In this Letter, we first formulate an effective theory, which generally captures long-range proximity effects of a surface system weakly coupled to an s-wave superconductor. The long-range proximity effects include both the emergent long-range pairing and hopping interactions in the surface system. We then model the Rashba spin-orbit-coupled nanowire in proximity with an s-wave superconductor by taking into account the emergent nonlocal effects in the weak-coupling limit. In this limit the induced superconducting pair potential is found much smaller than that of the host superconductor, which is in good agreement with recent experiments. Compared with the previously considered strong coupling limit with local proximity effects, the long-range interactions can significantly modify the topological phase diagram, and considerably lower the threshold magnetic field for the emergence of Majorana zero modes.
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 …
We study the behavior of Cooper pair amplitudes that emerge when a two-dimensional superconductor is coupled to two parallel nanowires, focusing on the conditions for realizing odd-frequency pair amplitudes in the absence of spin-orbit coupling or magnetism. In general, any finite tunneling between the superconductor and the two nanowires induces odd-frequency spin-singlet pair amplitudes in the substrate as well as a substantial odd-frequency interwire pairing, both of which vanish locally. Interestingly, in the regime of strong superconductor-nanowire tunneling, we find that the presence of …
We investigate pairing mechanism in multiband superconductors. To put our feet on firm ground, unbiased renormalization group analysis is carried out for iron-based superconductors. It is quite remarkable that, after integrating out quantum fluctuations, the renormalization-group flows agree exceedingly well with a mean-field Hamiltonian where interband pair hopping plays an essential role. Through interband pair hopping, electrons can overcome the repulsive interaction between them and form resonating Cooper pairs between different bands. Unlike the conventional superconductors, the pairing m…
Odd-frequency superconductivity, originally proposed by Berezinskii in 1974, is an exotic phase of matter in which Cooper pairing between electrons is entirely dynamical in nature. The pair potential is an odd function of frequency, leading to a vanishing static superconducting order parameter and exotic types of pairing seemingly inconsistent with Fermi statistics. Motivated by recent experimental progress in the realization of Majorana zero modes in semiconducting nanowires, we show that odd-frequency superconductivity generically appears in a spin-polarized nanowire coupled to Majorana zero…
We show that the topological Majorana modes in nanowires much longer than the superconducting coherence length are adiabatically connected with discrete zero-energy states generically occurring in short nanowires. We demonstrate that these zero-energy crossings can be tuned by an external magnetic field and are protected by the particle-hole symmetry. We study the evolution of the low-energy spectrum and the splitting oscillations as a function of magnetic field, wire length, and chemical potential, manifestly establishing that the low-energy physics of short wires is related to that occurring…
We study the effect of the correlated hopping term and the intersite Coulomb interaction term on principal features of the $d$-$wave$ superconducting (SC) state, in both the electron and hole doped regimes within the t-J-U model. In our analysis we use the approach based on the diagrammatic expansion of the Gutzwiller wave function (DE-GWF) which allows us to go beyond the renormalized mean field theory (RMFT). We show that the correlated hopping term enhances the pairing at the electron-doped side of the phase diagram. Moreover, the so-called non-BCS regime (which manifests itself by the nega…
We consider a system of two one-dimensional nanowires coupled via an $s$-wave superconducting strip, a geometry that is capable of supporting Kramers pairs of Majorana fermions. By performing an exact analytical diagonalization of a tunneling Hamiltonian describing the proximity effect (via a Bogoliubov transformation), we show that the excitation gap of the system varies periodically on the scale of the Fermi wavelength in the limit where the interwire separation is shorter than the superconducting coherence length. Comparing with the excitation gaps in similar geometries containing only dire…
The mean field Green function solution of the two-band singlet-hole Hubbard model for high-$T\sb{c}$ superconductivity in cuprates (Plakida, N.M. et al., Phys. Rev. B51, 16599 (1995), JETP 97, 331 (2003)) involves expressions of higher order correlation functions describing respectively the singlet hopping and the superconducting pairing. Rigorous derivation of their values is reported based on the finding that specific invariant classes of polynomial Green functions in terms of the Wannier overlap coefficients $ν\sb{ij}$ exist.
Using numerical diagonalization of a 4x4 cluster, we calculate on-site s, extended s and d pairing correlation functions (PCF) in an effective generalized Hubbard model for the cuprates, with nearest-neighbor correlated hopping and next nearest-neighbor hopping t'. The vertex contributions (VC) to the PCF are significantly enhanced, relative to the t-t'-U model. The behavior of the PCF and their VC, and signatures of anomalous flux quantization, indicate superconductivity in the d-wave channel for moderate doping and in the s-wave channel for high doping and small U.
Uniaxial strain affects pairing symmetry states in superconductors by changing the lattice symmetry, and by altering Fermi surface topology. Here, we present a systematic study of these effects within a one-band negative-U Hubbard model for s, p and d-wave pairing states. We consider a general 2D model that can be applied to superconductors under uniaxial strain, modelled via hopping anisotropy, on a square lattice. The results presented here model an in plane compression along the x-axis, which reduces the lattice symmetry from a tetragonal to an orthorhombic crystal space group. The effects …
We consider the Kitaev chain model with finite and infinite range in the hopping and pairing parameters, looking in particular at the appearance of Majorana zero energy modes and massive edge modes. We study the system both in the presence and in the absence of time reversal symmetry, by means of topological invariants and exact diagonalization, disclosing very rich phase diagrams. In particular, for extended hopping and pairing terms, we can get as many Majorana modes at each end of the chain as the neighbors involved in the couplings. Finally we generalize the transfer matrix approach useful…
Odd-frequency superconductivity is an exotic phase of matter in which Cooper pairing between electrons is entirely dynamical in nature. Majorana zero modes exhibit pure odd-frequency superconducting correlations due to their specific properties. Thus, by tunnel-coupling an array of Majorana zero modes to a spin-polarized wire, it is in principle possible to engineer a bulk one-dimensional odd-frequency spinless $s$-wave superconductor. We here point out that each tunnel coupling element, being dependent on a large number of material-specific parameters, is generically complex with sample varia…
We study the pairing properties of the two-dimensional $t$-$t'$-$t''$-$J$ model, where $t'$ and $t''$ are second and third neighbor hoppings, at a doping level $x\approx0.1$. Recent studies of the $t$-$t'$-$J$ model find strong pairing for $t'>0$, associated with electron doping, but an absence of pairing for $t'<0$ associated with hole doping. This is in contrast to the cuprates, where the highest transition temperatures appear for hole doping. Model parameterizations for the cuprates estimate a $t''$ comparable to $t'$, which, in principle, might fix this discrepancy. However, we find that i…
The recent experimental support for the presence of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase in the CeCoIn5 directed the attention towards the mechanisms responsible for this type of superconductivity. We investigate the FFLO state in a model where on--site/inter--site pairing coexists with repulsive pair hopping interaction. The latter interaction is interesting in that it leads to pairing with nonzero momentum of the Cooper pairs even in the absence of the external magnetic field (the so-called eta-pairing). It turns out that depending on the strength of the pair hopping interaction…
We study signatures of superconductivity in a 2--leg "checkerboard" Hubbard ladder model, defined as a one--dimensional (period 2) array of square plaquettes with an intra-plaquette hopping $t$ and inter-plaquette hopping $t'$, using the density matrix renormalization group method. The highest pairing scale (characterized by the spin gap or the pair binding energy, extrapolated to the thermodynamic limit) is found for doping levels close to half filling, $U\approx 6t$ and $t'/t \approx 0.6$. Other forms of modulated hopping parameters, with periods of either 1 or 3 lattice constants, are also …
We theoretically investigate the transport properties of a quasi one dimensional ferromagnet-superconductor junction where the superconductor consists of mixed singlet and triplet pairings. We show that the relative orientation of the stoner field ($\bf{\tilde{h}}$) in the ferromagnetic lead and the $\bf{d}$ vector of the superconductor acts like a on-off switch for the zero bias conductance of the device. In the regime, where triplet pairing amplitude dominates over the singlet counterpart (topological phase), a pair of Majorana zero modes appear at each end of the superconducting part of the…
It is difficult to unambiguously confirm the existence of Majorana zero modes (MZMs) due to the absence of smoking-gun signatures in charge transport measurements. Recent studies suggest that the spin degree of freedom of MZMs may provide an alternative detection method. We study the spin properties of the superconducting state in Majorana nanowires and the associated unconventional Josephson effect with realistic experimental parameters taken from [Zhang {\it et al.}, Nature 556, 74 (2018)]. For a superconducting thin film with in-plane polarized spin-triplet pairing, an out-of-plane electric…
First-Order Topological FFLO Transition and Superconducting Diode Sign Reversal in Altermagnetic Nanowires
School of Sciences, Great Bay University（大湾大学理学院）；Department of Physics, The University of Hong Kong（香港大学物理系）；Hefei National Laboratory（合肥国家实验室）；School of Emerging Technology, University of Science and Technology of China（中国科学技术大学新兴技术学院）
研究交变磁纳米线中拓扑 FFLO 态，发现与零净磁化 d 波交变磁体接近的自旋轨道耦合纳米线，通过一阶相变实现该态，伴有超导二极管效应，金兹堡 - 朗道理论解释机制，提供无场途径并确定输运指纹。
Determinant Quantum Monte Carlo (DQMC) is used to determine the pairing and magnetic response for a Hubbard model built up from four-site clusters -a two-dimensional square lattice consisting of elemental 2x2 plaquettes with hopping $t$ and on-site repulsion $U$ coupled by an inter-plaquette hopping $t' \leq t$. Superconductivity in this geometry has previously been studied by a variety of analytic and numeric methods, with differing conclusions concerning whether the pairing correlations and transition temperature are raised near half-filling by the inhomogeneous hopping or not. For $U/t=4$, …
We discuss pairing signatures in mesoscopic nanowires with variable attractive pairing interaction. Depending on wire length, density, and interaction strength, these systems realize a simultaneous bulk-to-mesoscopic and BCS-BEC crossover, which we describe in terms of the parity parameter that quantifies the odd-even energy difference and generalizes the bulk Cooper pair binding energy to mesoscopic systems. We show that the parity parameter can be extracted from recent measurements of conductance oscillations in SrTiO$_3$ nanowires by G. Cheng et al. [Nature 521, 196 (2015)], where it marks …
We consider time-reversal-symmetric two-channel semiconducting quantum wires proximity coupled to an s-wave superconductor. We analyze the requirements for a nontrivial topological phase and find that necessary conditions are 1) the determinant of the pairing matrix in channel space must be negative, 2) inversion symmetry must be broken, and 3) the two channels must have different spin-orbit couplings. The first condition can be implemented in semiconducting nanowire systems where interactions suppress intra-channel pairing, while the inversion symmetry can be broken by tuning the chemical pot…
A microscopic theory of superconductivity is formulated within an effective $p$-$d$ Hubbard model for a CuO2 plane. By applying the Mori-type projection technique, the Dyson equation is derived for the Green functions in terms of Hubbard operators. The antiferromagnetic exchange caused by interband hopping results in pairing of all carries in the conduction subband and high Tc proportional to the Fermi energy. Kinematic interaction in intraband hopping is responsible for the conventional spin-fluctuation pairing. Numerical solution of the gap equation proves the d-wave gap symmetry and defines…
We analyze influence of a dimerization on the topological phases of the Rashba nanowire proximitized to a superconducting substrate. We find that periodic alternations of the hopping integral and spin-orbit coupling can lead to band inversion, inducing transition to topologically nontrivial superconducting phase that hosts the Majorana zero-energy modes. This "dimerization-induced topological superconductivity" completely repels the topological phase of the uniform nanowire, whenever they happen to overlap. We provide an analytical justification for this puzzling behavior based on the symmetry…
For the newly discovered ironbased superconductors we propose a one band model for conduction band with first and secondnearestneighbor electron hoppings with two types of d wave and three types of s wave pairing symmetries for the superconductivity. We have calculated the Greens functions from which the temperature dependent superconducting (SC) gap and tunneling conductance are calculated and computed numerically. The evolution of the SC gap and the tunneling conductance spectra are investigated for different model parameters of the system.
In this article we review recent progress in the understanding of multiband superconductivity and its relationship to odd-frequency pairing. We begin our discussion by reviewing the emergence of odd-frequency pairing in a simple two-band model, providing a brief pedagogical overview of the formalism. We then examine several examples of multiband superconducting systems in each case describing, both, the origin of the band degree of freedom and the nature of the odd-frequency pairing. Throughout, we attempt to convey a unified picture of how odd-frequency pairing emerges in these materials and …
We study a double-nanowire setup proximity coupled to an $s$-wave superconductor and search for the bulk signatures of the topological phase transition that can be observed experimentally, for example, with an STM tip. Three bulk quantities, namely, the charge, the spin polarization, and the pairing amplitude of intrawire superconductivity are studied in this work. The spin polarization and the pairing amplitude flip sign as the system undergoes a phase transition from the trivial to the topological phase. In order to identify promising ways to observe bulk signatures of the phase transition i…
The novel idea that spin-orbit coupling (SOC) and $s$-wave pairing can lead to induced $p$-wave pairing at strong magnetic limit has stimulated widespread interests in the whole community for the searching of Majorana Fermions (MFs), a self-hermitian particle, in semiconductor-superconductor hybrid structures. However, this system has several inherent limitations that prohibit the realization and identification of MFs with the major advances of semiconductor nanotechnology. We show that these limitations can be resolved by replacing the $s$-wave superconductor with type-II Fulde-Ferrell (FF) s…
The bilayer Hubbard model with an intra-layer hopping $t$ and an inter-layer hopping $t_\perp$ provides an interesting testing ground for several aspects of what has been called unconventional superconductivity. One can study the type of pair structures which arise when there are multiple Fermi surfaces. One can also examine the pairing for a system in which the structure of the spin-fluctuation spectral weight can be changed. Using a dynamic cluster quantum Monte Carlo approximation, we find that near half-filling, if the splitting between the bonding and anti-bonding bands $t_\perp/t$ is sma…
We propose a microscopical theory of superconductivity in CuO$_2$ layer within the effective two-band Hubbard model in the strong correlation limit. By applying a projection technique for the matrix Green function in terms of the Hubbard operators, the Dyson equation is derived. It is proved that in the mean-field approximation d-wave superconducting pairing mediated by the conventional exchange interaction occurs. Allowing for the self-energy corrections due to kinematic interaction, a spin-fluctuation d-wave pairing is also obtained. $T\sb{c}$ dependence on the hole concentration and $\bf k$…
Both FeSe and cuprate superconductors are quasi 2D materials with high transition temperatures and local fermion pairs. Motivated by such systems, we investigate real space pairing of fermions in an anisotropic lattice model with intersite attraction, V, and strong local Coulomb repulsion, U, leading to a determination of the optimal conditions for superconductivity from Bose-Einstein condensation. Our aim is to gain insight as to why high temperature superconductors tend to be quasi 2D. We make both analytically and numerically exact solutions for two body local pairing applicable to intermed…
Among the different platforms to engineer Majorana fermions in one-dimensional topological superconductors, topological insulator nanowires remain a promising option. Threading an odd number of flux quanta through these wires induces an odd number of surface channels, which can then be gapped with proximity induced pairing. Because of the flux and depending on energetics, the phase of this surface pairing may or may not wind around the wire in the form of a vortex. Here we show that for wires with discrete rotational symmetry, this vortex is necessary to produce a fully gapped topological supe…
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…
We study the influence of the inverse proximity effect on the superconductivity nucleation in hybrid structures consisting of the semiconducting nanowires placed in contact with a thin superconducting film and discuss the resulting restrictions on the operation of Majorana-based devices. A strong paramagnetic effect for electrons entering the semiconductor together with spin-orbit coupling and van Hove singularities in the electronic density of states in the wire are responsible for the suppression of superconducting correlations in the low field domain and for the reentrant superconductivity …
A two-orbital model for Fe-pnictide superconductors is investigated using computational techniques on two-dimensional square clusters. The hopping amplitudes are derived from orbital overlap integrals, or by band structure fits, and the spin frustrating effect of the plaquette-diagonal Fe-Fe hopping is remarked. A spin 'striped' state is stable in a broad range of couplings in the undoped regime, in agreement with neutron scattering. Adding two electrons to the undoped ground state of a small cluster, the dominant pairing operators are found. Depending on parameters, two pairing operators were…
Exotic superconductors, such as high T$_C$, topological, and heavy-fermion superconductors, require phase sensitive measurements to determine the underlying pairing. Here we investigate the proximity-induced superconductivity in nanowires of SnTe, where an $s\pm is^{\prime}$ superconducting state is produced that lacks the time-reversal and valley-exchange symmetry of the parent SnTe. This effect, in conjunction with a ferroelectric distortion of the lattice at low temperatures, results in a marked alteration of the properties of Josephson junctions fabricated using SnTe nanowires. This work e…
In this paper, we investigate the relation between odd-frequency pairing and proximity effect in non-uniform Kitaev chain systems with a particular interest in the topological critical point. First, we correlate the odd-frequency pairing and Majorana fermion in a semi-infinite Kitaev chain, where we find that the spatial dependence of the odd-frequency pair amplitude coincides with that of the local density of states at low frequencies. Second, we demonstrate that, contrary to the standard view, the odd-frequency pair amplitude spreads into the bulk of a semi-infinite Kitaev chain at the topol…
We discuss possible pairing symmetries in the hexagonal pnictide superconductor SrPtAs. The local lack of inversion symmetry of the two distinct conducting layers in the unit cell results in a special spin-orbit coupling with a staggered structure. We classify the pairing symmetry by the global crystal point group D_3d, and suggest some candidates for the stable state using a tight-binding model with an in-plane, density-density type pairing interaction. We may have some unconventional states like s+f-wave and a mixture of chiral d-wave and chiral p-wave. The spin orbit coupling is larger than…
We address some open questions regarding the phase diagram of the one-dimensional Hubbard model with asymmetric hopping coefficients and balanced species. In the attractive regime we present a numerical study of the passage from on-site pairing dominant correlations at small asymmetries to charge-density waves in the region with markedly different hopping coefficients. In the repulsive regime we exploit two analytical treatments in the strong- and weak-coupling regimes in order to locate the onset of phase separation at small and large asymmetries respectively.
We report an experimental study of proximity effect-induced superconductivity in crystalline Cu and Co nanowires and a nanogranular Co nanowire structure in contact with a superconducting W floating electrode which we call inducer. The nanowires were grown by electrochemical deposition in heavy-ion-track etched polycarbonate templates. The nanogranular Co structure was fabricated by focused electron beam induced deposition (FEBID), while the amorphous W inducer was obtained by focused ion beam induced deposition (FIBID). For electrical resistance measurements up to three pairs of Pt voltage le…
We study the conductivity of granular superconductors in the weak coupling insulating regime. We show that it is governed by the hopping of either electrons or Cooper pairs depending on the relation between the superconducting gap and the charging energy of a single granule. Local superconducting pairing plays an important role in both cases. In particular, in the case of the transport via electron hopping the superconducting gap suppresses the inelastic cotunneling processes. We determine transport characteristics of an array in different regimes and construct the transport phase diagram.
We determine the optimal conditions to achieve topological superconducting phases having spin-singlet pairing for a planar nanowire with finite lateral width in the presence of an in-plane external magnetic field. We employ a microscopic description that is based on a three-band electronic model including both the atomic spin-orbit coupling and the inversion asymmetric potential at the interface between oxide band-gap insulators. We consider amplitudes of the pairing gap, spin-orbit interactions and electronic parameters that are directly applicable to nanowires of LaAlO$_3$-SrTiO$_3$. The lat…
We have studied the enhancement of the superconducting transition temperature, Tc, in a t-J-U model of electrons moving on a square lattice in which anisotropic electronic hopping is introduced. The inclusion of such hopping mimics, in a approximate fashion, a potentially important characteristic of materials possessing stripelike charge and spin correlations. For this model we have calculated Tc for singlet pairing using the non self-consistent Thouless criterion, and find a dramatic enhancement of Tc induced by hopping anisotropies. Further, the maximum increase in Tc is obtained when the sy…
A one-dimensional spin-orbit coupled nanowire with proximity-induced pairing from a nearby s-wave superconductor may be in a topological nontrivial state, in which it has a zero energy Majorana bound state at each end. We find that the topological trivial phase may have fermionic end states with an exponentially small energy, if the confinement potential at the wire's ends is smooth. The possible existence of such near-zero energy levels implies that the mere observation of a zero-bias peak in the tunneling conductance is not an exclusive signature of a topological superconducting phase even i…
We introduce a Hubbard model on a particular class of geometries, and consider the effect of doping the highly spin-degenerate Mott-insulating state with a microscopic number of holes in the extreme strong-coupling limit. The geometry is quite general, with pairs of atomic sites at each superlattice vertex, and a highly frustrated inter-atomic connectivity: the one dimensional realization is a chain of edge-sharing tetrahedra. The sole model parameter is the ratio of intra-pair to inter-pair hopping matrix elements. If the intra-pair hopping is negligible then introducing a microscopic number …
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.
For superconducting nanowire with the pairing of extended s-type symmetry, Rashba spin-orbit interaction in a magnetic field, the influence of strong intersite charge correlations on single-particle Majorana excitations is analyzed. This problem is investigated on the basis of the density matrix renormalization group numerical method. It is shown that with an increase in the repulsion intensity of electrons located at the neighboring sites, two subbands emerge in the lower Hubbard band of the open system. Based on calculations of the Majorana polarization and degeneracy of the entanglement spe…
We propose a concrete model which exhibits ferromagnetism and electron-pair condensation simultaneously. The model is defined on two chains and consists of the electron hopping term, the on-site Coulomb repulsion, and a ferromagnetic interaction which describes ferromagnetic coupling between two electrons, one on a bond in a chain and the other on a site in the other chain. It is rigorously shown that the model has fullypolarized ferromagnetic pairing ground states. The higher dimensional version of the model is also presented.
We study the phase diagram and edge states of a two-dimensional p-wave superconductor with long-range hopping and pairing amplitudes. New topological phases and quasiparticles different from the usual short-range model are obtained. When both hopping and pairing terms decay with the same exponent, one of the topological chiral phases with propagating Majorana edge states gets significantly enhanced by long-range couplings. On the other hand, when the long-range pairing amplitude decays more slowly than the hopping, we discover new topological phases where propagating Majorana fermions at each …
In order to explore why the multi-layered cuprates have such high Tc's, we have examined various inter-layer processes. Since the inter-layer one-electron hopping has little effects on the band structure, we turn to the inter-layer pair hopping. The superconductivity in a double-layer Hubbard model with and without the inter-layer pair hopping, as studied by solving the Eliashberg equation with the fluctuation exchange approximation, reveals that the inter-layer pair hopping acts to increase the pairing interaction and the self-energy simultaneously, but that the former effect supersedes the l…
