, density-density type pairing interaction. We may have some unconventional states like s+f-wave and a mixture of chiral d-wave and chiral p-wave. The spin orbit coupling is larger than…
We review all symmetry-allowed spin-singlet and spin-triplet superconducting (SC) order parameters in graphene ($s$-wave, $d$-wave, $p$-wave, and $f$-wave) generated by generic onsite, nearest-neighbor (NN), and next-to-nearest-neighbor (NNN) pairing interactions in a tight-binding model. For each pairing channel, we calculate both the band structure and the dependence of the density of states on energy, chemical potential, and on the pairing strength. In particular, we distinguish between nodal superconducting states and fully gapped states and study the dependence of gap closing points on th…
Recently, superconductivity with spontaneous time-reversal or parity symmetry breaking is attracting much attention owing to its exotic properties, such as nontrivial topology and nonreciprocal transport. Particularly fascinating phenomena are expected when the time-reversal and parity symmetry are simultaneously broken. This work shows that time-reversal symmetry-breaking mixed-parity superconducting states generally exhibit an unusual asymmetric Bogoliubov spectrum due to nonunitary interband pairing. For generic two-band models, we derive the necessary conditions for the asymmetric Bogoliub…
We have investigated the superconducting state of the non-centrosymmetric compound Re6Zr using magnetization, heat capacity, and muon-spin relaxation/rotation (muSR) measurements. Re6Zr has a superconducting transition temperature, Tc = 6.75 K. Transverse-field muSR experiments, used to probe the superfluid density, suggest an s-wave character for the superconducting gap. However, zero and longitudinal-field muSR data reveal the presence of spontaneous static magnetic fields below Tc indicating that time-reversal symmetry is broken in the superconducting state and an unconventional pairing mec…
The possible pairing symmetries for BiS$_{2}-$based superconductors is investigated by using a minimal two-orbital model with onsite and nearest-neighbor intraorbital attractions $V_{0}$ and $V_{1}$, respectively. By using the mean-field approximation and solving the self-consistent equations, the phase diagram of the pairing symmetry is obtained. It is shown that the model allows three possible pairing symmetries, depending on the values of $V_{0}$ and $V_{1}$: the isotopic $s-$wave pairing [$Δ_{\mathbf{k}}=Δ_{s}$], the anisotropic $s-$wave pairing [$Δ_{\mathbf{k}}=Δ_{s}+\frac{Δ_{xs}}{2}(\cos…
Paramagnetic effects are shown to result in the appearance of a triplet component of order parameter in a vortex phase of a d-wave superconductor in the absence of impurities. This component, which breaks both parity and time-reversal symmetries of Cooper pairs, is expected to be of the order of unity in a number of modern superconductors such as organic, high-Tc, and some others. A generic phase diagram of such type-IV superconductors, which are singlet ones at H=0 and characterized by singlet-triplet mixed Copper pairs with broken time-reversal symmetry in a vortex phase, is discussed.
We report the study of spontaneous magnetization (i.e., spin-polarization) for time-reversal symmetry (TRS)-breaking superconductors with unitary pairing potentials, in the absence of external magnetic fields or Zeeman fields. Spin-singlet ($Δ_s$) and spin-triplet ($Δ_t$) pairings can coexist in superconductors whose crystal structure lacks inversion symmetry. The TRS can be spontaneously broken once a relative phase of $\pmπ/2$ is developed, forming a TRS-breaking unitary pairing state ($Δ_s\pm iΔ_t$). We demonstrate that such unitary pairing could give rise to spontaneous spin-polarization w…
We report a sudden reversal in the pressure dependence of Tc in the iron-based superconductor CsFe2As2, similar to that discovered recently in KFe2As2 [Tafti et al., Nat. Phys. 9, 349 (2013)]. As in KFe2As2, we observe no change in the Hall coefficient at the zero temperature limit, again ruling out a Lifshitz transition across the critical pressure Pc. We interpret the Tc reversal in the two materials as a phase transition from one pairing state to another, tuned by pressure, and investigate what parameters control this transition. Comparing samples of different residual resistivity, we find …
We introduce and study an extended "t-U-J" two-orbital model for the pnictides that includes Heisenberg terms deduced from the strong coupling expansion. Including these J terms explicitly allows us to enhance the strength of the (pi, 0)-(0, pi) spin order which favors the presence of tightly bound pairing states even in the small clusters that are here exactly diagonalized. The A1g and B2g pairing symmetries are found to compete in the realistic spin-ordered and metallic regime. The dynamical pairing susceptibility additionally unveils low-lying B1g states, suggesting that small changes in pa…
Fermi-surface diagnosis for topological superconductivity with $s$-wave-like pairing symmetries
Hong Kong University of Science and Technology（香港科技大学）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Kavli Institute for Theoretical Sciences, Chinese Academy of Sciences（中国科学院理论科学卡弗里研究所）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理研跨学科理论与数学科学中心(iTHEMS)）
Giant and nil proximity effects and unconventional symmetry of cuprate superconductors are explained as a result of the strong Froehlich interaction of holes with c-axis polarised optical phonons acting together with an anisotropic nonlocal deformation potential.
We show that singlet superconductivity in the Abrikosov vortex phase is absolutely unstable with respect to the appearance of a chiral triplet component of a superconducting order parameter. This chiral component, p_x + i p_y, breaks time-reversal, parity, and spin rotational symmetries of the internal order parameter, responsible for a relative motion of two electrons in the Cooper pair. We demonstrate that the symmetry breaking Pauli paramagnetic effects can be tuned by a magnetic field strength and direction and can be made of the order of unity in organic and high-temperature layered super…
The results from muon spin relaxation experiments on the non-centrosymmetric intermetallic superconductor LaNiC$_2$ are reported. We find that the onset of superconductivity coincides with the appearance of spontaneous magnetic fields, implying that in the superconducting state time reversal symmetry is broken. An analysis of the possible pairing symmetries suggests only four triplet states compatible with this observation, all of which are non-unitary. They include the intriguing possibility of triplet pairing with the full point group symmetry of the crystal, which is only possible in a non-…
To identify the superconducting gap symmetry in CeCoIn5 (Tc=2.3 K), we performed angle-resolved specific heat (C_ϕ) measurements in a field rotated around the c-axis down to very low temperatures 0.05Tc and detailed theoretical calculations. In a field of 1 T, a sign reversal of the fourfold angular oscillation in C_ϕhas been observed at T ~ 0.1Tc on entering a quasiclassical regime where the maximum of C_ϕcorresponds to the antinodal direction, coinciding with the angle-resolved density of states (ADOS) calculation. The C_ϕbehavior, which exhibits minima along [110] directions, unambiguously …
We study the possibility of topological superconductivity in the noncentrosymmetric monolayer and trilayer TaS$_2$ with out-of-plane mirror symmetry. A gapless time-reversal invariant f+s-wave pairing state with even mirror parity is found to be a promising candidate. This mixing state holds 12(36) nodes at the Fermi pockets around $Γ$ for monolayer(trilayer) case and its unconventional superconductivity is consistent with the STM experiments observed in 2$H$-TaS$_2$ thin flakes. Furthermore, with doping or under uniaxial pressure for trilayer 2$H$-TaS$_2$, large-Chern-number time-reversal sym…
Vortex phase in a singlet superconductor in the absence of impurities is shown to be absolutely unstable with respect to the appearance of a triplet component which breaks both inversion and time-reversal symmetries of Cooper pairs. Symmetry breaking paramagnetic effects are demonstrated to be of the order of unity if the orbital upper critical field, Hc2(0), is of the order of Clogston paramagnetic limiting field, Hp. We suggest a generic phase diagram of such type-IV superconductor, which is singlet one at H=0 and characterized by mixed singlet-triplet order parameter with broken time-revers…
We here examine the relation between odd-frequency spin-triple even-parity (OTE) Cooper pairs and anomalous surface magnetic response in time-reversal invariant (TRI) spin-triplet superfluids and superconductors. The spin susceptibility generally consists of two contributions: Even-frequency odd-parity pair amplitudes and odd-frequency even-parity pair amplitudes. The OTE pair amplitudes are absent in the bulk region, but ubiquitously exist in the surface and interface region as Andreev bound states. We here clarify that additional discrete symmetries, originating from the internal symmetry an…
Thin sheets deposited on a substrate and interfaces of correlated materials offer a plethora of routes towards the realization of exotic phases of matter. In these systems, inversion symmetry is broken which strongly affects the properties of possible instabilities -- in particular in the superconducting channel. By combining symmetry and energetic arguments, we derive general and experimentally accessible selection rules for Cooper instabilities in noncentrosymmetric systems which yield necessary and sufficient conditions for spontaneous time-reversal-symmetry breaking at the superconducting …
We study fermions in a lattice, with on-site and nearest neighbor attractive interactions between two spin species. We consider two geometries: both spins in a triangular lattice, and a mixed geometry with up-spins in honeycomb and down-spins in triangular lattices. We focus on the interplay between spin-population imbalance, on-site and valence bond pairing, and order parameter symmetry. The mixed geometry leads to a rich phase diagram of topologically non-trivial phases. In both geometries, we predict order parameters with simultaneous time-reversal and translational symmetry breaking.
We investigate a possibility of holon pairing for bose condensation based on the Bethe-Salpeter equation obtained from the use of the t-J Hamiltonians of both t he U(1) and SU(2) slave-boson symmetries. It is shown that the vertex function contributed from ladder diagram series involving holon-holon scattering channel in the Bethe-Salpeter equation leads to a singular behavior at a critical temperature at each hole doping concentration, showing the instability of the normal state against holon pairing. We find that the holon pairing instability occurs only in a limited range of hole doping, by…
We report observation of the nonlinear Meissner effect (NLME) in Nb films by measuring the resonance frequency of a planar superconducting cavity as a function of the magnitude and the orientation of a parallel magnetic field. Use of low power rf probing in films thinner than the London penetration depth, significantly increases the field for the vortex penetration onset and enables NLME detection under true equilibrium conditions. The data agree very well with calculations based on the Usadel equations. We propose to use NLME angular spectroscopy to probe unconventional pairing symmetries in …
In order to explore how superconductivity arises when charge fluctuations and spin fluctuations coexist, we have obtained a phase diagram against the off-site repulsion V and band filling n for the extended, repulsive Hubbard model on the square lattice with the fluctuation exchange approximation. We have found the existence of (i) a transition between dxy and dx2-y2 pairing symmetries, (ii) f-pairing in between the dx2-y2 and CDW phases for intermediate 0.5<n<1.0 and large V, and (iii) for anisotropic cases the pairing symmetry changing, in agreement with a previously proposed ``generic phase…
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.
