Wave Pairing Research Articles

The dynamics of current-driven vortex in two-band superconductor with s+d wave pairing

Two-band mesoscopic superconductors are known to exhibit specific vortex states. In this paper, finite element method (FEM) is used to numerically solve the time-dependent Ginzburg-Landau (TDGL) equation with s+d wave coupling for a 3D mesoscopic superconducting strip. We discuss the effect of external magnetic field H and applied current I on s+d wave coupling at the temperatures Ts<T<Td, where Ts and Td are the critical temperatures of s- and d-bands, respectively. The obtained results show that this coupling is enhanced by the applied current and reflected by the vortex dynamics in two bands. A small amount of Cooper pairs in the s-band form a series of different vortex and antivortex states. Mutual attraction of vortex and antivortex induces simultaneous motion of s- and d-vortices at applied current I. This work provides theoretical background for the design of mesoscopic superconducting devices.

Competing orders and cascade of degeneracy lifting in doped Bernal bilayer graphene

Motivated by recent experiments [H. Zhou et al., Science 375, 774 (2022) and S. C. de la Barrera et al., arXiv:2110.13907], here we propose a general mechanism for valley and/or spin degeneracy lifting of the electronic bands in doped Bernal bilayer graphene, subject to electric displacement ($D$) fields. A $D$-field induced layer polarization (LP), when accompanied by a Hubbard repulsion-driven layer antiferromagnet (LAF) and next-nearest-neighbor repulsion-driven quantum anomalous Hall (QAH) orders, lifts the fourfold degeneracy of electronic bands, yielding a quarter metal for small doping, as also observed in ABC trilayer graphene. With the disappearance of the QAH order, electronic bands recover twofold valley degeneracy, thereby forming a conventional or compensated (with majority and minority carriers) half metal at moderate doping, depending on the relative strength of LP and LAF. At even higher doping and for a weak $D$ field only LAF survives and the Fermi surface recovers fourfold degeneracy. We also show that a pure repulsive electronic interaction mediated triplet $f$-wave pairing emerges from a parent correlated nematic liquid or compensated half metal when an in-plane magnetic field is applied to the system.

Enhancement of d -wave pairing in the striped phase with nearest neighbor attraction

Recently, the experimental results by the angle-resolved photoemission spectroscopy suggested that an additional strong nearest neighbor attraction in the Hubbard model might be significant to describe the properties of doped cuprates more accurately. The stripe-ordered patterns, formed by the inhomogeneous distribution of spin, charge and pairing correlations in the CuO$_{2}$ planes, is a known feature of doped cuprates. In this work, the effect of the nearest neighbor attraction and the stripe phase are examined by using the constrained path quantum Monte Carlo method within the repulsive Hubbard model on two-dimensional square lattice. The ground state spin correlations along and cross the stripe regions, and the $d$-wave pairing correlation are calculated. It is found that the spin-spin correlation is the highest when the interstripe region is fairly close to half-filling, and $d$-wave superconducting correlation on neighboring sites could be enhanced in the presence of stripe pattern and strong nearest neighbor attraction, which reveals their crucial roles on superconductivity in the doped cuprates.

Lifshitz transition enhanced triplet pz -wave superconductivity in hydrogen-doped KCr3As3

The recently synthesized air-insensitive hydrogen-doped ${\mathrm{KCr}}_{3}{\mathrm{As}}_{3}$ superconductor has aroused great research interest. This material has, for the first time in the research area of the quasi-one-dimensional (quasi-1D) Cr-based superconductivity (SC), realized a tunability through charge doping, which will potentially significantly push the development of this area. Here, based on the band structure from first-principles calculations, we construct a six-band tight-binding model equipped with multiorbital Hubbard interactions and adopt the random-phase-approximation approach to study the hydrogen-doping dependence of the pairing symmetry and superconducting ${T}_{c}$. Under the rigid-band approximation, our pairing phase diagram is occupied by the triplet ${p}_{z}$-wave pairing throughout the hydrogen-doping regime $x\ensuremath{\in}(0.4,1)$ in which SC has been experimentally detected. Remarkably, the $x$ dependence of ${T}_{c}$ shows a peak at the three-dimensional--quasi-1D Lifshitz transition point, although the total density of states exhibits a dip there. The corresponding doping level is near the experimental estimation of the optimal doping level. A thorough investigation of the band structure reveals type-II van Hove singularities (VHSs) in the $\ensuremath{\gamma}$ band, which favor the formation of the triplet SC. It turns out that the $\ensuremath{\gamma}$ Fermi surface (FS) comprises two flat quasi-1D FS sheets almost parallel to the ${k}_{z}=0$ plane and six almost perpendicular tubelike FS sheets, and the type-II VHS just lies in the boundary between these two FS parts. Furthermore, the $|{k}_{z}|$ of the van Hove planes reaches the maximum near the Lifshitz-transition point, which pushes the ${T}_{c}$ of the ${p}_{z}$-wave SC to the maximum. Our results appeal for more experimental access into this intriguing superconductor.

Doping dependence of the electron-phonon coupling in two families of bilayer superconducting cuprates

While electron-phonon coupling (EPC) is crucial for Cooper pairing in conventional superconductors, its role in high-$T_c$ superconducting cuprates is debated. Using resonant inelastic x-ray scattering at the oxygen $K$-edge, we studied the EPC in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212) and Nd$_{1+x}$Ba$_{2-x}$Cu$_3$O$_{7-\delta}$ (NBCO) at different doping levels ranging from heavily underdoped ($p =0.07$) to overdoped ($p=0.21$). We analyze the data with a localized Lang-Firsov model that allows for the coherent excitations of two phonon modes. While electronic band dispersion effects are non-negligible, we are able to perform a study of the relative values of EPC matrix elements in these cuprate families. In the case of NBCO, the choice of the excitation energy allows us to disentangle modes related to the CuO$_3$ chains and the CuO$_2$ planes. Combining the results from the two families, we find the EPC strength decreases with doping at $\mathbf{q_\parallel}=(-0.25, 0)$ r.l.u., but has a non-monotonic trend as a function of doping at smaller momenta. This behavior is attributed to the screening effect of charge carriers. We also find that the phonon intensity is enhanced in the vicinity of the charge-density-wave (CDW) excitations while the extracted EPC strength appears to be less sensitive to their proximity. By performing a comparative study of two cuprate families, we are able to identify general trends in the EPC for the cuprates and provide experimental input to theories invoking a synergistic role for this interaction in $d$-wave pairing.

Se77 -NMR evidence for spin-singlet superconductivity with exotic superconducting fluctuations in FeSe

Although the nature of the superconducting state has been intensively studied in FeSe, many fundamental issues including the pairing symmetry and superconducting fluctuations are still highly controversial. Here, we report a revised $^{77}\mathrm{Se}$-nuclear magnetic resonance (NMR) study on the superconducting state of $^{77}\mathrm{Se}$-enriched bulk FeSe. Below the superconducting temperature (${T}_{c}$), by carefully avoiding the rf heating effect, a remarkable linewidth broadening and obvious reduction of the Knight shift are observed under external magnetic field along both in-plane and out-of-plane directions, suggesting an intrinsic superconducting nature. These exotic results unambiguously rule out the possibility of chiral $p$-wave pairing, and favor a pairing scenario with the mixing of ${s}^{\ifmmode\pm\else\textpm\fi{}}$ and $d$ wave. A slight decrease of the Knight shift well above ${T}_{c}$ is also revealed under a moderated external magnetic field, suggesting exotic superconducting fluctuations beyond phase fluctuations in the two-dimensional limit. These renewed NMR results provide valuable constraints for the theoretical models on the exotic superconductivity in FeSe.

Enhancement of pairing correlations due to nearly flat bands in the plaquette-Lieb Hubbard model

Motivated by recent research works on the flat-band correlated electron systems, we construct a plaquette-Lieb lattice, which exhibits nearly flat bands close to the Fermi level around half-filling, when the inter-plaquette hopping integral is much smaller than the intra-plaquette one. A detailed comparison study with the well-known checkerboard lattice indicates that there exist significant differences for the pairing property between the two lattices, and the nearly flat bands on the plaquette-Lieb lattice strongly enhance the [Formula: see text]-wave pairing correlation for repulsive on-site Hubbard interaction and [Formula: see text]-wave pairing correlation for attractive on-site Hubbard interaction. Our study not only provides a new lattice with nearly flat bands, but also deepens the understanding of electronic structure on superconductivity.

Metals, fractional metals, and superconductivity in rhombohedral trilayer graphene

Combining mean-field and renormalization group analyses, here we unveil the nature of recently observed superconductivity and parent metallic states in chemically doped rhombohedral trilayer graphene, subject to external electric displacement fields ($D$) [H. Zhou, \emph{et al.}, Nature (London) {\bf 598}, 434 (2021)]. We argue that close to the charge neutrality, on site Hubbard repulsion favors layer antiferromagnet, which when combined with the $D$-field induced layer polarization, produces a spin-polarized, but valley-unpolarized half-metal, conducive to the nucleation of spin-triplet $f$-wave pairing (SC2). At larger doping valence bond order emerges as a prominent candidate for isospin coherent paramagent, boosting condensation of spin-singlet Cooper pairs in the $s$-wave channel (SC1), manifesting a "selection rule" among competing orders. Responses of these paired states to displacement and in-plane magnetic fields show qualitative similarities with experimental observation. With the onset of the quantum anomalous Hall order, the valley degeneracy of half-metal gets lifted, forming a quarter-metal at lower doping [H. Zhou, \emph{et al.}, Nature (London) {\bf 598}, 429 (2021)].

Projected BCS theory for the unification of antiferromagnetism and strongly correlated superconductivity

A close connection between antiferromagnetism and superconductivity is at the core of high-temperature superconductivity. Here, we put forward the projected BCS theory for the unification of antiferromagnetism and strongly correlated superconductivity. Specifically, it is shown that, with the $d$-wave pairing symmetry, the projected BCS theory can provide excellent trial states at general doping for the exact ground states of the $t\text{\ensuremath{-}}J$ model in the square lattice, generating a unified theory of high-temperature superconductivity as a continuous function of hole concentration. A key to the success of the projected BCS theory is an accurate treatment of the strong correlation between Cooper pairs, which not only causes the breakdown of superconductivity itself at half filling but also defines the precise nature of strongly correlated superconductivity at moderate doping. Also, via the proper incorporation of particle number fluctuations, the projected BCS theory allows direct computation of the superconducting order parameter, shedding important light on the pseudogap phenomenon.

Superconducting pairing symmetry in the kagome-lattice Hubbard model

The dominating superconducting pairing symmetry of the kagome-lattice Hubbard model is investigated using the determinant quantum Monte Carlo method. The superconducting instability occurs when doping the correlated insulators formed by the Hubbard interaction near the Dirac filling, and the superconducting state exhibits an electron-hole asymmetry. Among the pairing symmetries allowed, we demonstrate that the dominating channel is d-wave in the hole-doped case. This opens the possibility of condensation into an unconventional $d_{x^2-y^2}+id_{xy}$ phase, which is characterized by an integer topological invariant and gapless edge states. In contrast, the $s^*$-wave channel, which has no change of sign in the pairing function, is favored by electron doping. We further find the dominating $s^*$-wave pairing persists up to the Van Hove singularity. The results are closely related to the recent experimental observations in kagome compounds AV3Sb5(A: K, Rb, Cs), and provide insight into the pairing mechanism of their superconducting states.

Controlling Majorana modes by p -wave pairing in two-dimensional p+id topological superconductors

We show that corner Majorana zero modes in a two-dimensional $p+id$ topological superconductor can be controlled by the manipulation of the parent $p-$wave superconducting order. Assuming that the $p$-wave superconducting order is in either a chiral or helical phase, we find that when a $d_{x^2-y^2}$ wave superconducting order is induced, the system exhibits quite different behavior depending on the nature of the parent $p$-wave phase. In particular, we find that while in the helical phase, a localized Majorana mode appears at each of the four corners, in the chiral phase, it is localized only along two of the four edges. We furthermore demonstrate that the Majoranas can be directly controlled by the form of the edges, as we explicitly show in case of the circular edges. We argue that the application of strain may provide additional means of fine-tuning the Majorana zero modes in the system, in particular, it can partially gap them out. Our findings may be relevant for probing the topology in two-dimensional mixed-pairing superconductors.

Pattern-dependent proximity effect and Majorana edge mode in one-dimensional quasicrystals

The Majorana edge states of the Kitaev chain model have attracted extensive attention on their stability and experimental realization. One of the interesting aspects is finding the exotic proximity effect, which guarantees the presence of the Majorana modes, further enables efficient braidings between them. In this paper, we explore the superconducting proximity effect for quasiperiodic quantum wires and discuss how quasiperiodic patterns affect the stability of the Majorana modes. Considering the Kitaev chain model of the one-dimensional quasiperiodic system, we discuss the pattern-dependent proximity effects. First, we argue that the presence of quasiperiodic hoppings energetically induces the $p$-wave pairing also to be quasiperiodic rather than uniform pairing. More interestingly, when the normal metallic wire is adjacent to the quasiperiodic superconducting wire, we have found that the Majorana edge modes are transferred to the edge of the normal metallic side with enhanced stability. Finally, we discover the proximity effect on the strengths of the quasiperiodicities with a general power-law relationship, whose power depends on the tiling pattern. Our results show how quasiperiodic patterns play a crucial role in the Kitaev chain and the stabilization of the Majorana mode.

Topological phase transition driven by magnetic field in one-dimensional topological superconductor rings

We study the energy spectrum and transport property of a one-dimensional Kitaev quantum ring in a threading magnetic field. It is demonstrated that the magnetic field can effectively induce topological phase transitions for the ring in the topologically nontrivial phase at the zero magnetic field. However, for the ring in the topologically trivial phase at the zero field, there is no topological phase transition, and the energy spectrum of the system is always gapped. The magnetic field can control the appearance and disappearance of Majorana zero-energy states in the Kitaev quantum ring, when one half of the ring is in the topologically nontrivial phase and the other half is in the topologically trivial phase. Furthermore, we calculate the transport properties of the ring connected by two semi-infinite leads. It is found that the resonant peaks of transmission coefficient ${T}_{\text{QT}}$ correspond to the critical points of topological phase transition. In addition, we extend our findings to a more realistic quantum ring adopting a semiconductor nanowire with high spin-orbit coupling, superconducting $s$-wave pairing, and Zeeman splitting, and prove that our findings are universal.

The 1S0 Pairing Gap in Neutron Matter

We report ab initio calculations of the S wave pairing gap in neutron matter calculated using realistic nuclear Hamiltonians that include two- and three-body interactions. We use a trial state, properly optimized to capture the essential pairing correlations, from which we extract ground state properties by means of auxiliary field diffusion Monte Carlo simulations. We extrapolate our results to the thermodynamic limit by studying the finite-size effects in the symmetry-restored projected Bardeen-Cooper-Schrieffer (PBCS) theory and compare our results to other ab initio studies done in the past. Our quantum Monte Carlo results for the pairing gap show a modest suppression with respect to the mean-field BCS values. These results can be connected to cold atom experiments, via the unitarity regime where fermionic superfluidity assumes a unified description, and they are important in the prediction of thermal properties and the cooling of neutron stars.

Topological superconductors and exact mobility edges in non-Hermitian quasicrystals

We study a class of non-Hermitian topological superconductors described by one-dimensional Aubry-Andr\'e Harper and mosaic quasiperiodic models with $p$-wave superconducting pairing, where the non-Hermiticity is introduced by on-site complex quasiperiodic potentials. We generalize two topological invariants, one is based on the transfer matrix method and the other is the generalized Majorana polarization, to characterize the topological superconducting phases and verify the existence of Majorana zero modes in non-Hermitian quasiperiodic superconductors. By combing the Lyapunov exponent, the fractional dimension of wave functions, and topological invariants, we investigate the localization phenomena, topological superconductivity, and topological phase transitions. In the non-Hermitian Aubry-Andr\'e Harper model with $p$-wave pairing, the system undergoes an extended-critical-localized phase transition with increasing the complex phase. The localization transition is consistent with the topological phase transition and unconventional real-complex transition of eigenenergy. In the non-Hermitian mosaic model, we provide an analytical expression of mobility edges and prove the intrinsic relation between the mobility edges and unconventional real-complex transitions. Our discoveries unveil the richness of topological and localization phenomena in non-Hermitian quasicrystals with $p$-wave pairing.

Surface Bogoliubov-Dirac cones and helical Majorana hinge modes in superconducting Dirac semimetals

In the presence of certain symmetries, three-dimensional Dirac semimetals can harbor not only surface Fermi arcs, but also surface Dirac cones. Motivated by the experimental observation of rotation-symmetry-protected Dirac semimetal states in iron-based superconductors, we investigate the potential intrinsic topological phases in a $C_{4z}$-rotational invariant superconducting Dirac semimetal with $s_{\pm}$-wave pairing. When the normal state harbors only surface Fermi arcs on the side surfaces, we find that an interesting gapped superconducting state with a quartet of Bogoliubov-Dirac cones on each side surface can be realized, even though the first-order topology of its bulk is trivial. When the normal state simultaneously harbors surface Fermi arcs and surface Dirac cones, we find that a second-order time-reversal invariant topological superconductor with helical Majorana hinge states can be realized. The criteria for these two distinct topological phases have a simple geometric interpretation in terms of three characteristic surfaces in momentum space. By reducing the bulk material to a thin film normal to the axis of rotation symmetry, we further find that a two-dimensional first-order time-reversal invariant topological superconductor can be realized if the inversion symmetry is broken by applying a gate voltage. Our work reveals that diverse topological superconducting phases and types of Majorana modes can be realized in superconducting Dirac semimetals.

Enhancing d -wave superconductivity with nearest-neighbor attraction in the extended Hubbard model

Motivated by the recent discovery of the anomalously nearest-neighbor attraction arising from the electron-phonon coupling, we quantitatively investigate the enhancing effects of this additional attractive channel on the $d$-wave SC based on dynamic cluster quantum Monte Carlo calculations of doped two-dimensional extended Hubbard model with nearest-neighbor attraction $-V$. Focusing on the range of $0<-V/t \le 2$, our simulations indicate that the dynamics of $d$-wave projected pairing interaction is attractive at all frequencies and increases with $|V|$. Moreover, turning on $-V$ attraction enhances the $(\pi,\pi)$ spin fluctuations but only enhances (suppresses) the charge fluctuations for small (large) momentum transfer. Thus, at $V/t=-1$ relevant to ``holon folding branch'', the charge fluctuations are insufficient to compete with $d$-wave pairing interaction strengthened by enhanced spin fluctuations. Our work suggest the underlying rich interplay between the spin and charge fluctuations in giving rise to the superconducting properties.

Supercurrents and spontaneous time-reversal symmetry breaking by nonmagnetic disorder in unconventional superconductors

Recently, a theoretical study [Z.-X. Li et al., npj Quantum Mater. 6, 36 (2021)] investigated a model of a disordered $d$-wave superconductor, and reported local time-reversal symmetry breaking current loops for sufficiently high disorder levels. Since the pure $d$-wave superconducting state does not break time-reversal symmetry, it is surprising that such persistent currents arise purely from nonmagnetic disorder. Here, we perform a detailed theoretical investigation of such disorder-induced orbital currents, and show that the occurrence of the currents can be traced to the emergence of local (extended) $s$-wave order coexisting with underlying disordered $d$-wave pairing, making it favorable to generate local $s\ifmmode\pm\else\textpm\fi{}id$ regions. We discuss the energetics leading to such regions of $s\ifmmode\pm\else\textpm\fi{}id$ order, which support spontaneous local current loops in the presence of inhomogeneous density modulations.

Even-denominator fractional quantum Hall state in bilayer graphene

At a half-filled Landau level, composite fermions with chiral &lt;i&gt;p&lt;/i&gt;-wave pairing will form a Moore-Read state which hosts charge-&lt;i&gt;e&lt;/i&gt;/4 fractional excitation. This excitation supports non-Abelian statistics and has potential to enable topological quantum computation. Owing to the &lt;i&gt;SU&lt;/i&gt;(4) symmetry of electron and electric-field tunability, the bilayer graphene becomes an ideal platform for exploring physics of multi-component quantum Hall state and is candidate for realizing non-Abelian statistics. In this work, high-quality bilayer graphene/hBN heterostructure is fabricated by using dry-transfer technique, and electric transport measurement is performed to study quantum Hall state behavior in bilayer graphene under electric field and magnetic field. Under strong magnetic field, the sequences of incompressible state with quantized Hall conductivity are revealed at –5/2, –1/2, 3/2 filling of Landau level. The feature of even-denominator quantum Hall state is more visible then weaker with increasing magnetic field, and this corresponds to the polarization of Landau level wave function. The experimental results indicate that the observed even-denominator fractional quantum Hall state belongs to the topological phase described by Pfaffian wavefunction.

Correlation-induced d -wave pairing in a quantum dot square lattice

We consider an electrostatically induced square lattice of quantum dots and study the role of electron-electron correlations in the resulting electronic features of the system. We utilize the Wannier functions methodology in order to construct Hamiltonian for interacting fermions and find that the change of the depth of the quantum dot confining potential results in a transition from a moderately-, to strong-correlated regime of the system. We obtain the approximate ground state by means of Variational Monte-Carlo method for a wide range of dopings. The values of microscopic parameters, charge gap as well as spin- and pair-correlation functions obtained in the strongly-correlated regime signify the presence of antiferromagnetic spin-ordering and the realization of the Mott insulator phase. Moreover, we report on a two dome structure of the emerging $d$-wave paired state residing on both sides of the half filled case. The obtained results are discussed in view of the well known families of unconventional superconducting materials such as copper based compounds.