Pairing Symmetry Research Articles

Fermi surfaces of the topological semimetal CaSn3 probed through de Haas van Alphen oscillations

In the search of topological superconductors, nailing down the Fermiology of the normal state is as crucial a prerequisite as unraveling the superconducting pairing symmetry. In particular, the number of time-reversal-invariant momenta (TRIM) in the Brillouin zone enclosed by Fermi surfaces is closely linked to the topological class of time-reversal-invariant systems, and can experimentally be investigated. We report here a detailed study of de Haas van Alphen quantum oscillations in single crystals of the topological semimetal CaSn3 with torque magnetometry in high magnetic fields up to 35 T. In conjunction with density functional theory based calculations, the observed quantum oscillations frequencies indicate that the Fermi surfaces of CaSn3 enclose an odd number of TRIM, satisfying one of the proposed criteria to realize topological superconductivity. Nonzero Berry phases extracted from the magnetic oscillations also support the nontrivial topological nature of CaSn3.

Fabrication of Corner-Like Josephson Junctions Based on Pnictide Single crystals

In order to study fundamental properties of pnictide superconductors such as the pairing symmetry, we have fabricated corner-like Josephson devices. Their configuration contains two SNS* (superconductor/barrier/superconductor*) thin-film junctions in different crystal directions. We have used multi-band Co-doped Ba-122 single crystals as the base electrode (S), evaporated lead films as a counter electrode (S*) made of a conventional s-wave superconductor and normal-metal (N) layers as the barrier. The junctions exhibit symmetric I-V characteristics with IcRN values about 0.975 μV. Ic is the critical current, and RN is the normal-state resistance. The observed Fraunhofer pattern-like dependence of Ic on the magnetic field indicates homogeneity of the junctions studied.

Study on the Reduction of the Resonant Stress of Turbine Blades Caused by the Stage Interaction Force (Simultaneous Optimization of Blade Resonant Stress and Amount of Unbalance)

Abstract Although the bladed disks of turbomachinery are nominally designed to be cyclically symmetric (tuned system), the vibration characteristics of individual blades on a disk vary slightly owing to manufacturing tolerances, deviations in material properties, wear during operation, etc. These small variations break the cyclic symmetry and split eigenvalue pairs. Actual bladed disks with small variations are called mistuned systems. The resonant stress on a mistuned bladed disk may become large and cause a blade failure due to high cycle fatigue. Especially for a bladed disk with a free-standing blade structure, it is essential to use a sufficiently large safety factor at the design stage, considering the mistuning effect. Traditionally, blade designers have adopted various countermeasures to reduce the resonant stress at the design stage. Besides the countermeasures adopted in blade design, for a variable speed engine, which cannot avoid resonance, it is thought that a practical optimization method for a mistuned system (bladed disk with a free-standing blade structure) is to sort the blades so that the resonant stress is minimized. In this study, a simultaneous method for optimizing the blade resonant stress and the amount of unbalance causing rotor vibration is proposed. In this method, first, the natural frequencies and weights of all blades on a disk are measured. Then, a mistuned system is assembled and the analysis model is generated. Next, the resonant stress and the amount of unbalance in the mistuned system are analyzed. To reduce the computation time, the reduced-order model known as fundamental mistuning model (FMM) is used to calculate the resonant stress in the mistuned system. The analyses of the resonant stress and the amount of unbalance are carried out repeatedly, sorting the blades on the disk, and the optimal solution is explored using Monte Carlo simulations (MCS) or discrete differential evolution (DDE). As an example, a mistuned bladed disk of an aero-engine was analyzed, and the validity of the proposed method was verified.

Impurity effect as a probe for the pairing symmetry of graphene-based superconductors

We study theoretically the single impurity effect on graphene-based superconductors. Four different pairing symmetries are discussed. Sharp in-gap resonant peaks are found near the impurity site for the d + id pairing symmetry and the p + ip pairing symmetry when the chemical potential is large. As the chemical potential decreases, the in-gap states are robust for the d + id pairing symmetry while they disappear for the p + ip pairing symmetry. Such in-gap peaks are absent for the fully gapped extended s-wave pairing symmetry and the nodal f-wave pairing symmetry. The existence of the ingap resonant peaks can be explained well based on the sign-reversal of the superconducting gap along different Fermi pockets and by analyzing the denominator of the T-matrix. All of the features may be checked by the experiments, providing a useful probe for the pairing symmetry of graphene-based superconductors.

Two-orbital model for interference of pseudo-gaps in the superconducting state of iron-based superconductors

The coexistence of different order parameters and the pairing symmetry is the major issue in the study of iron-based superconductors. We adopt a two-orbital tight-binding model, for the coexistence of superconductivity, Jahn–Teller distortion and antiferromagnetism in s±-wave symmetry, which is consistent and most favourable for the nearest-neighbour pairing interaction, and derive a set of gap equations which are solved numerically and self-consistently. The effects of the order parameters on one another are studied. We have also calculated the electronic specific heat, density of states and band energies. The specific heat jump at the transition temperatures showing the second order phase transitions is observed and the coefficient of specific heat is also calculated. The density of states shows the defined interference peaks on either side of the Fermi level corresponding to the order parameters with van Hove singularities. The band energies which show the saddle point shift of the bands are also studied.

Odd-parity pairing correlations in a d -wave superconductor

We theoretically study the effects of spin-orbit interactions on symmetry of a Cooper pair in a spin-singlet d-wave superconductor in two-dimension. The pairing symmetry is analyzed in terms of the anomalous Green's function which is obtained by solving the Gor'kov equation analytically. A spin-orbit interaction induces a spin-triplet p-wave pairing correlation in a uniform superconductor. An odd-frequency spin-triplet s-wave pairing correlation appears at a surface of such superconductor as a result of breaking inversion symmetry locally. We also discuss a close relationship among the odd-frequency pairing correlation, chirality of surface bound states at the zero energy, and the anomalous proximity effect. The obtained results enable us to design a superconductor which causes the strong anomalous proximity effect.

Theory of optical responses in clean multi-band superconductors

Electromagnetic responses in superconductors provide valuable information on the pairing symmetry as well as physical quantities such as the superfluid density. However, at the superconducting gap energy scale, optical excitations of the Bogoliugov quasiparticles are forbidden in conventional Bardeen-Cooper-Schrieffer superconductors when momentum is conserved. Accordingly, far-infrared optical responses have been understood in the framework of a dirty-limit theory by Mattis and Bardeen for over 60 years. Here we show, by investigating the selection rules imposed by particle-hole symmetry and unitary symmetries, that intrinsic momentum-conserving optical excitations can occur in clean multi-band superconductors when one of the following three conditions is satisfied: (i) inversion symmetry breaking, (ii) symmetry protection of the Bogoliubov Fermi surfaces, or (iii) simply finite spin-orbit coupling with unbroken time reversal and inversion symmetries. This result indicates that clean-limit optical responses are common beyond the straightforward case of broken inversion symmetry. We apply our theory to optical responses in FeSe, a clean multi-band superconductor with inversion symmetry and significant spin-orbit coupling. This result paves the way for studying clean-limit superconductors through optical measurements.

Nodal superconducting gap in LiFeP revealed by NMR: Contrast with LiFeAs* *Project supported by the Natioanl Natural Science Foundation of China (Grant Nos. 11904023, 11974405, 11674377, and 11634015), the Fundamental Research Funds for the Central Universities, China (Grant No. 2018NTST22), the National Key R&D Program of China (Grant Nos. 2016YFA0300502 and 2017YFA0302904), and the Strategic Priority Research Program of the

Identifying the uniqueness of FeP-based superconductors may shed new lights on the mechanism of superconductivity in iron-pnictides. Here, we report nuclear magnetic resonance (NMR) studies on LiFeP and LiFeAs which have the same crystal structure but different pnictogen atoms. The NMR spectrum is sensitive to inhomogeneous magnetic fields in the vortex state and can provide the information on the superconducting pairing symmetry through the temperature dependence of London penetration depth λ L. We find that λ L saturates below T ∼ 0.2 T c in LiFeAs, where T c is the superconducting transition temperature, indicating nodeless superconducting gaps. Furthermore, by using a two-gaps model, we simulate the temperature dependence of λ L and obtain the superconducting gaps of LiFeAs, as Δ 1 = 1.2 k B T c and Δ2 = 2.8 k B T c, in agreement with previous result from spin-lattice relaxation. For LiFeP, in contrast, λ L does not show any saturation down to T ∼ 0.03 T c, indicating nodes in the superconducting gap function. Finally, we demonstrate that strong spin fluctuations with diffusive characteristics exist in LiFeP, as in some cuprate high temperature superconductors.

Dominant superconducting pairing symmetry in doped staggered-flux Hubbard model on a square lattice

A constrained path quantum Monte Carlo simulation is performed to investigate the superconducting pairing correlation in doped staggered-flux Hubbard model on a square lattice, which is an interesting platform to host Dirac fermions. For various electron fillings and interaction strengths, the [Formula: see text] pairing dominates over other pairing forms since its long-range correlation is larger over mostly distances. We deepen our study by including the nearest-neighbor interaction [Formula: see text] and found the [Formula: see text] pairing correlation is slightly suppressed by increasing [Formula: see text].

Note on generalized symmetries, gapless excitations, generalized symmetry protected topological states, and anomaly

We consider quantum many body systems with generalized symmetries, such as the higher form symmetries introduced recently, and the ‘tensor symmetry’. We consider a general form of lattice Hamiltonians which allow a certain level of nonlocality. Based on the assumption of dual generalized symmetries, we explicitly construct low energy excited states. We also derive the ’t Hooft anomaly for the general Hamiltonians after ‘gauging’ the dual generalized symmetries. A 3d system with dual anomalous 1-form symmetries can be viewed as the boundary of a 4d generalized symmetry protected topological (SPT) state with 1-form symmetries. We also present a prototype example of a 4d SPT state with mixed 1-form and 0-form symmetry topological response theory as well as its physical construction. The boundary of this SPT state can be a 3d anomalous QED state, or an anomalous 1-form symmetry-enriched topological order. Insights are gained by dimensional compatification/reduction. After dimensional compactification, the 3d system with N pairs of dual 1-form symmetries reduces to a 1d system with 2N pairs of dual U(1) global symmetries, which is the boundary of an ordinary 2d SPT state, while the 3d system with tensor symmetry reduces to a 1d Lifshitz theory, which is protected by the center of mass conservation of the system.

Pairing symmetry in monolayer of orthorhombic CoSb

Ferromagnetism and superconductivity are generally considered to be antagonistic phenomena in condensed matter physics. Here, we theoretically study the interplay between the ferromagnetic and superconducting orders in a recent discovered monolayered CoSb superconductor with an orthorhombic symmetry and net magnetization, and demonstrate the pairing symmetry of CoSb as a candidate of non-unitary superconductor with time-reversal symmetry breaking. By performing the group theory analysis and the first-principles calculations, the superconducting order parameter is suggested to be a triplet pairing with the irreducible representation of 3B2u, which displays intriguing nodal points and non-zero periodic modulation of Cooper pair spin polarization on the Fermi surface topologies. These findings not only provide a significant theoretical insight into the coexistence of superconductivity and ferromagnetism, but also reveal the exotic spin polarized Cooper pairing driven by ferromagnetic spin fluctuations in a triplet superconductor.

Jahn–Teller Distortion in Two-Orbital Model for s±-Wave Iron-Based Superconductors

The theoretical aspects of research on the role of Jahn–Teller (JT) interactions in quasi-two-dimensional iron-based high-[Formula: see text] superconductors are analyzed. A model Hamiltonian for the JT distortion in iron-based superconductors is proposed by a tight-binding two-orbital model. We have proposed here a [Formula: see text]-wave pairing symmetry of the form [Formula: see text] in the model in mean-field approximation. The model is solved by Zubarev’s double-time Green’s function technique to find the self-consistent gap equation and is solved self-consistently. The lattice strain energy, specific heat, conduction electron density of states and energy band structure for the system are calculated.

Superconductivity in Two-Orbital Model for s±-Wave Iron-Based Superconductors

In high-[Formula: see text] superconductors, electrons form pairs and electric transport becomes dissipation-less at low temperatures. The iron-based superconductors (FeSCs) have the highest superconducting (SC) transition temperature next to copper oxides. The gap structure and pairing mechanism for FeSCs are hotly discussed as a central issue since their discovery. A model Hamiltonian for the superconductivity in FeSCs is proposed by a tight-binding two-orbital model. The SC gap, conduction electron density of states, specific heat and energy band structure for the system are calculated. We have proposed here a [Formula: see text]-wave pairing symmetry of the form [Formula: see text] in the model in the mean-field approximation. The model is solved by Zubarev’s double-time Green’s function technique to find the self-consistent gap equation and is solved self-consistently.

Interplay between nematicity and Bardasis-Schrieffer modes in the short-time dynamics of unconventional superconductors

Motivated by the recent experiments suggesting the importance of nematicity in the phase diagrams of ironbased and cuprate high-Tc superconductors, we study the influence of nematicity on the collective modes inside the superconducting state in a non-equilibrium. In particular, we consider the signatures of collective modes in short-time dynamics of a system with competing nematic and s- and d-wave superconducting orders. In the rotationally symmetric state, we show that the Bardasis-Schrieffer mode, corresponding to the subdominant pairing, hybridizes with the nematic collective mode and merges into a single in-gap mode, with the mixing vanishing only close to the phase boundaries. For the d-wave ground state, we find that nematic interaction suppresses the damping of the collective oscillations in the short-time dynamics. Additionally, we find that even inside the nematic s+d-wave superconducting state, a Bardasis-Schrieffer-like mode leads to order parameter oscillations that strongly depend on the competition between the two pairing symmetries. We discuss the connection of our results to the recent pump-probe experiments on high-Tc superconductors.

Geometry-induced topological superconductivity

Intrinsic topological superconductors with p-wave pairing are rare in nature. Its underlying reason is due to the fact that it is usually difficult to change the relative strength between the singlet and triplet channels for the electron-electron interaction in material. Here we show that by considering superconductivity occurring on surfaces of topological insulators (TIs), the relative strength between the singlet and triplet channels can be changed by geometry and sizes of TIs. Specifically, we show that pairing of electrons at different locations on the surface of a topological insulator generally tends to favor the triplet pairing and can induce topological superconductivity by controlling the surface curvature and size of the topological insulator. We illustrate the effects in two configurations, thin film geometry and the spherical geometry with a sphere or a hemisphere, and find that topological superconductivity arises with the $p \pm ip$ pairing symmetry dominated in nanoscale size of the TI. As a consequence, vortices can spontaneously form on surfaces of topological insulators with roughness of appropriate curvature. These vortices support a Majorana zero mode inside each core and can be used as a platform to host Majorana zero modes without invoking real magnetic fields. Our theoretical discovery opens a new route to realize topological superconductivity in material.

The pairing symmetries in the two-dimensional Hubbard model

Using finite-temperature determinant quantum Monte Carlo algorithm, we study the effective pairing susceptibility of different pairing channels in the two-dimensional Hubbard model, to determine the dominant pairing channels in different parameter regimes. At half-filling, we can access low enough temperatures and find the dominant pairing is the dx2−y2-wave. While in the doping case, we are limited to some finite temperatures due to the sign problem of the determinant quantum Monte Carlo algorithm. When the doping is not very heavy, the dominant pairing channel keeps to be the dx2−y2-wave. While at low electron densities, the dominant pairing channel switches to the dxy-wave. In additional, there's a small region of the s⁎-wave pairing at weak interaction strength and close to half-filling. Except for the s⁎-wave regime, the interaction strength and the temperature seem to only have slight effect on the dominant pairing channels.

Superconductivity and Jahn-Teller Distortion in s±-Wave Iron-Based Superconductors

We discuss here a few possibilities of gap structure and pairing mechanism for iron-based superconductors with more than one orbital symmetry contributing to the pairing. We have proposed a two-orbital tight-binding model for the coexistence of superconductivity and Jahn-Teller distortion in s±-wave pairing symmetry of the form coskx × cosky in the mean-field approximation. The self-consistent gap equations are calculated by Zubarev’s double-time Green’s function technique which are solved numerically and self-consistently. The specific heat, density of states and band energies are also studied for the proposed model.

Shadowed triplet pairings in Hund's metals with spin-orbit coupling

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 triplet nature associated with this interorbital pairing, which we dub a "shadowed triplet". Applying this concept to Sr$_{2}$RuO$_{4}$, we first derive several forms of SOC with $d$-wave form factors from a microscopic model, and subsequently we show that for a range of SOC parameters, a pairing state with $s+id_{xy}$ symmetry can be stabilized. Such a pairing state is distinct from pure spin-singlet and -triplet pairings due to its unique character of pseudospin-energy locking. We discuss experimental probes to differentiate the shadowed triplet pairing from conventional pseudospin-triplet and -singlet pairings.

Quantum Monte Carlo study of the Hubbard model with next-nearest-neighbor hopping t′: pairing and magnetism

Using the finite-temperature determinant quantum Monte Carlo (DQMC) algorithm, we study the pairing symmetries of the Hubbard Hamiltonian with next-nearest-neighbor (NNN) hopping t′ on square lattices. By varying the value of t′, we find that the d-wave pairing is suppressed by the onset of t′, while the p + ip-wave pairing tends to emerge for low electron density and t′ around −0.7. Together with the calculation of the anti-ferromagnetic and ferromagnetic spin correlation function, we explore the relationship between anti-ferromagnetic order and the d-wave pairing symmetry, and the relationship between ferromagnetic order and the p + ip-wave pairing symmetry. Our results may be useful for the exploration of the mechanism of the electron pairing symmetries, and for the realization of the exotic p + ip-wave superconductivity.

Emergence of the Unconventional Type-II Nambu-Goldstone Modes with Topological Origin in Bose Superfluids.

The Nambu-Goldstone (NG) modes in a nonrelativistic system can be classified into two types from their characteristic features: being of either an odd (type I) or an even (type II) power energy-momentum dispersion. Conventionally, the type-II NG modes may universally arise from spontaneous breaking of noncommutative symmetry pairs. Here, we predict a novel type of quadratically dispersed NG modes that emerges in mixed s and p band Bose superfluids in an optical lattice and, unlike the conventional type-II NG modes, cannot be solely interpreted with the celebrated symmetry-based argument. Instead, we show that the existence of such modes has a profound connection to the topological transition on projective complex order-parameter space. The detection scheme is also proposed. Our Letter reveals a new universal mechanism for emergence of type-II NG modes, which bridges intrinsically the Landau symmetry-breaking and topological theories.