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 multiorbital $d$-wave superconductors in the same manner that the negative-$U$ Hubbard model is widely applied to the study of the properties of $s$-wave single-orbital superconductors. In particular, we show that by splitting the $e_g$ orbitals and working at three-quarters filling, such that the $x^2-y^2$ orbital dominates at the Fermi level but the $3z^2-r^2$ orbital contribution is nonzero, the $d$-wave pairing state found here phenomenologically reproduces several properties of the superconducting state of the high $T_c$ cuprates.
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…
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 differ…
The present work describes the theoretical analysis of isotope effect coefficient as a function of transition temperature in two orbital per site model Hamiltonian in iron based superconducting system. The expression of isotope effect coefficient has been computed numerically and self-consistently by employing Green's function technique within the BCS-mean-field approximation. It is observed that the isotope effect coefficient increases with the increase of the hybridization while with the increase in Coulomb interaction it starts decreasing. On increasing the carrier density per site in two o…
The tetragonal compound Sr2RuO4 exhibits a chiral p-wave superconducting (SC) state of its three t2g-type conduction bands. The characteristics of unconventional gap structure are known from experiment, in particular field-angle resolved specific heat measurements and from microscopic theories. A rotated extremal structure on the main active SC band with respect to the nodal gaps on the passive bands was concluded. We propose that this gap structure can be further specified by applying the STM- quasiparticle interference (QPI) method. We calculate the QPI spectrum within a three band and chira…
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…
Unconventional superconductors represent one of the most intriguing quantum states of matter. In particular, multiorbital systems have the potential to host exotic non-unitary superconducting states. While the microscopic origin of non-unitarity is not yet fully solved, competing interactions are suggested to play a crucial role in stabilizing such states. The interplay between charge order and superconductivity has been a recurring theme in unconventionally superconducting systems, ranging from cuprate-based superconductors to dichalcogenide systems and even to twisted van der Waals materials…
We apply a Hartree-Fock approximation to a two-orbital model proposed for Fe pnictide superconductors. It is found that the antiferromagnetic (AFM) order with the ordering vector Q=(pi,0) is realized. The AFM order appears simultaneously with ferro-orbital order, the latter leads to a secondary lattice distortion. We also investigate the influence of doping on the AFM order. The size of the AFM moment changes continuously for lightly doped cases, but when the amount of doped carriers exceeds a certain value the AFM state is suddenly destroyed. We also show that Fermi surfaces remain and change…
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 …
The origin of uniaxial and hydrostatic pressure effects on $T_c$ in the single-layered cuprate superconductors is theoretically explored. A two-orbital model, derived from first principles and analyzed with the fluctuation exchange approximation gives axial-dependent pressure coefficients, $\partial T_c/\partial P_a>0$, $\partial T_c/\partial P_c<0$, with a hydrostatic response $\partial T_c/\partial P>0$ for both La214 and Hg1201 cuprates, in qualitative agreement with experiments. Physically, this is shown to come from a unified picture in which higher $T_c$ is achieved with an "orbital dist…
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…
There is increasing recognition that the multiorbital nature of the 3d electrons is important to the proper description of the electronic states in the normal state of the iron-based superconductors. Earlier studies of the pertinent multiorbital Hubbard models identified an orbital-selective Mott phase, which anchors the orbital-selective behavior seen in the overall phase diagram. An important characteristics of the models is that the orbitals are kinetically coupled -- i.e. hybridized -- to each other, which makes the orbital-selective Mott phase especially nontrivial. A U(1) slave-spin meth…
The role of electronic correlations in Condensed Matter is at the heart of various important systems, like magnetic materials, superconductors, topological materials, optical lattices, etc. Electronic correlations are those which change the motion of individual electrons when considering the interaction with other electrons in the material. Among the available systems to study electronic correlation effects, in this thesis I focus on unconventional superconductors, specifically in high-$T_c$ iron-based superconductors, and on two-dimensional materials, like the recent magic-angle twisted bilay…
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…
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…
Multiorbital character of the density wave in trilayer nickelate superconductors
Max-Planck-Institute for Solid State Research（马克斯·普朗克固体研究所）；Facultad de Ciencias Exactas, Ingeniería y Agrimensura and Instituto de Física Rosario (UNR-CONICET)（罗萨里奥大学精确科学、工程与测量学院及罗萨里奥物理研究所（UNR-CONICET））；RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心 (iTHEMS)）
The orbital-selective Mott phase (OSMP) of multiorbital Hubbard models has been extensively analyzed before using static and dynamical mean-field approximations. In parallel, the properties of Block states (antiferromagnetically coupled ferromagnetic spin clusters) in Fe-based superconductors have also been much discussed. The present effort uses numerically exact techniques in one-dimensional systems to report the observation of Block states within the OSMP regime, connecting two seemingly independent areas of research, and providing analogies with the physics of Double-Exchange models.
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…
Determining the symmetry of the order parameter of unconventional superconductors remains a recurrent topic and non-trivial task in the field of strongly correlated electron systems. Here we show that the behavior of Dirac points away from the Fermi energy is a potential tool to unveil the orbital structure of a superconducting state. In particular, we show that gap openings in such Dirac crossings are a signature of non-unitary multiorbital superconducting order. Consequently, also spectral features at higher energy can help us to identify broken symmetries of superconducting phases and the o…
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 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…
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）
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…
We analyze the magnetic excitations and the spin-mediated superconductivity in iron-based superconductors within a low-energy model that operates in the band basis but fully incorporates the orbital character of the spin excitations. We show how the orbital selectivity, encoded in our low-energy description, simplifies substantially the analysis and allows for analytical treatments, while retaining all the main features of both spin-excitations and gap functions computed using multiorbital models. Importantly, our analysis unveils the orbital matching between the hole and electron pockets as t…
Multiorbital systems such as the iron-based superconductors provide a new avenue to attack the longstanding problem of superconductivity in strongly correlated systems. In this work we study the superconductivity driven by a generic bosonic mechanism in a multiorbital model including the full dynamical electronic correlations induced by the Hubbard U and the Hund's coupling. We show that superconductivity survives much more in a Hund's metal than in an ordinary correlated metal with the same degree of correlation. The crucial role of the redistribution of spectral weight in the Hund's metal re…
Multi-orbital Hubbard model Hamiltonians for the undoped parent compounds of the Fe-pnictide superconductors are here investigated using mean-field techniques. For a realistic four-orbital model, our results show the existence of an intermediate Hubbard U coupling regime where the mean-field ground state has spin stripe magnetic order, as in neutron scattering experiments, while remaining metallic, due to the phenomenon of band overlaps. The angle-resolved photoemission intensity and Fermi surface of this magnetic and metallic state are discussed. Other models are also investigated, including …
We investigate the Josephson transport properties in a Josephson junction consisting of a conventional $s$-wave superconductor coupled to a multi-orbital noncentrosymmetric superconductor marked by an orbitally driven inversion asymmetry and isotropic interorbital spin-triplet pairing. Contrary to the canonical single band noncentrosymmetric superconductor, we demonstrate that the local interorbital spin-triplet pairing is tied to the occurrence of sign-changing spin-singlet pair amplitude on different bands with $d$-wave symmetry. Such multi-band $d^{\pm}$-wave state is a unique superconducti…
We study the mechanism of orbital/spin fluctuations due to multiorbital Coulomb interaction in iron-based superconductors, going beyond the random-phase-approximation. For this purpose, we develop a self-consistent vertex correction (SC-VC) method, and find that multiple orbital fluctuations in addition to spin fluctuations are mutually emphasized by the "multimode interference effect" described by the VC. Then, both the antiferro-orbital and ferro-orbital (=nematic) fluctuations simultaneously develop for $J/U \sim 0.1$, both of which contribute to the s-wave superconductivity. Especially, th…
The origin of the nematic state is an important puzzle to be solved in iron pnictides. Iron superconductors are multiorbital systems and these orbitals play an important role at low energy. The singular $C_4$ symmetry of $d_{zx}$ and $d_{yz}$ orbitals has a profound influence at the Fermi surface since the $Γ$ pocket has vortex structure in the orbital space and the X/Y electron pockets have $yz$/$zx$ components respectively. We propose a low energy theory for the spin--nematic model derived from a multiorbital Hamiltonian. In the standard spin--nematic scenario the ellipticity of the electron…
We study the collective magnetic excitations of the recently discovered $C_{4}$ symmetric spin-density wave states of iron-based superconductors with particular emphasis on their orbital character based on an itinerant multiorbital approach. This is important since the $C_{4}$ symmetric spin-density wave states exist only at moderate interaction strengths where damping effects from a coupling to the continuum of particle-hole excitations strongly modifies the shape of the excitation spectra compared to predictions based on a local moment picture. We uncover a distinct orbital polarization inhe…
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…
In a multiorbital model of the cuprate high-temperature superconductors soft antiferromagnetic (AF) modes are assumed to reconstruct the Fermi surface to form nodal pockets. The subsequent charge ordering transition leads to a phase with a spatially modulated transfer of charge between neighboring oxygen p_x and p_y orbitals and also weak modulations of the charge density on the copper d_{x^2-y^2} orbitals. As a prime result of the AF Fermi surface reconstruction, the wavevectors of the charge modulations are oriented along the crystalline axes with a periodicity that agrees quantitatively wit…
The electronic nematic state without magnetization emerges in various strongly correlated metals such as Fe-based and cuprate superconductors. To understand this universal phenomenon, we focus on the nematic state in Ti-oxypnictide BaTi2(As,Sb)2O, which is expressed as the three-dimensional 10-orbital Hubbard model. The antiferromagnetic fluctuations are caused by the Fermi surface nesting. Interestingly, we find the spin-fluctuation-driven orbital order due to the strong orbital-spin interference, which is described by the Aslamazov-Larkin vertex correction (AL-VC). The predicted intra-unit-c…
The discovery of iron pnictides and iron chalcogenides as a new class of unconventional superconductors in 2008 has generated an enourmous amount of experimental and theoretical work that identifies these materials as correlated metals with multiorbital physics, where magnetism, nematicity and superconductivity are competing phases that appear as a function of pressure and doping. A microscopic understanding of the appearance of these phases is crucial in order to determine the nature of superconductivity in these systems. Here we review our recent theoretical efforts to describe and understan…
We present evidence for the existence of a spontaneous instability towards an orbital loop-current phase in a multiorbital Hubbard model for the CuO$_2$ planes in cuprates. Contrary to the previously proposed $θ_{II}$ phase with intra-unit cell currents, the identified instability is towards a staggered pattern of intertwined current loops. The orbitally resolved current pattern thereby shares its staggered character with the proposal of d-density wave order. The current pattern will cause a Fermi surface reconstruction and the opening of a pseudogap. We argue that the pseudogap phase with tim…
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…
The main features in iron-pnictide superconductors are summarized as (i) the orthorhombic transition accompanied by remarkable softening of shear modulus, (ii) high-Tc superconductivity close to the orthorhombic phase, and (iii) stripe-type magnetic order induced by orthorhombicity. To present a unified explanation for them, we analyze the multiorbital Hubbard-Holstein model with Fe-ion optical phonons based on the orbital fluctuation theory. In the random-phase-approximation (RPA), a small electron-phonon coupling constant ($λ~ 0.2$) is enough to produce large orbital (=charge quadrupole) flu…
Alkali-fulleride superconductors with a maximum critical temperature Tc of 40 K exhibit similar electronic phase diagram with unconventional high-Tc superconductors where the superconductivity resides proximate to a magnetic Mott-insulating state. However, distinct from cuprate compounds, which superconduct through two-dimensional (2D) CuO2 planes, alkali fullerides are attributed to the three-dimensional (3D) members of high-Tc family. Here, we employ scanning tunneling microscopy to show that trilayer K3C60 displays fully gapped strong coupling s-wave superconductivity that coexists spatiall…
Orbital polarization and electronic correlation are two essential aspects in understanding the normal state and superconducting properties of multiorbital FeAs-based superconductors. In this Letter, we present a systematical study on the orbital polarization of iron pnictides from weak to strong Coulomb correlations within the KRSB approach. The magnetic phase diagram of the two-orbital model for LaFeAsO clearly shows that the striped antiferromagnetic metallic phase with considerable orbital polarization exists over a wide doping range. The reversal of the orbital polarization occurs in the i…
The bad metal behavior in the normal state of the iron-based superconductors suggests an intimate connection between the superconductivity and a proximity to a Mott transition. At the same time, there is strong evidence for the multi-orbital nature of the electronic excitations. It is then important to understand the orbital-dependent effects of electron correlations. In this paper we review the recent theoretical progresses on the metal-to-insulator transition in multiorbital models for the iron-based superconductors. These include studies of models that contain at least the 3d xy and 3d xz/y…
The electronic states near the Fermi level of recently discovered superconductor Ba$_2$CuO$_{4-δ}$ consist primarily of the Cu $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals. We investigate the electronic correlation effect and the orbital polarization of an effective two-orbital Hubbard model mimicking the low-energy physics of Ba$_2$CuO$_{4-δ}$ in the hole-rich regime by utilizing the dynamical mean-field theory with the Lanczos method as the impurity solver. We find that the hole-overdoped Ba$_2$CuO$_{4-δ}$ with $3d^8$ (Cu$^{3+}$) is in the orbital-selective Mott phase (OSMP) at half-filling, an…
Motivated by the nematic electronic fluid phase in Sr_{3}Ru_{2}O_{7}, we develop a combined scheme of the renormalization-group method and the random-phase-approximation-type method, and analyze orbital susceptibilities of the (d_{xz},d_{yz})-orbital Hubbard model with high accuracy. It is confirmed that the present model exhibits a ferro-orbital instability near the magnetic or superconducting quantum criticality, due to the Aslamazov-Larkin-type vertex corrections. This mechanism of orbital nematic order presents a natural explanation for the nematic order in Sr_{3}Ru_{2}O_{7}, and is expect…
High-temperature cuprate superconductors have been known to exhibit significant pressure effects. In order to fathom the origin of why and how Tc is affected by pressure, we have recently studied the pressure effects on Tc adoptig a model that contains two cupper d-orbitals derived from first-principles band calculations, where the dz2 orbital is considere on top of the usually considered dx2-y2 orbital. In that paper, we have identified two origins for the Tc enhancement under hydrostatic pressure: (i) while at ambient pressure the smaller the hybridization of other orbital components the hig…
To reveal the origin of the "nematic electronic fluid phase" in Sr$_3$Ru$_2$O$_7$, we apply the self-consistent vertex correction analysis to the ($d_{xz},d_{yz}$)-orbital Hubbard model. It is found that the Aslamazov-Larkin type vertex correction causes the strong coupling between spin and orbital fluctuations, which corresponds to the Kugel-Khomskii spin-orbital coupling in the local picture. Due to this mechanism, orbital nematic order with $C_2$ symmetry is induced by the magnetic quantum criticality in multiorbital systems, while this mechanism is ignored in the random-phase-approximation…
To understand the amazing variety of the superconducting states of Fe-based superconductors, we analyze the multiorbital Hubbard models for LaFeAsO and LiFeAs going beyond the random-phase approximation (RPA), by calculating the vertex correction (VC) and self-energy correction. Due to the spin+orbital mode coupling described by the VC, both orbital and spin fluctuations mutually develop, consistently with the experimental phase diagram with the orbital and magnetic orders. Due to both fluctuations, the s-wave gap function with sign-reversal ($s_{\pm}$-wave), without sign-reversal ($s_{++}$-wa…
The pairing mechanism and gap structure in Ba122 pnictides have been hotly discussed for long time as one of the central issues in Fe-based superconductors. Here, we attack this problem by taking account of the vertex corrections (VCs) for the Coulomb interaction $U$ ($U$-VCs), which are totally dropped in conventional Migdal-Eliashberg formalism. The $U$-VC in the charge susceptibility induces strong orbital fluctuations, and the $U$-VC also enlarges the orbital-fluctuation-mediated attractive interaction. By analyzing the effective multiorbital Hubbard model for Ba122 pnictides, we find that…
