We show that the phase information contained in coherent phonon oscillations generated by a laser pulse in a multi- orbital superconductor can be used as an experimental tool to probe super- conductivity and pairing symmetries. The phase difference between the normal and superconducting states is proportional to the superconducting order parameter just below the superconducting transition temperature, Tc. It also exhibits different behaviors for superconducting states with different pairing symmetries. In particular, if there is an orbital-dependent internal sign change state, the phase difference can have a discontinuous jump below Tc.
The discovery of high-$T_c$ superconductivity in the pnictides, materials with a Fermi surface determined by several bands, highlights the need to understand how superconductivity arises in multiband systems. In this effort, using symmetry considerations and mean-field approximations, we discuss how strong hybridization among orbitals may lead to both intra and interband pairing, and we present calculations of the spectral functions to guide the experimental search for this kind of state.
Pairing-channel reconstruction by Ising spin-orbit coupling in a multiorbital triangular-lattice model
Anhui Provincial Key Laboratory of Low-Energy Quantum Materials and Devices, High Magnetic Field Laboratory, HFIPS, Chinese Academy of Sciences（中国科学院合肥物质科学研究院强磁场科学中心安徽省低能量子材料与器件重点实验室）；Science Island Branch of Graduate School, University of Science and Technology of China（中国科学技术大学研究生院科学岛分院）
We theoretically study the superconductivity in multiorbital superconductors based on a three-orbital tight-banding model. With appropriate values of the nearest-neighbour exchange $J_{1}^{αβ}$ and the next-nearest-neighbour exchange $J_{2}^{αβ}$, we find a two-dome structure in the $T_{c}-n$ phase diagram: one dome in the doping range $n<3.9$ where the superconducting (SC) state is mainly $s_{x^{2} y^{2}}$ component contributed by inter-orbital pairing, the other dome in the doping range $3.9<n<4.46$ where the SC state is mainly $s_{x^{2} y^{2}}+s_{x^{2}+y^{2}}$ components contributed by intr…
Hund's pairing refers to Cooper pairing generated by onsite interactions that become attractive due to large Hund's exchange $J$. This is possible in multiorbital systems even when all local bare interactions are repulsive, since attraction in specific channels are given by certain linear combinations of interaction parameters. On the other hand, pairing processes such as the exchange of spin fluctuations, are also present. We compare these two mechanisms on an equal footing using electronic bands appropriate for different classes of multiorbital systems over a wide range of interaction parame…
Recent experiments in multiband Fe-based and heavy-fermion superconductors have challenged the long-held dichotomy between simple $s$- and $d$-wave spin-singlet pairing states. Here, we advance several time-reversal-invariant irreducible pairings that go beyond the standard singlet functions through a matrix structure in the band/orbital space, and elucidate their naturalness in multiband systems. We consider the $sτ_{3}$ multiorbital superconducting state for Fe-chalcogenide superconductors. This state, corresponding to a $d+d$ intra- and inter-band pairing, is shown to contrast with the more…
The heavy-fermion system $\mathrm{UTe_2}$ is a candidate for spin-triplet superconductivity, which is of considerable interest to quantum engineering. Among the outstanding issues is the nature of the pairing state. A recent surprising discovery is the observation of a resonance in the spin excitation spectrum at an antiferromagnetic wavevector [C. Duan {\it et al.}, Nature \textbf{600}, 636 (2021)], which stands in apparent contrast to the ferromagnetic nature of the interactions expected in this system. We show how the puzzle can be resolved by a multiorbital spin-triplet pairing constructed…
Weak-coupling approaches to the pairing problem in the iron pnictide superconductors have predicted a wide variety of superconducting ground states. We argue here that this is due both to the inadequacy of certain approximations to the effective low-energy band structure, and to the natural near-degeneracy of different pairing channels in superconductors with many distinct Fermi surface sheets. In particular, we review attempts to construct two-orbital effective band models, the argument for their fundamental inconsistency with the symmetry of these materials, and the comparison of the dynamic…
For the recently discovered cuprate superconductor $\mathrm{Ba_{2}CuO_{3+δ}}$, we propose a lattice structure which resembles the model considered by Lieb to represent the vastly oxygen-deficient material. We first investigate the stability of the Lieb-lattice structure, and then construct a multiorbital Hubbard model based on first-principles calculation. By applying the fluctuation-exchange approximation to the model and solving the linearized Eliashberg equation, we show that $s$-wave and $d$-wave pairings closely compete with each other, and, more interestingly, that the intra-orbital and …
We develop an efficient approach for computing two-particle response functions and interaction vertices for multiorbital strongly correlated systems based on fluctuation around rotationally-invariant slave-boson saddle-point. The method is applied to the degenerate three-orbital Hubbard-Kanamori model for investigating the origin of the s-wave orbital antisymmetric spin-triplet superconductivity in the Hund's metal regime, previously found in the dynamical mean-field theory studies. By computing the pairing interaction considering the particle-particle and the particle-hole scattering channels…
In the framework of a multiorbital Hubbard model description of superconductivity, a matrix formulation of the superconducting pairing interaction that has been widely used is designed to treat spin, charge and orbital fluctuations within a random phase approximation (RPA). In terms of Feynman diagrams, this takes into account particle-hole ladder and bubble contributions as expected. It turns out, however, that this matrix formulation also generates additional terms which have the diagrammatic structure of vertex corrections. Here we examine these terms and discuss the relationship between th…
We introduce a variational state for one-dimensional two-orbital Hubbard models that intuitively explains the recent computational discovery of pairing in these systems when hole doped. Our Ansatz is an optimized linear superposition of Affleck-Kennedy-Lieb-Tasaki valence bond states, rendering the combination a valence bond liquid dubbed Orbital Resonant Valence Bond. We show that the undoped (one electron/orbital) quantum state of two sites coupled into a global spin singlet is exactly written employing only spin-1/2 singlets linking orbitals at nearest-neighbor sites. Generalizing to longer…
The results of neutron scattering and angle-resolved photoemission experiments for the Fe-pnictide parent compounds, and their metallic nature, are shown to impose severe constraints on the range of values that can be considered "realistic" for the intraorbital Hubbard repulsion U and Hund coupling J in multiorbital Hubbard models treated in the mean-field approximation. Phase diagrams for three- and five-orbital models are here provided, and the physically realistic regime of couplings is highlighted, to guide future theoretical work into the proper region of parameters of Hubbard models. In …
We study the orbital-dependent superconducting pairing in a five-orbital t-J1-J2 model for iron pnictides. Depending on the orbital selectivity of electron correlations and the orbital characters along the Fermi surface, the superconducting gap in an A_{1g} pairing state may exhibit anisotropy. This anisotropy varies with the degree of J1-J2 magnetic frustration. We have also calculated the dynamical spin susceptibility in the superconducting state. The frequency dependence of the susceptibility at the antiferromagnetic wavevector (π,0) shows a resonance, whose width is enhanced by the orbital…
We introduce a two-orbital Hamiltonian on a square lattice that contains on-site attractive interactions involving the two $e_g$ orbitals. Via a canonical mean-field procedure similar to the one applied to the well-known negative-$U$ Hubbard model, it is shown that the new model develops $d$-wave ($B_{1g}$) superconductivity with nodes along the diagonal directions of the square Brillouin zone. This result is also supported by exact diagonalization of the model in a small cluster. The expectation is that this relatively simple attractive model could be used to address the properties of multior…
We develop a minimal multiorbital tight-binding model with realistic hopping parameters. The model breaks the symmetry of the tetragonal point group by lowering it from $C_4$ to $D_{2d}$, which accurately describes the Fermi surface evolution of the electron-doped BaFe$_{2-x}$Co$_x$As$_2$ and hole-doped Ba$_{1-y}$K$_y$Fe$_2$As$_2$ compounds. An investigation of the phase diagram with a mean-field $t$-$U$-$V$ Bogoliubov-de Gennes Hamiltonian results in agreement with the experimentally observed electron- and hole-doped phase diagram with only one set of $t$, $U$ and $V$ parameters. Additionally…
Starting from a two-orbital model and based on the random phase approximation, spin excitations in the superconducting state of the newly discovered BiS$_2$ superconductors with three possible pairing symmetries are studied theoretically. We show that spin response is uniquely determined by the pairing symmetry. Possible spin resonance excitations might occur for the d-wave symmetry at an incommensurate momentum about $(0.7π,0.7π)$. For the p-wave symmetry the transverse spin excitation near $(0,0)$ is enhanced. For the s-wave pairing symmetry there is no spin resonance signature. These distin…
In unconventional superconductors, understanding the form of the pairing interaction is the primary goal. In this regard, Raman spectroscopy is a very useful tool, as it identifies the ground state and also the subleading pairing channels by probing collective modes. Here we propose a general theory for multiband Raman response and identify new features in the spectrum that can provide a robust test for a pairing theory. We identify multiple Bardasis-Schrieffer type collective modes and connect the weights of these modes to the sub-leading gap structures within a microscopic pairing theory. Th…
We study the relation between the spin fluctuation and superconductivity in an heavily hole doped end material KFe$_2$As$_2$. We construct a five orbital model by approximately unfolding the Brillouin zone of the three dimensional ten orbital model obtained from first principles calculation. By applying the random phase approximation, we obtain the spin susceptibility and solve the linearized Eliashberg equation. The incommensurate spin fluctuation observed experimentally is understood as originating from interband interactions, where the multiorbital nature of the band structure results in an…
Even-parity interorbital spin-triplet pairing emerges as an intriguing candidate in multiorbital superconductors with significant Hund's and spin-orbit coupling. Within such a state, the pairing is dominated by the intraband pseudospin-singlet component via the spin-orbit coupling, distinguishing it from a pure spin triplet and motivating the name, shadowed triplet. With atomic spin-orbit coupling, it was shown that this pairing exhibits an anisotropic $s$-wave character, while higher angular momentum pairing symmetries such as $d$- or $g$-wave have been proposed based on phenomenological anal…
We present a general theory of multiorbital spin waves in magnetically ordered metallic systems. Motivated by the itinerant magnetism of iron-based superconductors, we compare the magnetic excitations for two different scenarios: when the magnetic order either sets in on the on-site orbital level; or when it appears as an electron-hole pairing between different bands of electron and hole character. As an example we treat the two-orbital model for iron-based superconductors. For small magnetic moments the spin excitations look similar in both scenarios. Going to larger interactions and larger m…
Both families of high $T_c$ superconductors, iron pnictides and cuprates, exhibit material dependence of superconductivity. Here, we study its origin within the spin fluctuation pairing theory based on multiorbital models that take into account realistic band structures. For pnictides, we show that the presence and absence of Fermi surface pockets is sensitive to the pnictogen height measured from the iron plane due to the multiorbital nature of the system, which is reflected to the nodeless/nodal form of the superconducting gap and $T_c$. Surprisingly, even for the cuprates, which is conventi…
In various multiorbital systems, the emergence of the orbital fluctuations and its role on the pairing mechanism attract increasing attention. To archive deep understanding on these issues, we perform the functional-renormalization-group (fRG) study for the two-orbital Hubbard model. The vertex corrections for the electron-boson coupling ($U$-VC), which are dropped in the Migdal-Eliashberg gap equation, are obtained by solving the RG equation. We reveal that the dressed electron-boson coupling for the charge-channel, $U_{eff}^c$, becomes much larger than the bare Coulomb interaction, $U^0$, du…
Bloch electrons in multiorbital systems carry nontrivial quantum geometric information characteristic of their orbital composition as a function of their wave vector. When such electrons form Cooper pairs, the resultant superconducting state naturally inherits aspects of the quantum geometry. In this paper, we study how this geometric character is revealed in the intrinsic optical response of the superconducting state. In particular, due to the superconducting gap opening, interband optical transitions involving states around the Fermi level are forbidden. This generally leads to an anomalous …
We investigate superconducting pairing driven by electron-electron interactions in a theoretical model for monolayer MoS$_2$ with the temperature-flow functional renormalization group(fRG). At low doping, the dominant instability is toward odd-parity pairing with $f$-wave Mo-nearest-neighbor structure. We compute the fRG phase diagram versus electron doping below the van Hove filling of the conduction band. In the superconducting regime, the critical temperature grows with doping, comparable to the experiments. Near van Hove filling the system favors a ferromagnetic state. We demonstrate that …
Probing Sign-Changing Order Parameters via Impurity States in unconventional superconductors: Implications for La$_3$Ni$_2$O$_7$ Superconductors with interlayer pairing
The topological properties of a multiband topological superconductor in two dimensions are studied, when the latter is obtained by introducing electron pairing in an otherwise topological insulator. The type of pairing, doping and Fermi surface topology play an essential role. Considering the Andreev reflection problem, we use a previously developed quantum waveguide theory for multiorbital systems and find that, when the Fermi surface has several pockets, this theory retrieves the correct number of Majorana fermion states as predicted by the topological index. By varying band structure parame…
The nature of the nuclear pairing condensate is an active topic of investigation, especially as regards its neutron-proton versus identical-particle character, which manifests as the difference between spin-singlet and spin-triplet pairing. In this work, we probe the recently proposed mixed-spin pairing condensates, using a phenomenological Hamiltonian and Hartree-Fock-Bogoliubov theory along with the gradient method. In addition to improving the solution of the many-body problem, we have calculated a series of physical quantities and examined the robustness of the mixed-spin pairing state as …
We study the spin dynamics in a p-wave superconductor at the nesting vector associated with $α$ and $β$ bands in Sr$_2$RuO$_4$. We find a collective mode at the nesting vector in the superconducting phase identified as the odd-parity pairing state which breaks time reversal symmetry. This mode in the spin channel only exists in the p-wave superconductor, not in s- or d-wave superconductors. We propose that probing this mode would clarify the pairing symmetry in Sr$_2$RuO$_4$. The possibility of second superconducting phase transition is also discussed.
We study stimulated scattering of polarized light in a two-component Fermi gas of atoms at zero temperature. Within the framework of Nambu-Gorkov formalism, we calculate the response function of superfluid gas taking into account the final state interactions. The dynamic structure factor deduced from the response function provides information about the pairing gap and the momentum distributions of atoms. Model calculations using local density approximation indicates that the pairing gap of trapped Fermi gas may be detectable by Bragg spectroscopy due to stimulated scattering.
Considering model Hamiltonians that respect the symmetry properties of the pnictides, it is argued that pairing interactions that couple electrons at different orbitals with an orbital-dependent pairing strength inevitably lead to interband pairing matrix elements, at least in some regions of the Brillouin zone. Such interband pairing has not been considered of relevance in multiorbital systems in previous investigations. It is also observed that if, instead, a purely intraband pairing interaction is postulated, this requires that the pairing operator has the form $Δ^+(k)=f(k) \sum_α d^+_{k,α,…
Recent experiments on Zn-doped 122-type iron pnictides, Ba(Fe$_{1-x-y}$Co$_y$Zn$_x$)$_2$As$_2$, are challenging our understanding of electron doping the 122s and the interplay between doping and impurity scattering. To resolve this enigma, we investigate the disorder effects of nonmagnetic Zn impurities in the strong (unitary) scattering limit on various properties of the system in the $s_{\pm}$-wave superconducting pairing state. The lattice Bogoliubov-de Gennes equation (BdG) is solved self-consistently based on a minimal two-orbital model with an extended range of impurity concentrations. W…
The thermoelectric response of SND configuration is considered within the generalized Ginzburg-Landau theory for a homogeneous admixture of s-wave and d-wave superconductors. The resulting thermopower (TP) is found to strongly depend on the relative phase θ=ϕ_s-ϕ_d between the two superconductors. Two independent mechanisms are shown to contribute to the TP. One, based on the charge imbalance between the quasiparticles and Cooper pairs (described by the corresponding chemical potentials) due to the normal metal insert, results in a pronounced maximum of the peak near θ=π/2 (the so-called s+id …
The interplay between electronic orders and superconductivity is central to the physics of unconventional superconductors, and is particularly pronounced in the iron-based superconductors. Motivated by recent experiments on FeSe, we study the superconducting pairing in its nematic phase in a multiorbital model with frustrated spin-exchange interactions. The electron correlations in the presence of the nematic order give rise to an enhanced orbital selectivity in the superconducting pairing amplitudes. This orbital-selective pairing produces a large gap anisotropy on the Fermi surface. Our resu…
Correlation-driven enhancement of pairing in a nematic Hund's metal
Budapest University of Technology and Economics（布达佩斯技术与经济大学）；HUN-REN-BME-BCE Quantum Technology Research Group（匈牙利研究与创新网络-布达佩斯技术与经济大学-布达佩斯理工大学量子技术研究组）；Istituto dei Sistemi Complessi (ISC-CNR)（复杂系统研究所）
The quasiparticle interference (QPI) in Sr$_{2}$RuO$_{4}$ is theoretically studied based on two different pairing models in order to propose an experimental method to test them. For a recently proposed two-dimensional model with pairing primarily from the $γ$ band, we found clear QPI peaks evolving with energy and their locations can be determined from the tips of the constant-energy contour (CEC). On the other hand, for a former quasi-one-dimensional model with pairing on the $α$ and $β$ bands, the QPI spectra are almost dispersionless and may involve off-shell contributions to the scattering…
We propose here an experiment aimed to determine whether there are superconducting pairing fluctuations in the pseudogap regime of the high-$T_c$ materials. In the experimental setup, two samples above $T_c$ are brought into contact at a single point and the differential AC conductivity in the presence of a constant applied bias voltage between the samples, $V$, should be measured. We argue the the pairing fluctuations will produce randomly fluctuating Josephson current with zero mean, however the current-current correlator will have a characteristic frequency given by Josephson frequency $ω_J…
To understand the role that degeneracy, hybridization, and nesting play in the magnetic and pairing properties of multiorbital Hubbard models we here study numerically two types of two- orbital models, both with hole-like and electron-like Fermi surfaces (FS's) that are related by nesting vectors (π, 0) and (0, π). In one case the bands that determine the FS's arise from strongly hybridized degenerate dxz and dyz orbitals, while in the other the two bands are determined by non-degenerate and non-hybridized s-like orbitals. Using a variety of techniques, in the weak coupling regime it is shown …
A crucial step in revealing the nature of unconventional superconductivity is to investigate the symmetry of the superconducting order parameter. Scanning tunneling spectroscopy has proven a powerful technique to probe this symmetry by measuring the quasiparticle interference (QPI) which sensitively depends on the superconducting pairing mechanism. A particularly well suited material to apply this technique is the stoichiometric superconductor LiFeAs as it features clean, charge neutral cleaved surfaces without surface states and a relatively high Tc~18K. Our data reveal that in LiFeAs the qua…
The possible superconducting states of strontium ruthenate (Sr$_2$RuO$_4$) are organized into irreducible representations of the point group $D_{4h}$, with a special emphasis on nodes occurring within the superconducting gap. Our analysis covers the cases with and without spin-orbit coupling and takes into account the possibility of inter-orbital pairing within a three-band, tight-binding description of Sr$_2$RuO$_4$. No dynamical treatment if performed: we are confining ourselves to a group-theoretical analysis. The case of uniaxial deformations, under which the point group symmetry is reduce…
We show, that in contrast to the free electron model (standard BCS model), a particular gap in the spectrum of multiband superconductors opens at some distance from the Fermi energy, if conduction band is composed of hybridized atomic orbitals of different symmetries. This gap has composite superconducting-hybridization origin, because it exists only if both the superconductivity and the hybridization between different kinds of orbitals are present. So for many classes of superconductors with multiorbital structure such spectrum changes should take place. These particular changes in the spectr…
An important issue in the study of the iron-arsenic based superconductors is the symmetry of the superconducting gap, a problem complicated by multiple gaps on different Fermi surface sheets. Electronic Raman scattering is a flexible bulk probe which allows one in principle to determine gap magnitudes and test for gap nodes in different regions of the Brillouin zone by employing different photon polarization states. Here we calculate the clean Raman intensity for $A_{1g}$, $B_{1g}$ and $B_{2g}$ polarizations, and discuss the peak structures and low-energy power laws which might be expected for…
We perform the numerical equivalent of a phase sensitive experiment on doped $t-J$ ladders. We apply proximity effect fields with different complex phases at both ends of an open system and we study the transport of Cooper pairs. Measuring the response of the system and the induced Josephson current, Density Matrix Renormalization Group calculations show how, depending on the doping fraction, the rung-leg parity of the pair field changes from minus to plus as the density of holes is increased. We also study the pair charge stiffness, and we observe a supression of the superconductivity in the …
High-Tc pairing mechanism absent of hole-pockets in heavily electron-doped FeSe is one of the key unsolved problems in Fe-based superconductors. Here, this problem is attacked by focusing on the higher-order many-body effects neglected in conventional Migdal-Eliashberg formalism. We uncover two significant many-body effects for high-Tc superconductivity: (i) Due to the "vertex correction", the dressed multiorbital Coulomb interaction acquires prominent orbital dependence for low-energy electrons. The dressed Coulomb interaction not only induces the orbital fluctuations, but also magnifies the …
Using the Density Matrix Renormalization Group and two-leg ladders, we investigate an electronic two-orbital Hubbard model including plaquette diagonal hopping amplitudes. Our goal is to search for regimes where charges added to the undoped state form pairs, presumably a precursor of a superconducting state.For the electronic density $ρ=2$, i.e. the undoped limit, our investigations show a robust $(π,0)$ antiferromagnetic ground state, as in previous investigations. Doping away from $ρ=2$ and for large values of the Hund coupling $J$, a ferromagnetic region is found to be stable. Moreover, whe…
