We study the effects of phonon interaction on the superconducting pairing in the background of a d-wave gap, mediated by antiferromagnetic (AFM) spin fluctuations, using coupled BCS gap equations. We found that phonon interaction can induce a s-wave component to the d-wave gap in the (D+S) form with an interaction anisotropy and in the (D+$i$S) form without anisotropy, respectively. In either case, however, T$_c$ is not enhanced compared to the pure d-wave pairing without phonon interaction. On the other hand, anisotropic phonon interaction can dramatically enhance the d-wave pairing itself and therefore T$_c$, together with the AFM spin fluctuation interaction. This (D$_{AFM}$ + D$_{ph}$) type pairing exhibits strongly reduced isotope coefficient despite the large enhancement of T$_c$ by phonon interaction. Finally, we study the combined type of (D$_{AFM}$ + D$_{ph}$ +$i$S)) gap and calculate the penetration depth and specific heat to be compared with the experiments.
We give an extensive treatment of the pairing symmetry in the ferromagnetic superconductor $UGe_{2}$. We show that one can draw important conclusions concerning the superconducting state, considering only the transformation properties of the pairing function, without assumptions about the form of the pairing amplitudes.
We investigate the effect of interchain repulsive interaction on the pairing symmetry competition in quasi-one-dimensional organic superconductors (TMTSF)$_2$X by applying random phase approximation and quantum Monte Carlo calculation to an extended Hubbard model. We find that interchain repulsive interaction enhances the $2k_F$ charge fluctuations, thereby making the possibility of spin-triplet $f$-wave pairing dominating over spin-singlet d-wave pairing realistic.
We examine the effects of hole-phonon interaction on the formation of hole pairs in the 2D Holstein t--J model. Using finite-lattice diagonalization techniques, we present exact results for the two-hole binding energy and the s-- and d-wave pairing susceptibilities.
We analyze the influence of phonons on the d-wave pairing instability in the Hubbard model on the two-dimensional square lattice at weak to moderate interaction U, using a functional renormalization group scheme with frequency-dependent interaction vertices. As measured by the pairing scale, the B1g buckling mode enhances the pairing, while other phonon modes decrease the pairing. When various phonon modes are included together, the net effect on the scale is small. However, in situations where d-wave superconductivity and other tendencies, e.g. antiferromagnetism, are closely competing, the c…
We propose a microscopic mechanism for triplet pairing due to spin-orbit-assisted electron interaction with optical phonons in a crystal with a complex unit cell. Using two examples of electrons with symmetric Fermi surfaces in crystals with either a cubic or a layered square lattice, we show that spin-orbit-assisted electron-phonon coupling can, indeed, generate triplet pairing and that, in each case, it predetermines the tensor structure of a p-wave order parameter.
The t-t'-t''-J model of electrons interacting with three phonon modes (breathing, apical breathing, and buckling) is considered. The wave-vector dependence of the matrix elements of the electron-phonon interaction leads to opposite contributions to the pairing potential with the d-symmetry: the buckling mode facilitates electron pairing, while the breathing mode suppresses it. As a result, the critical temperature of La{2 - x}Sr{x}CuO{4} that is associated with the magnetic mechanism is lowered when phonons are taken into account.
Effects of short-range correlations on the Coulomb screening, the phonons, and the pairing interactions are examined in electron-phonon systems. First, we derive a model Hamiltonian of Coulomb interactions which includes both the long-range part v_q and the short-range part U. It is found from the expression of the dielectric function that the strong on-site correlations weaken the Coulomb screening. Secondly, we examine the screened phonons and the interaction mediated by phonons. In a consistent picture, we derive an expression of the effective interaction which includes (1) the screened Cou…
Density-matrix renormalization group is used to study the pairing when both of electron-electron and electron-phonon interactions are strong in the Holstein-Hubbard model at half-filling in a region intermediate between the adiabatic (Migdal's) and antiadiabatic limits. We have found: (i) the pairing correlation obtained for a one-dimensional system is nearly degenerate with the CDW correlation in a region where the phonon-induced attraction is comparable with the electron-electron repulsion, but (ii) pairing becomes dominant when we destroy the electron-hole symmetry in a trestle lattice. Thi…
A review is given on theoretical studies concerning the pairing symmetry in organic superconductors. In particular, we focus on (TMTSF)$_2$X and $κ$-(BEDT-TTF)$_2$X, in which the pairing symmetry has been extensively studied both experimentally and theoretically. Possibilities of various pairing symmetry candidates and their possible microscopic origin are discussed. Also some tests for determining the actual pairing symmtery are surveyed.
The nonadiabatic electron-phonon corrections for the superconducting pairing are investigated for a specific tight-binding model corresponding to a 2d square lattice. This permits to investigate the role of various specific properties like the band filling, nesting effects and a realistic van Hove singularity on the superconducting effective pairing beyond Migdal's limit. The main results are: (i) Starting from a momentum independent electron-phonon coupling the nonadiabatic effects lead to an effective pairing which is strongly dependent on frequency and momentum. (ii) If instead the electron…
In this paper we present a real space pairing glue for the iron-based layered superconductors. It is shown that two static electrons embedded symmetrically into two adjacent Fe plaquettes of the superconductor can be bounded due to the Coulombic interaction. The pairing mechanism favors the existence of the pseudogap in the underdoped FeAs superconductors. A criterion is introduced to distinguish whether or not the pseudogap can open in a material.
In this work, we establish a few exact identities through commutation of intra-orbital and inter-orbital on-site pairings with a two-orbital model describing newly discovered FeAs-based superconductors. Applying the conclusion drawn from rigorous relation and physical interpretation, we give constraints on the possible symmetries of the superconducting pairing of the model. Hence the favorable pairings in newly discovered high-temperature oxypnictide superconductors are proposed.
The Hamiltonian describing a system of strongly correlated electrons coupled to dispersionless phonons was solved numerically for a ring of 8 atoms using the density matrix renormalization group (DMRG) method. It was found that electron correlation and electron-phonon coupling compete against each other, and strong electron correlations suppress the charge-ordered insulating state. This allows extended polarons to form in the strong electron-phonon coupling regime. It is shown that in this regime two polarons may form a pair via phonon mediated charge-fluctuation interaction. Based on the resu…
We use the microscopic weak coupling theory to predict the pairing state in superconductors of cubic, hexagonal, or tetragonal symmetry, where the order parameter is multicomponent, i.e., transforms according to either a 2-dimensional or a 3-dimensional representation of the crystal point group. We show that the superconducting phase usually breaks the time-reversal symmetry for singlet multicomponent superconductors. The superconducting order parameter for triplet superconductors in most cases turns out to be non-magnetic.
Pairing symmetry is important to indentify the pairing mechanism. The analysis becomes particularly timely and important for the newly discovered iron-based multi-orbital superconductors. From group theory point of view we classified all pairing matrices (in the orbital space) that carry irreducible representations of the system. The quasiparticle gap falls into three categories: full, nodal and gapless. The nodal-gap states show conventional Volovik effect even for on-site pairing. The gapless states are odd in orbital space, have a negative superfluid density and are therefore unstable. In c…
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.
It is developed a microscopic description of superconductivity in ferromagnetic materials with triplet pairing triggered by the exchange of magnetic fluctuations. Instead widely used paramagnon model we work with phenomenological spectrum of fluctuations in the orthorhombic ferromagnet with strong magnetic anisotropy. Depending of the field orientation parallel or perpendicular to the direction of spontaneous magnetization the effective amplitude of pairing interaction proves to be decreasing or increasing function of magnetic field that allows to explain the drastic difference in magnitudes o…
We discuss a many-electron Hamiltonian with Hubbard-like repulsive interaction and linear coupling to the phonon branches, having the Cu-O plane of the superconducting cuprates as a paradigm. A canonical transformation extracts an effective two-body problem from the many-body theory. As a prototype system we study the $\cu$ cluster, which yields electronic pairing in the Hubbard model; moreover, a standard treatment of the Jahn-Teller effect predicts distortions that destroy electronic pairing. Remarkably, calculations that keep all the electronic spectrum into account show that vibrations are…
Recent experiments on Bi-based cuprate superconductors have revealed an unexpected enhancement of the pairing correlations near the interstitial oxygen dopant ions. Here we propose a possible mechanism -- based on local screening effects -- by which the oxygen dopants do modify the electronic parameters within the CuO_2 planes and strongly increase the superexchange coupling J. This enhances the spin pairing effects locally and may explain the observed spatial variations of the density of states and the pairing gap.
I review recent works on the symmetry and the structure of the superconducting gap in Fe-based superconductors and on the underlying pairing mechanism in these systems. The experimental data on superconductivity show very rich behavior, with potentially different symmetry of a superconducting state for different compositions of the same material. The variety of different pairing states raised the issue whether the physics of Fe-based superconductors is model-dependent or is universal, governed by a single underlying pairing mechanism. I argue that the physics is universal and that all pairing …
Josephson junctions between a $FeAs$-based superconductor with antiphase s-wave pairing and a conventional s-wave superconductor are studied. The translational invariance in a planar junction between a single crystal pnictide and an aluminum metal greatly enhances the relative weight of electron pockets in the pnictide to the critical current. In a wide doping region of the pnictide, a planar and a point contact junctions have opposite phases, which can be used to design a tri-junction ring with $π$ phase to probe the antiphase pairing.
In this Letter, we develop a microscopic theory to describe the close proximity between the insulating antiferromagnetic (AF) order and the d-wave superconducting (dSC) order in cuprates. We show that the cuprate ground states form a configuration of coherent pairing states consisting of extended singlet Cooper pairs and triplet $π$ pairs, which can simultaneously describe AF and dSC orders.
The spin-fluctuation mechanism of superconductivity usually results in the presence of gapless or nodal quasiparticle states in the excitation spectrum. Nodal quasiparticle states are well established in copper-oxide, and heavy-fermion superconductors, but not in iron-based superconductors. Here, we study the pairing symmetry and mechanism of a new class of plutonium-based high-Tc superconductors and predict the presence of a nodal $s^{+-}$ wave pairing symmetry in this family. Starting from a density-functional theory (DFT) based electronic structure calculation we predict several three-dimen…
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…
Using functional renormalization group (FRG) approach, we analyze the $η$ pairing in a five d-orbital model for iron-pncitides. We find that the $η$ pairing between two hole pockets is a leading instability at low energy when the two hole pockets are close to a nesting condition. The result suggests that the system can spontaneously enter into a superconducting state with both normal pairing and $η$ paring and the $η$ pairing is driven by the inter-orbital antiferromagnetic exchange coupling while the normal pairing is driven by the intra-orbital antiferromagnetic ones. An effective model at l…
We investigate the interplay of phonons and correlations in superconducting pairing by introducing a model Hamiltonian with on-site repulsion and couplings to several vibration branches having the Cu-O plane of the cuprates as a paradigm. We express the electron-phonon coupling (EP) through two force constants for O-Cu and O-O bond stretchings. Without phonons, this reduces to the Hubbard Model, and allows purely electronic W=0 pairing. A W=0 pair is a two-body singlet eigenstate of the Hubbard Hamiltonian, with no double occupancy, which gets bound from interactions with background particles.…
We present a self-consistent real space formulation of spin-fluctuation mediated d-wave pairing. By calculating all relevant inhomogeneous spin and charge susceptibilities in real space within the random phase approximation (RPA), we obtain the effective pairing interaction and study its spatial dependence near both local potential and hopping impurities. A remarkably large enhancement of the pairing interaction may be obtained near the impurity site. We discuss the relevance of our result to inhomogeneities observed by scanning tunneling spectroscopy on the surface of cuprate superconductors.
We present a microscopic theory of fluctuation-mediated pairing mechanism in organic superconductors ${(TMTSF)}_{2}X$, where the experimentally observed coexistence of $2k_{F}$ charge fluctuation and $2k_{F}$ spin fluctuation is naturally taken into account. We have studied, within the random phase approximation, the extended Hubbard model at quarter-filling on a quasi-one-dimensional lattice, where we consider the off-site repulsive interaction up to third next nearest neighbors along with the on-site repulsion. The results show that spin-triplet $f$-wave-like pairing can be realized in this …
The behavior of some high temperature superconductors (HTSC) such as $\rm La_{2-x}Sr_{x}CuO_{4}$ and $\rm Bi_{2}Sr_{2-x}La_xCuO_{6+δ}$, at very high magnetic field, is similar to that of thin films of amorphous InOx near the magnetic field-tuned superconductor-insulator transition. Analyzing the InOx data at high fields in terms of persisting local pairing amplitude, we argue by analogy that local pairing amplitude also persists well into the dissipative state of the HTSCs, the regime commonly denoted as the "normal state" in very high magnetic field experiments.
We perform a systematic quantum Monte Carlo study of the pairing correlation in the $S_4$ symmetric microscopic model for iron-based superconductors. It is found that the pairing with an extensive s-wave symmetry robustly dominates over other pairings at low temperature in reasonable parameter region. The pairing susceptibility, the effective pairing interaction and the $(π,0)$antiferromagnetic correlation strongly increase as the on-site Coulomb interaction increases, indicating the importance of the effect of electron-electron correlation. Our non-biased numerical results provide a unified u…
Two predictions are made for properties of the ferromagnetic superconductors discovered recently. The first one is that spin-triplet, p-wave pairing in such materials will give the magnons a mass inversely proportional to the square of the magnetization. The second one is based on a specific mechanism for p-wave pairing, and predicts that the observed broad anomaly in the specific heat of URhGe will be resolved into a split transition with increasing sample quality. These predictions will help discriminate between different possible mechanisms for ferromagnetic superconductivity.
We discuss the influence on spin-fluctuation pairing theory of orbital selective strong correlation effects in Fe-based superconductors, particularly Fe chalcogenide systems. We propose that a key ingredient for an improved itinerant pairing theory is orbital selectivity, i.e., incorporating the reduced coherence of quasiparticles occupying specific orbital states. This modifies the usual spin-fluctuation via suppression of pair scattering processes involving those less coherent states and results in orbital selective Cooper pairing of electrons in the remaining states. We show that this parad…
The common wisdom that the phonon mechanism of electron pairing in the weak-coupling Bardeen-Cooper-Schrieffer (BCS) superconductors leads to conventional s-wave Cooper pairs is revised. An inevitable anisotropy of sound velocity in crystals makes the phonon-mediated attraction of electrons non-local in space providing unconventional Cooper pairs with a nonzero orbital momentum in a wide range of electron densities. As a result of this anisotropy quasi-two dimensional charge carriers undergo a quantum phase transition from an unconventional d-wave superconducting state to a conventional s-wave…
Identifying the mechanism of superconductivity in the high-temperature cuprate superconductors is one of the major outstanding problems in physics. We report local measurements of the onset of superconducting pairing in the high-transition temperature (Tc) superconductor Bi2Sr2CaCu2O8+d using a lattice-tracking spectroscopy technique with a scanning tunneling microscope. We can determine the temperature dependence of the pairing energy gaps, the electronic excitations in the absence of pairing, and the effect of the local coupling of electrons to bosonic excitations. Our measurements reveal th…
The local density of states (LDOS) around a vortex core is studied theoretically in terms of the odd-frequency (odd-omega) Cooper pairing induced around the center of the vortex core. We find a robust zero-energy peak in the LDOS at a vortex core in a chiral p-wave superconductor originating from an odd-omega s-wave pair amplitude. We suggest how to discriminate a spin-triplet pairing symmetry and spatial chiral-domain structure by scanning tunneling spectroscopy via odd-omega pair amplitudes inside vortex cores.
We propose that the superconducting pairing symmetry of organic superconductors $κ$-(BEDT-TTF)$_{2}$X can be determined by measuring the position in momentum space of the incommensurate peaks of the spin susceptibility. We show that the positions of these peaks are uniquely determined by the symmetry of the superconducting state and the geometry of Fermi surface. Based on the effective Hubbard model on an anisotropic triangular lattice, we demonstrate the different incommensurate patters of spin responses for $d_{x^{2}-y^{2}}$- and $d_{xy}$-like pairing states. In addition, we predict the none…
We argue that the recent Raman spectroscopy observation of a sharp mode in s-wave superconducting V$_3$Si is due to a competing d-wave pairing state. We present microscopic arguments for the origin of this d-wave order. We further argue that the d-wave order explains the observed shrinking of the vortex core structure at anomalously low magnetic fields and the large anisotropy observed in the upper critical field.
We present the ground state pairing correlations in the $S_4$ symmetric microscopic model for iron-based superconductors, computed with the constrained-path Monte Carlo method. For various electron fillings and interaction strengths, we find that the $s_{xy}$ pairing dominates over other pairing correlations and is positive when the pair separation exceeds several lattice constants, whatever for iron pnictides and iron chlcogenides. These ground state properties, especially the long range part pairing correlations re-confirm the previous finite temperature results published in Phys. Rev. Lett.…
We review briefly several approaches used to investigate the stability of stripe phases in high temperature superconductors, where charge inhomogeneities arise from competing kinetic and magnetic energies. The mechanism of stripe formation, their consequences for the normal state and enhancement of pairing interaction triggered by charge inhomogeneities are briefly summarized. Finally, we demonstrate that orbital degeneracy ($i$) leads to a more subtle mechanism of stripe formation, and ($ii$) plays an important role and determines the symmetry of the superconducting state in pnictides.
In iron selenide superconductors only electron-like Fermi pockets survive, challenging the $S^{\pm}$ pairing based on the quasi-nesting between the electron and hole Fermi pockets (as in iron arsenides). By functional renormalization group study we show that an in-phase $S$-wave pairing on the electron pockets ($S^{++}_{ee}$) is realized. The pairing mechanism involves two competing driving forces: The strong C-type spin fluctuations cause attractive pair scattering between and within electron pockets via Cooperon excitations on the virtual hole pockets, while the G-type spin fluctuations caus…
The basic feature of the BCS theory is that pairing occurs between electrons in states with opposite momentum and opposite spins, e.g., between states (k) and (-k), in which the symbols of vectors are neglected, and later theories followed the BCS theory. However, since the center of mass for any pair is not always static, we must consider the pairing between states (k) and (-k+q). In this letter, we derive a set of self-consistent equations for the correlation functions associated with the pairs that have non-opposite momentums, in which we find some new results. This means that some physics …
We report a scaling relation between $ρ_s(0)$ and the product of the pseudogap temperature $T^{*}$, $σ_{\rm DC}$ and $T_c$, with $T^{*}$ rescaled by the maximum transition temperature $T_c^{\max}$, which holds universally in the entire doping range for the hole-doped cuprates.
We derive the effective action for superconducting fluctuations in a four-band model for pnictides, discussing the emergence of a single critical mode out of a dominant interband pairing mechanism. We then apply our model to calculate the paraconductivity in two-dimensional and layered three-dimensional systems, and compare our results with recent resistivity measurements in SmFeAsOF
A model with interlayer pairing is proposed to explain the sinusoidal modulation of the resonant neutron scattering in high temperature superconductors. It is found that the interlayer pairing has s-wave symmetry in the CuO$_2$ plane and has comparable magnitude with the d-wave intralayer pairing. It is also found that the interlayer pairing mainly affects momentum close to the hot spots on the Fermi surface while its effect on the gap nodes is negligible. It is pointed out that these characteristics of the interlayer pairing can be understood in a model in which the superconducting pairing or…
We show that an additional triplet component of the order parameter is generated in the vortex phase of the d-wave superconductor. Spatial variations of the triplet component are analyzed for a strong spin-orbit coupling. Corrections to the London equation and an unusual temperature dependence of the upper critical field, are obtained in the case of the weak spin-orbit coupling.
Scanning tunneling microscopy (STM) has made tremendous progress in the study and understanding of both classical and unconventional superconductors. This has motivated a rapidly growing effort to apply the same techniques to the iron-based high-Tc superconductors since their discovery in 2008. Five years have brought exciting advances in imaging and spectroscopic investigation of this new class of materials. In this review, we focus on several recent STM contributions to the identification of the gap symmetry and pairing glue. We highlight the unique capabilities and challenges still ahead fo…
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.
We analyze Cooper pairing instabilities in strongly driven electron-phonon systems. The light-induced non-equilibrium state of phonons results in a simultaneous increase of the superconducting coupling constant and the electron scattering. We demonstrate that the competition between these effects leads to an enhanced superconducting transition temperature in a broad range of parameters. Our results may explain the observed transient enhancement of superconductivity in several classes of materials upon irradiation with high intensity pulses of terahertz light, and may pave new ways for engineer…
We investigate how hybridization (single-quasiparticle scattering) between two superconducting bands induces odd-frequency superconductivity in a multiband superconductor. An explicit derivation of the odd-frequency pairing correlation and its full frequency dependence is given. We find that the density of states is modified, from the sum of the two BCS spectra, at higher energies by additional hybridization gaps with strong coherence peaks when odd-frequency pairing is present.
Its direct momentum sensitivity confers to angle-resolved photoemission spectroscopy (ARPES) a unique perspective in investigating the superconducting gap of multi-band systems. In this review we discuss ARPES studies on the superconducting gap of the high-temperature Fe-based superconductors. We show that while Fermi-surface-driven pairing mechanisms fail to provide a universal scheme for the Fe-based superconductors, theoretical approaches based on short-range interactions lead to a more robust and universal description of superconductivity in these materials. Our findings are also discussed…
The real-space localized hole pair is constructed in the Cu-O plane of the hole-doped cuprate superconductors. We prove analytically and numerically that two electrons, due to the nearest-neighbor Coulomb repulsive confinement effect, can be in pairing inside a single plaquette with the d-wave symmetry. The scenario supports the `no glue' pairing picture for the cuprates. Based on the scenarios, the physical origin of the Fermi pocket (or Fermi arc) and the two pseudogap behavior are provided. Our framework leads directly to a unified linear relationship between the pseudogap temperature T* an…
This Letter theoretically discusses quasiparticle states and nuclear spin relaxation rates $T_1^{-1}$ in a quasi-one-dimensional superconductor (TMTSF)$_2$PF$_6$ under a magnetic field applied parallel to the conduction chains. We study the effects of Josephson-type vortices on $T_1^{-1}$ by solving the Bogoliubov de Gennes equation for $p$-, $d$- or $f$-wave pairing interactions. In the presence of line nodes in pairing functions, $T_1^{-1}$ is proportional to $T$ in sufficiently low temperatures because quasiparticles induced by vortices at the Fermi energy relax spins. We also try to identi…
We introduce a generalized joint density of states (GJDOS), which incorporates the coherent factor into the JDOS, to study quasiparticle interference (QPI) in superconductors. The intimate relation between the Fourier-transformed local density of states and GJDOS is revealed: they corre- spond respectively to the real and imaginary parts of a generalized impurity-response function, and particularly share the same angular factors and singular boundaries, as seen from our approximate analytic results for d-wave superconductors. Remarkably, our numerical GJDOS analysis agrees well with the QPI pa…
We have analyzed data from angle resolved photoemission spectroscopy, Fourier transform scanning tunnelling spectroscopy, and low-temperature thermal conductivity for optimally doped Bi_2Sr_2CaCu_2O_8+y in order to discriminate between d-wave and extended s-wave pairing symmetry. The combined data are inconsistent with d-wave symmetry, but quantitatively consistent with extended s-wave symmetry with eight line nodes. We also consistently explain all the phase-sensitive experiments.
Quantum simulations can provide new insights into the physics of strongly correlated electronic systems. A well studied system, but still open in many regards, is the Hubbard-Holstein Hamiltonian, where electronic repulsion is in competition with attraction generated by the electron-phonon coupling. In this context we study the phase diagram of four quantum dots in a suspended carbon nanotube and coupled to its flexural degrees of freedom. The system is described by a Hamiltonian of the Hubbard-Holstein class, where electrons on different sites interact with the same phonon. We find that the s…
We show that the newly discovered BiS$_2$-based superconductors may have a dominant triplet pairing component, in addition to a subdominant singlet component arising from the spin-orbital coupling. The pairing respects time-reversal symmetry. The dominant triplet gap causes gap sign changes between the spin-split Fermi pockets. Within a pocket, the gap function respects $d^*_{x^2-y^2}$-wave symmetry, where the star indicates joint spin-lattice rotations. Below the Lifshitz filling level the gap is nodelss and the superconducting state is weak topological. Above the Lifshitz points the gap beco…
We propose that anisotropic $p$-, $d$-, or $f$-wave pairing symmetries can be distinguished from a tunneling spectroscopy in the presence of magnetic fields, which is exemplified here for a model organic superconductor ${(TMTSF)}_{2}X$. The shape of the Fermi surface (quasi-one-dimensional in this example) affects sensitively the pairing symmetry, which in turn affects the shape (U or V) of the gap along with the presence/absence of the zero-bias peak in the tunneling in a subtle manner. Yet, an application of a magnetic field enables us to identify the symmetry, which is interpreted as an eff…
