We analyze antiferromagnetism and superconductivity in novel $Fe-$based superconductors within the itinerant model of small electron and hole pockets near $(0,0)$ and $(π,π)$. We argue that the effective interactions in both channels logarithmically flow towards the same values at low energies, {\it i.e.}, antiferromagnetism and superconductivity must be treated on equal footings. The magnetic instability comes first for equal sizes of the two pockets, but looses to superconductivity upon doping. The superconducting gap has no nodes, but changes sign between the two Fermi surfaces (extended s-wave symmetry). We argue that the $T$ dependencies of the spin susceptibility and NMR relaxation rate for such state are exponential only at very low $T$, and can be well fitted by power-laws over a wide $T$ range below $T_c$.
The interplay between magnetism and superconductivity in the newly discovered heavy-fermion superconductor CePt$_3$Si has been investigated using the zero-field $μ$SR technique. The $μ$SR data indicate that the whole muon ensemble senses spontaneous internal fields in the magnetic phase, demonstrating that magnetism occurs in the whole sample volume. This points to a microscopic coexistence between magnetism and heavy-fermion superconductivity.
Specific heat studies of CeRhIn5 as functions of pressure and magnetic field have been used to explore the relationship between magnetism and unconventional superconductivity, both of which involve the 4f electron of Ce. Results of these studies cannot be understood as a simple competition for Fermi-surface states and require a new conceptual framework.
We analyze the competition of magnetism and superconductivity in the two-dimensional Hubbard model with a moderate interaction strength, including the possibility of incommensurate spiral magnetic order. Using an unbiased renormalization group approach, we compute magnetic and superconducting order parameters in the ground state. In addition to previously established regions of Neel order coexisting with d-wave superconductivity, the calculations reveal further coexistence regions where superconductivity is accompanied by incommensurate magnetic order.
Using the numerical unrestricted Hartree-Fock approach, we study the ground state of a two-orbital model describing newly discovered FeAs-based superconductors. We observe the competition of a $(0, π)$ mode spin-density wave and the superconductivity as the doping concentration changes. There might be a small region in the electron-doping side where the magnetism and superconductivity coexist. The superconducting pairing is found to be spin singlet, orbital even, and mixed s$_{xy}$ + d$_{x^{2}-y^{2}}$ wave (even parity).
The Nd1-xCexCoIn5 alloys evolve from local moment magnetism (x = 0) to heavy fermion superconductivity (x =1). Magnetic order is observed over a broad range of x. For a substantial range of x (0.83 <= x <= 0.95) in the temperature - composition phase diagram we find that superconductivity may coexist with spin - density wave magnetic order at the Fermi surface. We show that a delicate balance betwen superconducting and magnetic instabilities can be reversibly tuned by both the Ce/Nd ratio and magnetic field, offering a new and unique model electronic system.
The interplay of magnetism and unconventional superconductivity (d singlet wave or p triplet wave) in strongly correlated electronic system (SCES) is discussed with recent examples found in heavy fermion compounds. A short presentation is given on the formation of the heavy quasiparticle with the two sources of a local and intersite enhancement for the effective mass. Two cases of the coexistence or repulsion of antiferromagnetism and superconductivity are given with CeIn3 and CeCoIn5. A spectacular example is the emergence of superconductivity in relatively strong itinerant ferromagnets UGe2 …
We performed zero and transverse field MuSR experiments on a large number of YBa2Cu3O6+x samples. We detect coexistence of antiferro-type (AF) short range magnetism with superconductivity below T_f~10 K in compositions 0.37<x<0.39. Most muons experience local AF fields, even when SQUID detects a full SC volume fraction, which points to a local minimal interference organization of short AF stripes embedded in the superconductor. A detailed phase diagram is produced and the consequences of the minimal interference are discussed.
The properties of the superconducting ferromagnets RuSr2(Gd/Eu)Cu2O8 and RuSr2(Ln/Ce)2Cu2O10 are reviewed. The problem of coexistence and competition between magnetic and superconducting states is discussed. A model of phase separation into ferromagnetic and anti-ferromagnetic structures is proposed.
The nonadiabatic Heisenberg model presents a nonadiabatic mechanism generating Cooper pairs in narrow, roughly half-filled "superconducting bands" of special symmetry. Here we show that this mechanism may be understood as the outcome of a special spin structure in the reciprocal space, hereinafter referred to as k-space magnetism. The presented picture permits a vivid depiction of this new mechanism highlighting the height similarity as well as the essential difference between the new nonadiabatic and the familiar Bardeen-Cooper-Schrieffer mechanism.
We investigated the evolution of the temperature-composition phase diagram of Fe$_{1+y}$Te upon Se substitution. In particular, the effect of Se substitution on the two-step, coupled magneto-structural transition in Fe$_{1+y}$Te single crystals is investigated. To this end, the nominal Fe excess was kept at $y$ = 0.12. For low Se concentrations, the two magneto-structural transitions displayed a tendency to merge. In spite of the high Fe-content, superconductivity emerges for Se concentrations, $x \geq$ 0.1. We present a temperature-composition phase diagram to demonstrate the interplay of str…
Recent discovery of high T_c superconductivity in Fe-based compounds may have opened a new pathway to the room temperature superconductivity. The new materials feature FeAs layers instead of the signature CuO_2 planes of much-studied cuprates. A model Hamiltonian describing FeAs layers is introduced, highlighting the crucial role of puckering of As atoms in promoting d-electron itinerancy and warding off large local-moment magnetism of Fe ions, the main enemy of superconductivity. Quantum many-particle effects in charge, spin and multiband channels are explored and a nesting-induced spin densi…
In order to examine the interplay between magnetism and superconductivity, we monitor the non- superconducting chalcogenide FeTe and follow its transitions under insertion of oxygen, doping with Se and vacancies of Fe using spin-polarized band structure methods (LSDA with GGA) starting from the collinear and bicollinear magnetic arrangements. We use a supercell of Fe8Te8 as our starting point so that it can capture local changes in magnetic moments. The calculated values of magnetic moments agree well with available experimental data while oxygen insertions lead to significant changes in the b…
Superconducting pnictides are widely found to feature spin-singlet pairing in the vicinity of an antiferromagnetic phase, for which nesting between electron and hole Fermi surfaces is crucial. LiFeAs differs from the other pnictides by (i) poor nesting properties and (ii) unusually shallow hole pockets. Investigating magnetic and pairing instabilities in an electronic model that incorporates these differences, we find antiferromagnetic order to be absent. Instead we observe almost ferromagnetic fluctuations which drive an instability toward spin-triplet p-wave superconductivity.
We provide and analyze a periodic Anderson model for studying magnetism and superconductivity in UTe$_2$, a recently-discovered candidate for a topological spin-triplet superconductor. The 24-band tight-binding model reproduces the band structure obtained from a DFT$+U$ calculation consistent with an angle-resolved photoemission spectroscopy. The Coulomb interaction of $f$-electrons enhances Ising ferromagnetic fluctuation along the $a$-axis and stabilizes spin-triplet superconductivity of either $B_{3u}$ or $A_{u}$ symmetry. When effects of pressure are taken into account in hopping integrals…
We briefly overview the importance of Hubbard and Anderson-lattice models as applied to explanation of high-temperature and heavy-fermion superconductivity. Application of the models during the last two decades provided an explanation of the paired states in correlated fermion systems and thus extended essentially their earlier usage to the description of itinerant magnetism, fluctuating valence, and the metal-insulator transition. In second part, we also present some of the new results concerning the unconventional superconductivity and obtained very recently in our group. A comparison with e…
We review magnetic, superconducting and non-Fermi-liquid properties of the structurally layered heavy-fermion compounds Ce$_n$M$_m$In$_{3n+2m}$ (M=Co, Rh, Ir). These properties suggest d-wave superconductivity and proximity to an antiferromagetic quantum-critical point.
The recent discovery of pressure induced superconductivity in the binary helimagnet CrAs has attracted much attention. How superconductivity emerges from the magnetic state and what is the mechanism of the superconducting pairing are two important issues which need to be resolved. In the present work, the suppression of magnetism and the occurrence of superconductivity in CrAs as a function of pressure ($p$) were studied by means of muon spin rotation. The magnetism remains bulk up to $p\simeq3.5$~kbar while its volume fraction gradually decreases with increasing pressure until it vanishes at …
A consistent picture on the mean-field level of the magnetic properties and electronic structure of the superconducting itinerant ferromagnet UGe$_2$ is shown to require inclusion of correlation effects beyond the local density approximation (LDA). The "LDA+U" approach reproduces both the magnitude of the observed moment, composed of strongly opposing spin and orbital parts, and the magnetocrystalline anisotropy. The largest Fermi surface sheet is comprised primarily of spin majority states with orbital projection $m_{\ell}$=0, suggesting a much simpler picture of the pairing than is possible …
Ferromagnetism and superconductivity are in most cases adverse. However, recent experiments reveal that they coexist at interfaces of LaAlO3 and SrTiO3. We analyze the magnetic state within density functional theory and provide evidence that magnetism is not an intrinsic property of the two-dimensional electron liquid at the interface. We demonstrate that the robust ferromagnetic state is induced by the oxygen vacancies in SrTiO3- or in the LaAlO3-layer. This allows for the notion that areas with increased density of oxygen vacancies produce ferromagnetic puddles and account for the previous o…
We carried out specific-heat and ac-susceptibility experiments under hydrostatic pressure to investigate the interplay of spin-density-wave antiferromagnetism (A) and superconductivity (S) in single-crystalline AS-type CeCu2Si2. We find evidence for a line of magnetic-field- and pressure-tuned quantum critical points in the normal state in the zero-temperature magnetic field-pressure plane. Our analysis suggests an extension of this line into the superconducting state and corroborates the close connection of the underlying mechanisms leading to the formation of the antiferromagnetic and the su…
We report a thermodynamic and transport study of the phase diagram of CeRh1-xIrxIn5. Superconductivity is observed over a broad range of doping, 0.3 < x < 1, including a substantial range of concentration (0.3 < x <0.6) over which it coexists with magnetic order (which is observed for 0 < x < 0.6). The anomalous transition to zero resistance that is observed in CeIrIn5 is robust against Rh substitution. In fact, the observed bulk Tc in CeRh0.5Ir0.5In5 is more than double that of CeIrIn5, whereas the zero-resistance transition temperature is relatively unchanged for 0.5 < x < 1.
Single crystals of SrFe2-xPtxAs2 (0 < x < 0.36) were grown using the self flux solution method and characterized using x-ray crystallography, electrical transport, magnetic susceptibility, and specific heat measurements. The magnetic/structural transition is suppressed with increasing Pt concentration, with superconductivity seen over the range 0.08 < x < 0.36 with a maximum transition temperature Tc of 16 K at x = 0.16. The shape of the phase diagram and the changes to the lattice parameters are similar to the effects of other group VIII elements Ni and Pd, however the higher transition tempe…
We find an exotic spin excitation in a magnetically ordered system with spin orbit magnetism in 2D, where the order parameter has a net spin current and no net magnetization. Starting from a Fermi liquid theory, similar to that for a weak ferromagnet, we show that this excitation emerges from an exotic magnetic Fermi liquid state that is protected by a generalized Pomeranchuck condition. We derive the propagating mode using the Landau kinetic equation, and find that the dispersion of the mode has a $\sqrt q$ behavior in leading order in 2D. We find an instability toward superconductivity induc…
We present a low-energy muon-spin-rotation study of the magnetic and superconducting properties of YBa2Cu3O7/PrBa2Cu3O7 trilayer and bilayer heterostructures. By determining the magnetic-field profiles throughout these structures we show that a finite superfluid density can be induced in otherwise semiconducting PrBa2Cu3O7 layers when juxtaposed to YBa2Cu3O7 "electrodes" while the intrinsic antiferromagnetic order is unaffected.
Superconductivity and magnetism in the non-centrosymmetric heavy fermion compound CePt$_3$Si and related materials are theoretically investigated. Based on the randam phase approximation (RPA) analysis for the extended Hubbard model we describe the helical spin fluctuation induced by the Rashba-type anti-symmetric spin-orbit coupling and identify the two stable superconducting phases with either dominantly p-wave ($s$ + $P$-wave) or d-wave ($p$ + $D$ + $f$-wave) symmetry. The influcnce of the coexistent anti-ferromagnetic order is investigated in both states. The SC order parameter, quasiparti…
The model of local electron pairs and itinerant fermions coupled via charge exchange mechanism, which mutually induces superconductivity in both subsystems is studied for anisotropic pairing symmetry. The phase diagram is presented and the phase fluctuations effects are analyzed within the Kosterlitz-Thouless scenario.
The discovery of EuFeAs2, currently the only charge-neutral parent phase of the 112-type iron-pnictide system, provides a new platform for the study of elemental doping effects on magnetism and superconductivity (SC). In this study, a series of polycrystalline EuFe1-yCoyAs2 and Eu0.9Pr0.1Fe1-yCoyAs2 samples are synthesized through solid-state reaction, and the evolutions of SC and magnetism with Co doping in EuFeAs2 and Eu0.9Pr0.1FeAs2 are investigated by electrical transport and magnetic susceptibility measurements. For EuFe1-yCoyAs2, the Eu-related antiferromagnetic (AFM) transition around 4…
By bosonizing the electronic t-J model exactly on any two-dimensional (2D) lattices, and integrating out the gauge fluctuations combined to slave particles beyond mean fields, we get a theory in terms of physical Cooper pair and spin condensates. In the sense of mutual Berry phase they turns out to be dual to each other. The mutual-duality is the missing key in the resonant-valance-bond idea\cite{rvb} to work as a paradigm of doped 2D Mott insulators. We argue that essential aspects of high-$T_c$ phenomenology find natural solutions in the theory. We also provide interesting predictions for sy…
This article reviews recent results of magnetotransport and magnetization measurements performed on highly oriented pyrolitic graphite (HOPG) and single crystalline Kish graphite samples. Both metal-insulator and insulator-metal transitions driven by magnetic field applied perpendicular to the basal planes of graphite were found and discussed in the light of relevant theories. The results provide evidence for the existence of localized superconducting domains in HOPG even at room temperature, as well as an interplay between superconducting and ferromagnetic correlations. We also present experi…
I present a microscopic theory I proposed recently to describe high-T_c superconductivity in cuprates. I show that coherent pairing states consisting of extended singlet Cooper pairs and triplet $π$ pairs can manifest both the Mott insulating antiferromagnetic order and the d-wave superconducting order. From this configuration of coherent pairing states, I can describe both the single electron properties and the low energy collective excitations in high T_c superconductivity in the same framework. The quasiparticle can be derived directly with respect to the coherent pairing states. While, the…
The magnetization rotation transition occurs in the itinerant ferromagnet URhGe when the field about 12T is applied in direction perpendicular to spontaneous magnetization in the plane of the smallest magnetic anisotropy energy.The transition is accompanied by the maximum of resistivity in the normal state and by reentrance of superconductivity at lower temperatures in the field interval between 8 and 13 Tesla [F.Levy et al, Science, to be published (2005)]. We discuss the magnetization orientation transition and the modification of triplet pairing superconducting state coexisting with ferroma…
We investigate the influence of critical spin fluctuations on electromagnetic responses in the odd-frequency superconductivity. It is shown that the Meissner kernel of the odd-frequency superconductivity is strongly reduced by the critical spin fluctuation or the massless spin wave mode in the antiferromagnetic phase. These results imply that the superfluid density is reduced, and the London penetration depth is lengthened for the odd-frequency pairing. It is also shown that the zero field Flude-Ferrell-Larkin-Ovchinnikov pairing is spontaneously realized both for even- and odd-frequency in th…
The effect of magnetic moments on superconductivity has long been a controversial subject in condensed matter physics. While Matthias and collaborators experimentally demonstrated the destruction of superconductivity in La by the addition of magnetic moments (Gd), it has since been suggested that magnetic fluctuations are in fact responsible for the development of superconducting order in other systems. Currently this debate is focused on several families of unconventional superconductors including high-Tc cuprates, borocarbides as well as heavy fermion systems where magnetism and superconduct…
EuBiS$_{2}$F is a self-doped superconductor due to the mixed valence of Eu. Here we report that, with the Ce substitution for Eu by 50 at.\%, the material exhibits ferromagnetic ordering at 8 K for the Ce-4$f$ moment, superconductivity at 2.2 K in the BiS$_2$ layers, and possibly antiferromagnetic ordering at 2.1 K for the Eu-4$f$ spins. The Eu valence is essentially divalent with the Ce incorporation. We tentatively interpret the coexistence of ferromagnetism and superconductivity by considering different Bi-6$p$ orbitals that are responsible for superconductivity itself and for mediating the…
This Colloquium reviews the 25 year quest for understanding the continuous (second-order) mean-field-like phase transition occurring at 17.5 K in URu2Si2. About ten years ago, the term hidden order (HO) was coined and has since been utilized to describe the unknown ordered state, whose origin cannot be disclosed by conventional solid-state probes, such as x rays, neutrons, or muons. HO is able to support superconductivity at lower temperatures (Tc ~ 1.5 K), and when magnetism is developed with increasing pressure both the HO and the superconductivity are destroyed. Other ways of probing the HO…
To understand the rich electronic phase diagram in FeSe under pressure that vividly demonstrates the strong interplay between the nematicity, magnetism and superconductivity, we analyze the electronic states by including the higher-order many-body effects called the vertex correction (VC). We predict the pressure-induced emergence of xy-orbital hole-pocket based on the first-principles analysis. Due to this pressure-induced Lifshitz transition, the spin fluctuations on the xy orbital are enhanced, whereas those on xz,yz orbitals are gradually reduced. For this reason, nonmagnetic orbital order…
The magnetic ordering temperature of some rare earth based heavy fermion compounds is strongly pressure-dependent and can be completely suppressed at a critical pressure, p$_c$, making way for novel correlated electron states close to this quantum critical point. We have studied the clean heavy fermion antiferromagnets CePd$_2$Si$_2$ and CeIn$_3$ in a series of resistivity measurements at high pressures up to 3.2 GPa and down to temperatures in the mK region. In both materials, superconductivity appears in a small window of a few tenths of a GPa on either side of p$_c$. We present detailed mea…
We report Eu-local-spin magnetism and Ni-doping-induced superconductivity (SC) in a 112-type ferroarsenide system Eu(Fe$_{1-x}$Ni$_{x}$)As$_2$. The non-doped EuFeAs$_2$ exhibits two primary magnetic transitions at $\sim$100 and $\sim$ 40 K, probably associated with a spin-density-wave (SDW) transition and an antiferromagnetic ordering in the Fe and Eu sublattices, respectively. Two additional successive transitions possibly related to Eu-spin modulations appear at 15.5 and 6.5 K. For the Ni-doped sample with $x$ = 0.04, the SDW transition disappears, and SC emerges at $T_\mathrm{c}$ = 17.5 K. …
We systematically studied the transport properties of single crystals of $Eu_{1-x}Sr_xFe_{2-y}$Co$_{y}As_2$. Co doping can suppress the spin-density wave (SDW) ordering and induces a superconducting transition, but a resistivity reentrance due to the antiferromagnetic ordering of $Eu^{2+}$ spins is observed, indicating the competition between antiferromagnetism (AFM) and superconductivity. It is striking that the resistivity reentrance can be completely suppressed by external magnetic field (H) because a metamagnetic transition from antiferromagnetism to ferromagnetism for $Eu^{2+}$ spins is i…
With understood exceptions, conventional superconductivity does not coexist with long-range magnetic order[1]. In contrast, unconventional superconductivity develops near a boundary separating magnetically ordered and magnetically disordered phases[2,3]. A maximum in the superconducting transition temperature Tc develops where this boundary extrapolates to T=0 K, suggesting that fluctuations associated with this magnetic quantum-critical point are essential for unconventional superconductivity[4,5]. Invariably though, unconventional superconductivity hides the magnetic boundary when T < Tc, pr…
We report the local magnetic response of an electron-doped cuprate superconductor, Pr$_{0.89}$LaCe$_{0.11}$CuO$_{4}$ ($T_c\simeq26$ K), studied by muon spin rotation (\msr) over a field range from 200 Oe to 50 kOe. In the paramagnetic state, the muon Knight shift along the crystal c-axis ($K_μ^z$) is proportional to the in-plane susceptibility $χ_{ab}$. More surprisingly, $K_μ^z$ is strongly enhanced by the occurrence of superconductivity (below $\sim$40 kOe), changing its sign from positive to negative around $\sim1$ kOe with increasing field. This finding can be understood by considering the…
Topological Chiral Superconductivity from Doping Correlated Charge Transfer Insulator with Orbital Magnetism
Great Bay Institute for Advanced Study, Great Bay University（大湾区先进科技学院）；Great Bay University（大湾区大学）；Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China（中国科学院微观尺度物理科学国家研究中心及中国科学技术大学物理科学学院）
The particular shape of the Fermi surface can give rise to a number of collective quantum phenomena in solids, such as density wave orderings or even superconductivity. In many new iron superconductors this shape, the 'nested' Fermi surface, is indeed observed, but its role in the formation of spin-density waves or superconductivity is not clear. We have studied the electronic structure of the non-magnetic LiFeAs (Tc~18K) superconductor using angle-resolved photoemission spectroscopy. We find a notable absence of the Fermi surface nesting, strong renormalization of the conduction bands by a fa…
We report the phase diagram of the doping series U_2Pt_xRh_(1-x)C_2, studied through measurements of resistivity, specific heat and magnetic susceptibility. The Neel temperature of U_2RhC_2 of ~ 22 K is suppressed with increasing Pt content, reaching zero temperature close to x=0.7, where we observed signatures of increased quantum fluctuations. In addition, evidence is presented that the antiferromagnetic state undergoes a spin-reorientation transition upon application of an applied magnetic field. This transition shows non-monotonic behaviour as a function of x, peaking at around x=0.3. Supe…
We report local probe ($μ$SR) measurements on the recently discovered tetragonal FeS superconductor which has been predicted to be electronically very similar to superconducting FeSe. Most remarkably, we find that low moment ($10^{-2}-10^{-3}μ_\mathrm{B})$ disordered magnetism with a transition temperature of $T_\mathrm{N}\approx 20$ K microscopically coexists with bulk superconductivity below $T_\mathrm{c}=4.3(1)$ K. From transverse field $μ$SR we obtain an in-plane penetration depth $λ_\mathrm{ab}(0)=223(2)$ nm for FeS. The temperature dependence of the corresponding superfluid density $λ_\m…
Confirmation of the phononic origin of Cooper pair formation in superconductors came with the demonstration that the interaction was retarded and that the corresponding energy scales were associated with phonons. Using cellular dynamical mean-field theory for the two-dimensional Hubbard model, we identify such retardation effects in d-wave pairing and associate the corresponding energy scales with short-range spin fluctuations. We find which frequencies are relevant for pairing as a function of interaction strength and doping and show that the disappearance of superconductivity on the overdope…
The recent observation of fully-gapped superconductivity in Yb doped CeCoIn5 poses a paradox, for the disappearance of nodes suggests that they are accidental, yet d-wave symmetry with protected nodes is we ll established by experiment. Here, we show that composite pairing provides a natural resolution: in this scenario, Yb doping drives a Lifshitz transition of the nodal Fermi surface, forming a fully-gapped d-wave molecular superfluid of composite pairs. The T4 dependence of the penetration depth associated with the sound mode of this condensate is in accord with observation.
The interplay between magnetic and superconducting states on a square lattice is studied using the extended Hubbard model, which takes into account the attraction of electrons located at nearest neighbor sites. Ferro-, antiferro-, and spiral magnetic states with all possible sets of spiral wave vectors, as well as singlet superconducting states with s-wave and d-wave pairing order parameters, are considered. Formation of a state with the intermediate s+id-symmetry and phase separation between different phases are allowed. The results of the Hartree-Fock and slave boson approximations are compa…
To pinpoint the microscopic mechanism for superconductivity has proven to be one of the most outstanding challenges in the physics of correlated quantum matter. Thus far, the most direct evidence for an electronic pairing mechanism is the observation of a new symmetry of the order-parameter, as done in the cuprate high-temperature superconductors. Like distinctions based on the symmetry of a locally defined order-parameter, global, topological invariants allow for a sharp discrimination between states of matter that cannot be transformed into each other adiabatically. Here we propose an unconv…
We report on zero-field muon spin relaxation studies of cerium based heavy-fermion materials CeRh_{1-x}Ir_xIn_5. In the superconducting x=0.75 and 1 compositions muon spin relaxation functions were found to be temperature independent across T_c; no evidence for the presence of electronic magnetic moments was observed. The x=0.5 material is antiferromagnetic below T_N=3.75 K and superconducting below T_c=0.8 K. Muon spin realxation spectra show the gradual onset of damped coherent oscillations characteristic of magnetic order below T_N. At 1.65 K the total oscillating amplitude accounts for at …
The recently discovered superconductivity in Nd$_{1-x}$Sr$_x$NiO$_2$ provides a new opportunity for studying strongly correlated unconventional superconductivity. The single-hole Ni$^+$ ($3d^9$) configuration in the parent compound NdNiO$_2$ is similar to that of Cu$^{2+}$ in cuprates. We suggest that after doping, the intra-orbital spin-singlet and inter-orbital spin-triplet double-hole (doublon) configurations of Ni$^{2+}$ are competing, and we construct a two-band Hubbard model by including both the $3d_{x^2-y^2}$ and $3d_{xy}$-orbitals. The effective spin-orbital super-exchange model in th…
