We study a toy model for a superconductor on a bipartite lattice, where intrinsic pairing inhomogeneity is produced by two different coupling constants on the sublattices. The simplicity of the model allows for analytic solutions and tests of the consequences of atomic-scale variations in pairing interactions which have been considered recently in the cuprates. We present results for the transition temperature, density of states, and thermodynamics of the system over a phase diagram in the plane of two pairing coupling constants. For coupling constants of alternating sign, a gapless superconducting state is stable. Inhomogeneity is generally found to enhance the critical temperature, and at the same time the superfluid density is remarkably robust: at T=0, it is suppressed only in the gapless phase.
We study properties of the zero energy Andreev bound state in a superconductor-normal metal-superconductor(SNS) junction consisting of two intrinsic degrees of freedom. The superconductors on either sides of the normal metal are assumed to have two sublattices with an $intra$-sublattice pairing with a phase of zero or $π$ between the two sublattices. In addition, we add a uniform $inter$-sublattice pairing and study its effect on the local density of states (LDOS). In particular, we find that as the inter-sublattice pairing is turned on, the zero bias peak (ZBP) is unstable (robust) when the p…
Previous studies have shown that bipartite Hubbard systems with inhomogeneous hopping amplitudes can exhibit higher pair-binding energies than the uniform model. Here we examine whether this result holds for systems with a more generic band structure. To this end, we use exact diagonalization and the density matrix renormalization group method to study the 4x4 Hubbard cluster and the two-leg Hubbard ladder with checkerboard-modulated nearest-neighbor hopping, t, and next-nearest-neighbor (diagonal) hopping, t_d. We find that the strongest pairing continues to occur at an intermediate level of …
Scanning tunneling spectroscopy has recently discovered a positive correlation between the magnitude of the superconducting gap and positions of dopant oxygen atoms in Bi-based cuprates. We propose a microscopic mechanism that could be responsible for this effect. In particular, we demonstrate that the dopant-induced spatial variation of the atomic levels always enhances the superexchange interaction.
A new theoretical model is presented to study the nanoscale electronic inhomogeneity in high-$T_c$ cuprates. In this model, we argue that the randomly distributed out-of-plane interstitial oxygen dopants induces locally the off-diagonal (i.e., hopping integral) disorder. This disorder modulates the superexchange interaction resulting from a large-$U$ Hubbard model, which in turns changes the local pairing interaction. The microscopic self-consistent calculations shows that the large gap regions are registered to the locations of dopants. Large gap regions exhibit small and broader coherence pe…
Striped phases, in which spin, charge, and pairing correlations vary inhomogeneously in the CuO_2 planes, are a known experimental feature of cuprate superconductors, and are also found in a variety of numerical treatments of the two dimensional Hubbard Hamiltonian. In this paper we use determinant Quantum Monte Carlo to show that if a stripe density pattern is imposed on the model, the d-wave pairing vertex is significantly enhanced. We attribute this enhancement to an increase in antiferromagnetic order which is caused by the appearence of more nearly half-filled regions when the doped holes…
In this review we examine the experimental evidence for dynamic inhomogeneity, defining the length, time and energy scales of the relevant elementary excitations. The dynamics of the objects below and above Tc are examined in detail with femtosecond spectroscopy and compared with magnetic and other measurements. The dynamically inhomogeneous state is described theoretically by considering an interaction between electrons, spins and the lattice. By symmetry, only electrons in degenerate states can couple to the lattice and spins to give an anisotropic, d-wave symmetry interaction. The proposed …
A t-J-like model for inhomogeneous superconductivity of cuprate oxides is presented, in which local anisotropic magnetic terms are essential. We show that this model predicts pairing, consistent with experiments, and argue how the macroscopic phase-coherent state gradually grows upon lowering of the temperature. We show that appropriate inhomogeneities are essential in order to have significant pair binding in the thermodynamic limit. Particularly, {\it local} breaking of SU(2) spin symmetry is an efficient mechanism for inducing pairing of two holes, as well as explaining the magnetic scatter…
We study the quasiparticle spectrum of 2D topological $s$-wave superconductors with the Zeeman magnetic field and the Rashba spin-orbit coupling in the presence of spatial inhomogeneity. Solving the real-space Bogoliubov-de Gennes equations, we focus on the excitations within the superconducting gap amplitude, i.e., the appearance of mid-gap states. Two kinds of potential functions, line-type (a chain of impurities) and point-type (a single impurity) ones are examined to take spatial inhomogeneity into account. The line setting shows a link of the mid-gap states with the gapless surface modes …
We construct minimum effective models to investigate the pairing symmetry in the newly discovered quasi-one-dimensional superconductor K$_2$Cr$_3$As$_3$. We show that a minimum three-band model based on the $d_{z^2}$, $d_{xy}$ and $d_{x^2-y^2}$ orbitals of one Cr sublattice can capture the band structures near Fermi surfaces. In both weak and strong coupling limits, the standard random phase approximation (RPA) and mean-field solutions consistently yield the triplet $p_z$-wave pairing as the leading pairing symmetry for physically realistic parameters. The triplet pairing is driven by the ferr…
The Su-Schrieffer-Heeger model describes fermions that hop on a one-dimensional chain with staggered hopping amplitude, where the unit cell contains two sites, or two sublattices. In this work we consider the Su-Schrieffer-Heeger model with superconducting pairing and show that the sublattice index acts as an additional quantum number in the classification of Cooper pairs, giving rise to inter- and intra-sublattice odd-frequency pair correlations in the bulk. Interestingly, this system behaves as a two band superconductor where the bulk odd-frequency correlations depend solely on the intrinsic…
We study the pairing and superconducting properties of the attractive Hubbard model in two quasi one-dimensional topological lattices: the Creutz and sawtooth lattices. They share two peculiar properties: each of their band structures exhibits a flat band with a non-trivial winding number. The difference, however, is that only the Creutz lattice is genuinely topological, owing to a chiral (sub-lattice) symmetry, resulting in a quantized winding number and zero energy edge modes for open boundary conditions. We use mean field and exact density matrix renormalization group in our work. Our three…
Inhomogeneous s-wave superconductivity is studied in the two-dimensional, square lattice attractive Hubbard Hamiltonian using the Bogoliubov-de Gennes (BdG) mean field approximation. We find that at weak coupling, and for densities mainly below half-filling, an inhomogeneous interaction in which the on-site interaction $U_i$ takes on two values, $U_i=0, 2U$ results in a larger zero temperature pairing amplitude, and that the superconducting $T_c$ can also be significantly increased, relative to a uniform system with $U_i=U$ on all sites. These effects are observed for stripe, checkerboard, and…
Scanning tunneling spectroscopy studies on Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ suggest the presence of electronic inhomogeneity with a large spatial variation in gap size. Andersen {\it et al} have modelled this variation by assuming a spatially-varying pairing interaction. We show that their calculated specific heat is incompatible with the experimental data which exhibit narrow transitions. This calls into question the now-common assumption of gap and pairing inhomogeneity.
We have constructed a quasiclassical framework on superconductors with strong spin-orbit couplings, applicable to CuxBi2Se3[Y. Nagai, H. Nakamura, and M. Machida: arXiv:1305.3025]. The notable point is that in this framework the Bogoliubov-de Gennes Hamiltonians with suggested odd-parity pairing states turn to quasiclassical ones with usual spin-triplet Cooper pairs. Using this quasiclassical theory, we can investigate inhomogeneity effects such as the phenomena with vortices and surfaces in this superconductors and shed light on the pairing state of topological superconductors. In this paper,…
The nanometer-scale gap inhomogeneity revealed by recent STM images of BSCCO surface suggests that the ``gap coherence length'' is of that order. Thus a robust pairing gap can develop despite the poorly screened Coulomb interaction. This can be taken as an evidence that the pairing in high $T_c$ materials occurs primarily in the neutral (spin) sector. We provide theoretical support for this point of view.
Using dynamic cluster quantum Monte Carlo simulations, we study the superconducting behavior of a 1/8 doped two-dimensional Hubbard model with imposed uni-directional stripe-like charge density wave modulation. We find a significant increase of the pairing correlations and critical temperature relative to the homogeneous system when the modulation length-scale is sufficiently large. With a separable form of the irreducible particle-particle vertex, we show that optimized superconductivity is obtained for moderate modulation strength due to a delicate balance between the modulation enhanced pai…
We study the effect of nonuniform transverse couplings on a quasi-one dimensional superconductor. We show that inhomogeneous couplings quite generally increase the superconducting (pairing) gap relative to the uniform system, but that beyond an ``optimal'' degree of inhomogeneity, they lead to a suppression of the tendency to phase coherence. The optimal conditions for superconductivity are derived. We also show that a {\it delocalized}, spin-gapped phase is stable against weak disorder in a four-leg-ladder with moderate repulsive interactions.
Under the assumption that superconducting pairing is driven by local d-p hybridization, we show that the superconducting state in iron-based superconductors is classified as an odd parity s-wave spin-singlet pairing state in a single trilayer FeAs/Se, the building block of the materials. In a low energy effective model with only d-orbitals in an iron square bipartite lattice, the superconducting order parameter in this state is a combination of a s-wave normal pairing between two sublattices and a s-wave $η$-pairing within the sublattices. Parity conservation was violated in proposed supercond…
We study the effective interactions between Dirac fermions on the surface of a three-dimensional topological insulator due to the proximity coupling to the magnetic fluctuations in a ferromagnetic or antiferromagnetic insulator. Our results show that the magnetic fluctuations can mediate attractive interactions between Dirac fermions of both Amperean and BCS type. In the ferromagnetic case, we find pairing between fermions with parallel momenta, so-called Amperean pairing, whenever the effective Lagrangian for the magnetic fluctuations does not contain a quadratic term. The pairing interaction…
Under what conditions does a superconductor inherit topologically protected nodes from its parent normal state? In the context of Weyl semimetals with broken time-reversal symmetry, the pairing order parameter is classified by monopole harmonics and necessarily nodal [Li and Haldane, Phys. Rev. Lett., 120, 067003 (2018)]. Here, we show that a similar conclusion could also apply to 2D Dirac semimetals, although the conditions for the existence of nodes are more complex, depending on the pairing matrix structure in the valley and sublattice space. We analytically and numerically analyze the Bogo…
The recent discovery of AV$_3$Sb$_5$ (A=K,Rb,Cs) has uncovered an intriguing arena for exotic Fermi surface instabilities in a kagome metal. Among them, superconductivity is found in the vicinity of multiple van Hove singularities, exhibiting indications of unconventional pairing. We show that the sublattice interference mechanism is central to understanding the formation of superconductivity in a kagome metal. Starting from an appropriately chosen minimal tight-binding model with multiple with multiple van Hove singularities close to the Fermi level for AV$_3$Sb$_5$, we provide a random phase…
CeRh$_2$As$_2$, a non-symmorphic heavy fermion material, was recently reported to host a remarkable phase diagram with two superconducting phases. In this material, the two inequivalent Ce sites per unit cell, related by inversion symmetry, introduce a sublattice structure corresponding to an extra internal degree of freedom. Here we propose a classification of the possible superconducting states in CeRh$_2$As$_2$ from the two Ce-sites perspective. Based on the superconducting fitness analysis and the quasiclassical Eilenberger equations, we discuss two limits: Rashba spin-orbit coupling and i…
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 $d_{…
We study the inhomogeneity of the electronic pairing gap observed by STM near the surface of Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$ to be correlated with interstitial O defects. We treat the problem in a slave boson mean field theory of a disordered $t-t^{\prime}-J$ model, and identify three aspects of the O defects related to the inhomogeneity: (1)the superexchange interaction is locally enhanced in their vicinity, which enhances the local pairing gap and reduces the coherence peak; (2)they donate holes into CuO$_2$ plane, which reduces the spinon density of states of and hence the average gap a…
Recent local tunneling data indicate strong nanoscale inhomogeneity of superconducting gap in high temperature superconductors. Strong local nanoscale inhomogeneity in the bosonic scattering mode has also been observed in the same samples. We argue that these two inhomogeneities directly related to each other. To address local boson scattering effects, we develop a local strong coupling model of superconducting pairing in a coarse grained superconducting state. Each patch is characterized by local coupling to the bosonic mode as well as by local mode energy. We find that local gap value on eac…
We study the emergence of odd-frequency superconducting pairing in UPt$_3$. Starting from a tight-binding model accounting for the nonsymmorphic crystal symmetry of UPt$_3$ and assuming an order parameter in the $E_{2u}$ representation, we demonstrate that odd-frequency pairing arises very generally, as soon as inter-sublattice hopping or spin-orbit coupling is present. We also show that in the low temperature superconducting $B$ phase, the presence of a chiral order parameter together with spin-orbit coupling, leads to additional odd-frequency pair amplitudes not present in the $A$ or $C$ pha…
The influence of the atomic-scale inhomogeneities of the pairing interaction on the superconducting order parameter distribution and the LDOS is studied in the framework of mean-field BCS theory for two-dimensional lattice model. It is found that the ratio of the local low-temperature gap in differential conductance to the local temperature of vanishing the gap $2Δ_g/T_p$ can take large enough values compared to the homogeneous case. This ratio practically does not depend on the location in the sample and is independent on the concentration of local pair interaction perturbations in wide range…
Recent experiments suggest a multi-component pairing function in Sr2RuO4, which appears to be inconsistent with the absence of an apparent cusp in the transition temperature (Tc) as a function of the uniaxial strain. We show, however, that the theoretical cusp in Tc for a multi-component pairing can be easily smeared out by the spatial inhomogeneity of strain, and the experimental data can be reproduced qualitatively by a percolation model. This shed new light on multi-component pairings. We then perform a thorough group-theoretical classification of the pairing functions, taking the spin-orbi…
Parity is a fundamental quantum number to classify a state of matter. Materials rarely possess ground states with odd parity. We show that the superconducting state in iron-based superconductors is classified as an odd parity s-wave spin-singlet pairing state in a single trilayer FeAs/Se, the building block of the materials. In a low energy effective model constructed on the Fe square bipartite lattice, the superconducting order parameter in this state is a combination of a s-wave normal pairing between two sublattices and a s-wave $η$-pairing within the sublattices. Parity conservation was vi…
We show that a spontaneous magnetic moment may appear at the edge of a spin-triplet superconductor if the system allows for pairing in a subdominant channel. To unveil the microscopic mechanism behind such effect we combine numerical solution of the Bogoliubov-De Gennes equations for a tight-binding model with nearest-neighbor attraction, and the symmetry based Ginzburg-Landau approach. We find that a potential barrier modulating the electronic density near the edge of the system leads to a non-unitary superconducting state close to the boundary where spin-singlet pairing coexists with the dom…
Inhomogeneity is introduced through random local interactions (Ui) in an attractive Hubbard model on a square lattice and studied using mean-field Bogoliubov-de Gennes formalism. Superconductivity is found to get suppressed by the random Ui contrary to the results of a bimodal distribution of Ui. The proximity effect of superconductivity is found to be strong, all sites develop non-zero pairing amplitude. The gap in the density of states is always non-zero and does not vanish even for strong disorder. When two such superconductors are coupled via a channel, the effect of one on the other is ne…
Pairing symmetry plays a central role in the study of superconductivity. It is usually characterized by integer partial-waves, for example, $s$-, $p$-, $d$-waves. In this article, we investigate a new class of topological superconductivity whose gap functions possess a half-odd-integer monopole charge and, therefore, fractionalized half-odd-integer partial-wave symmetry in three dimensions. This exotic pairing occurs between Fermi surfaces of which Chern numbers are differed by an odd integer. The corresponding superconducting gap function is represented by monopole harmonics with half-odd-int…
Silicene has been synthesized recently, with experimental evidence showing possible superconductivity in the doped case. The noncoplanar low-buckled structure of this material inspires us to study the pairing symmetry of the doped system under a perpendicular external electric field. Our study reveals that the electric field induces an interesting quantum phase transition from the singlet chiral $d + id'$-wave superconducting phase to the triplet $f$-wave one. The emergence of the $f$-wave pairing results from the sublattice-symmetry-breaking caused by the electric field and the ferromagnetic-…
Magnetic Correlations and Pairing Tendencies of the Hybrid Stacking Nickelate Superlattice La$_7$Ni$_5$O$_{17}$ (La$_3$Ni$_2$O$_7$/La$_4$Ni$_3$O$_{10}$) under Pressure
We study a two-orbital $t$-$J_1$-$J_2$ model, originally developed to describe iron-based superconductors at low energies, in the presence of bond disorder (via next-nearest-neighbor $J_2$-bond dilution). By using the Bogoliubov--de Gennes approach, we self-consistently calculate the local pairing amplitudes and the corresponding density of states, which demonstrate a change of dominant pairing symmetry from $s_\pm$ wave to $d$ wave when increasing disorder strength as long as $J_1\lesssim J_2$. Moreover, the combined pairing interaction and strong bond disorder lead to the formation of $s_\pm…
Using selection rules imposed by the Pauli principle, we classify pairing correlations according to their symmetry properties with respect to spin, momentum, and energy. We observe that inhomogeneity always leads to mixing of even- and odd-energy pairing components. We investigate the superconducting pairing correlations present near interfaces between superconductors and ferromagnets, with focus on clean systems consisting of singlet superconductors and either weak or half-metallic ferromagnets. Spin-active scattering in the interface region induces all of the possible symmetry components. In…
We investigate the influence of inhomogeneity in the pairing coupling constant $U(\vec r)$ on dirty BCS superconductors, focusing on $T_c$, the order parameter $Δ(\vec r)$, and the energy gap $E_g(\vec r)$. Within mean-field theory, we find that when the length-scale of the inhomogeneity is comparable to, or larger than the coherence length, the ratio $2E_g/T_c$ is significantly reduced from that of a homogeneous superconductor, while in the opposite limit this ratio stays unmodified. In two dimensions, when strong phase fluctuations are included, the Kosterlitz-Thouless temperature $T_{KT}$ i…
The influence of the atomic-scale inhomogeneities of the pairing interaction strength on the superconducting order parameter and the conductance spectra measurable by STM is studied in the framework of weak-coupling BCS-like theory for two-dimensional lattice model. First of all, it is found that the inhomogeneity having the form of atomic-scale regions of enhanced pair interaction increases the ratio of the local low-temperature gap in differential conductance spectra to the local temperature of vanishing the gap $2Δ_g/T_p$. Even in the framework of mean-field treatment this ratio is shown to…
Spin-triplet superconductivity is known to be a rare quantum phenomenon. Here we show that nonsymmorphic crystalline symmetries can dramatically assist spin-triplet superconductivity in the presence of spin-orbit coupling. Even with a weak spin-orbit coupling, the spin-triplet pairing can be the leading pairing instability in a lattice with a nonsymmorphic symmetry. The underlining mechanism is the spin-sublattice-momentum lock on electronic bands that are protected by the nonsymmorphic symmetry. We use the nonsymmorphic space group P4/nmm to demonstrate these results and discuss related exper…
We study the emergence of topological superconductivity in a two-dimensional (2D) Weyl system, composed of stacked Su-Schrieffer-Heeger (SSH) chains. A previous analysis of the model showed that the addition of an attractive Hubbard interaction between spinful electrons leads to a superconducting state that has an intricate pairing structure, but is topologically trivial. Here we consider a pairing interaction that couples spinless fermions on opposite sublattices within the same unit cell. We observe that this physically motivated, momentum-independent pairing interaction induces a topologica…
We investigate theoretically magnon-mediated superconductivity in a heterostructure consisting of a normal metal and a two-sublattice antiferromagnetic insulator. The attractive electron-electron pairing interaction is caused by an interfacial exchange coupling with magnons residing in the antiferromagnet, resulting in p-wave, spin-triplet superconductivity in the normal metal. Our main finding is that one may significantly enhance the superconducting critical temperature by coupling the normal metal to only one of the two antiferromagnetic sublattices employing, for example, an uncompensated …
Upon thermal annealing at or above room temperature (RT) and high pressure $\it P$ $\sim$ 155 GPa, H$_3$S exhibits superconductivity at $\it T_C$ $\sim$ 200 K. Various theoretical frameworks with strong electron-phonon coupling and Coulomb repulsion have reproduced this record-level $\it T_C$. Of particular relevance is that observed H-D isotopic correlations among $\it T_C$, $\it P$, and annealed order indicate limitations on the H-D isotope effect, leaving open for consideration unconventional high-$\it T_C$ superconductivity with electronic-based enhancements. The present work examines Coul…
We consider a generic Hamiltonian that is suitable for describing a uniform BCS superfluid on a lattice with a two-point basis, and study its collective excitations at zero temperature. For this purpose, we first derive an effective-Gaussian action for the pairing fluctuations, and then extract the low-energy dispersion relations for the in-phase Goldstone and out-of-phase Leggett modes along with the corresponding amplitude (i.e., the so-called Higgs) ones. We find that while the Goldstone mode is gapless at zero momentum and propagating in general, the Leggett mode becomes undamped only with…
