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.
It is developed a microscopic description of superconductivity in ferromagnetic materials with triplet pairing triggered by the exchange of magnetic fluctuations. Instead widely used paramagnon model we work with phenomenological spectrum of fluctuations in the orthorhombic ferromagnet with strong magnetic anisotropy. Depending of the field orientation parallel or perpendicular to the direction of spontaneous magnetization the effective amplitude of pairing interaction proves to be decreasing or increasing function of magnetic field that allows to explain the drastic difference in magnitudes o…
Within a multicomponent BCS-like framework we study the coexistence and competition of various CDW and SC order parameters in the presence of a strong ferromagnetic background. We find that the competition of unconventional CDW, also called orbital antiferromagnet, with SC at finite momentum (${\bf Q}$-triplet pairing) shows unique characteristics like an extreme sensitivity on the deviations from nesting. We argue that pressure in UGe$_2$ creates deviations from the nesting and report a phase diagram in qualitative agreement with the observed behavior of UGe$_2$.
We report that a novel type of superconducting order parameter has been realized in the ferromagnetic states in UGe$_2$ via $^{73}$Ge nuclear-quadrupole-resonance (NQR) experiments performed under pressure ($P$). Measurements of the nuclear spin-lattice relaxation rate $(1/T_1)$ have revealed an unconventional nature of superconductivity such that the up-spin band is gapped with line nodes, but the down-spin band remains gapless at the Fermi level. This result is consistent with that of a ferromagnetic spin-pairing model in which Cooper pairs are formed among ferromagnetically polarized electr…
The microscopic origin of triplet superconductivity in Sr$_2$RuO$_4$ is discussed, paying attention to the role of Coulomb interaction, $U_{pp}$, at the O site. It is shown on the $d$-$p$ model that $U_{pp}$ induces a ferromagnetic exchange interaction between "d-electrons" (molecular orbital with $d_{xy}$-symmetry) at adjacent Ru sites, leading to short-range ferromagnetic correlations and promoting Cooper pairing with ($\sin p_x\pm\ii\sin p_y$)-symmetry on the $γ$-band. The reason why such ferromagnetic correlations work effectively may be traced back to the fact that the level of 4d-electro…
We address the quantum-critical behavior of a two-dimensional itinerant ferromagnetic systems described by a spin-fermion model in which fermions interact with close to critical bosonic modes. We consider Heisenberg ferromagnets, Ising ferromagnets, and the Ising nematic transition. Mean-field theory close to the quantum critical point predicts a superconducting gap with spin-triplet symmetry for the ferromagnetic systems and a singlet gap for the nematic scenario. Studying fluctuations in this ordered phase using a nonlinear sigma model, we find that these fluctuations are not suppressed by a…
We study tunneling conductance in normal metal / insulator / ferromagnetic superconductor junctions. The tunneling spectra show a clear difference between spin-singlet s-wave pairing, spin-triplet opposite spin pairing and spin-triplet equal spin pairing: These pairings exhibit, respectively, gap struture, double peak structure and zero bias peak in the spectra. The obtained result may serve as a tool for determining the pairing symmetry of ferromagnetic superconductors.
The superconducting state in uranium compounds UGe2, URhGe and UCoGe is formed at temperatures far below the Curie temperature pointing on nonconventional nature of superconductivity in these materials - namely the superconductivity with triplet pairing. The emergence of superconductivity is accompanied by the slight magnetization expulsion typical for the type-II superconductors. Following classic Abrikosov paper I develop the theory of type-II superconductivity in application to two-band ferromagnetic metal with equal spin triplet pairing.
The self-energy, spectral functions and susceptibilities of 2D systems with strong ferromagnetic fluctuations are considered within the quasistatic approach. The self-energy at low temperatures T has a non-Fermi liquid form in the energy window w<Delta near the Fermi level, where Delta is the ground-state spin splitting for magnetically ordered ground state, and Delta ~ T^(1/2) ln^(1/2)(vF/T) in the quantum critical regime (vF is the Fermi velocity). Spectral functions have a two-peak structure at finite T above the magnetically ordered ground state, which implies quasi-splitting of the Fermi …
In the recently discovered Co-oxide superconductor Na_xCoO_2*yH_2O, the edge-shared CoO6 octahedra are trigonally contracted along the c-axis in the CoO2-plane. We study how this CoO6 distortion affects the magnetic properties and superconductivity in this compound by analyzing the multiorbital Hubbard model using the fluctuation-exchange approximation. It is shown that through generating the trigonal crystal field, the distortion pushes the Co e'g bands up and consequently gives rise to the hole-pocket Fermi surfaces, which have been predicted in the band calculations. As the distortion incre…
It is argued that the pairing symmetry realized in a ferromagnetic superconductor UGe$_2$ must be a non-unitary triplet pairing. This particular state is free from the Pauli limitation and can survive under a huge internal molecular filed. To check our identification we examine its basic properties and several experiments are proposed. In particular, the external field is used to raise $T_c$ by controlling the internal spontaneous dipole field.
The theory of triplet superconductivity in ferromagnetic metals based on electron-electron interaction by spin fluctuation exchange is developed. The equations for the upper critical field temperature dependence are derived. In contrast to the similar equations for the superconductivity in two band metals they contain the pairing amplitudes and the Fermi velocities depending on magnetic field. The critical field behaviour near the critical temperature and at T = 0 is established analytically.
A general phenomenological theory is presented for the phase behavior of ferromagnetic superconductors with spin-triplet electron Cooper pairing. The theory describes in details the temperature-pressure phase diagrams of real inter-metallic compounds exhibiting the remarkable phenomenon of coexistence of spontaneous magnetic moment of the itinerant electrons and spin-triplet superconductivity. The quantum phase transitions which may occur in these systems are also described. The theory allows for a classification of these itinerant ferromagnetic superconductors in two types: type I and type II…
The spin susceptibility in the uranium ferromagnet superconductors is calculated. There is shown that the absence of superconductivity paramagnetic limitation for the field directions perpendicular to the direction of the spontaneous magnetization is explained by the itinerant ferromagnet band splitting rather than by a rotation of magnetization toward the external field direction. The qualitative description of the upper critical field temperature dependence is given.
We propose that screened pairing interactions mediated by phonons could give rise to a spin triplet superconductivity in ferromagnetic compounds such as UGe_2. It is pointed out that the pairing interactions include anisotropic components such as those of p, d, f-waves in addition to dominant s-wave component due to the momentum dependence. Since the ferromagnetic long-range order coexists, there is a large splitting of the Fermi surfaces of up and down spin electrons, which suppresses singlet pairing. Therefore, triplet pairing occurs at last due to the sub-dominant anisotropic interactions, …
We study the ground-state properties of the double-chain Hubbard model coupled with ferromagnetic exchange interaction by using the weak-coupling theory, density-matrix renormalization group technique, and Lanczos exact-diagonalization method. We determine the ground-state phase diagram in the parameter space of the ferromagnetic exchange interaction and band filling. We find that, in high electron density regime, the spin gap opens and the spin-singlet $d_{xy}$-wave-like pairing correlation is most dominant, whereas in low electron density regime, the fully-polarized ferromagnetic state is st…
The coexistence of superconductivity and ferromagnetism has brought about the phenomena of ferromagnetic superconductors. The theory needed to understand the compatibility of such antagonistic phenomena cannot be built until the pairing symmetry of such superconductors is correctly identified. The proper and unambiguous identification of the pairing symmetry of such superconductors is the subject of this paper. This work shows that crossed Andreev reflection can be a very effective tool in order to identify the pairing symmetry of these superconductors.
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 …
We present a general thermodynamic theory that describes phases and phase transitions of ferromagnetic superconductors with spin-triplet electron Cooper pairing. The theory is based on extended Ginzburg-Landau expansion in powers of superconducting and ferromagnetic order parameters. We propose a simple form for the dependence of theory parameters on the pressure that allows correct theoretical outline of the temperature-pressure phase diagram for which at low temperatures a stable phase of coexistence of p-wave superconductivity and itinerant ferromagnetism appears. We demonstrate that the th…
We point out that ferromagnetic superconductors with triplet pairing and strong spin-orbit coupling are even in the simplest case at least two-band superconductors. The Gor'kov type formalism for such superconductors is developed and the Ginzburg-Landau equations are derived. The dependence of the critical temperature on the concentration of ordinary point-like impurities is found. Its nonuniversality could serve as a qualitative measure of the two-band character of ferromagnetic superconductors. The problem of the upper critical field determination is also discussed.
We investigate numerically the local density of states (LDOS) in the vicinity of a vortex core in a ferromagnetic superconductor. Specifically, we investigate how the LDOS is affected by the relative weight of the spin bands in terms of the superconducting pairing, and we also examine the effect of different pairing symmetries for the superconducting order parameter. Our findings are directly related to scanning tunneling microscopy measurements and may thus be highly useful to clarify details of the superconducting pairing in recently discovered ferromagnetic superconductors.
Ferromagnetism and superconductivity are generally considered to be antagonistic phenomena in condensed matter physics. Here, we theoretically study the interplay between the ferromagnetic and superconducting orders in a recent discovered monolayered CoSb superconductor with an orthorhombic symmetry and net magnetization, and demonstrate the pairing symmetry of CoSb as a candidate of non-unitary superconductor with time-reversal symmetry breaking. By performing the group theory analysis and the first-principles calculations, the superconducting order parameter is suggested to be a triplet pair…
A general quasi-phenomenological theory that describes phases and phase transitions of ferromagnetic superconductors with spin-triplet electron Cooper pairing is presented. The theory is based on extended Ginzburg-Landau expansion in powers of superconducting and ferromagnetic order parameters. A simple form for the dependence of theory parameters on the pressure ensures a correct theoretical outline of the temperature-pressure phase diagram where a stable phase of coexistence of p-wave superconductivity and itinerant ferromagnetism appears. This new theory is in an excellent agreement with th…
We study spin-half fermions in a one-dimensional extended Hubbard chain at low filling. We identify three triplet and one singlet pairing channels in the system, which are independently tunable as a function of nearest-neighbor charge and spin interactions. In a large-size system with translational invariance, we derive gap equations for the corresponding pairing gaps and obtain a Bogoliubov-de Gennes Hamiltonian with its non-trivial topology determined by the interplay of these gaps. In an open-end system with a fixed number of particles, we compute the exact many-body ground state and identi…
In order to determine the superconducting paring state in the ferromagnetic superconductor UCoGe, ^{59}Co NMR Knight shift, which is directly related to the microscopic spin susceptibility, was measured in the superconducting state under magnetic fields perpendicular to spontaneous magnetization axis: ^{59}K^{a, b}. ^{59}K^{a, b} shows to be constant, but does not decrease below a superconducting transition. These behaviors as well as the invariance of the internal field at the Co site in the superconducting state exclude the spin-singlet pairing, and can be interpreted with the equal-spin pai…
We study the interplay of the ferromagnetic (FM) state and the p-wave superconducting (SC) state observed in several materials such as UCoGe and URhGe in a totally nonperturbative manner. To this end, we introduce a lattice Ginzburg-Landau model that is a genuine generalization of the phenomenological Ginzburg-Landau theory proposed previously in the continuum and also a counterpart of the lattice gauge-Higgs model for the s-wave SC transition, and study it numerically by Monte-Carlo simulations. The obtained phase diagram has qualitatively the same structure as that of UCoGe in the region whe…
After reminding of properties of the condensate of the complex scalar field in the external uniform magnetic field $H$ focus is made on the study of phases of the complex neutral vector boson fields coupled with magnetic field by the Zeeman coupling and phases of the charged vector boson fields. These systems may behave as nonmagnetic and ferromagnetic superfluids and ordinary and ferromagnetic superconductors. Response of the ferromagnetic superfluid and superconducting systems occupying half of space on the external uniform static magnetic field $H$ is thoroughly studied. Then the spin-tripl…
Different models for doping of two-orbital chains with mobile $S=1/2$ fermions and strong, ferromagnetic (FM) Hund's rule couplings stabilizing the S=1 spins are investigated by density matrix renormalization group (DMRG) methods. The competition between antiferromagnetic (AF) and FM order leads to a rich phase diagram with a narrow FM region for weak AF couplings and strongly enhanced triplet pairing correlations. Without a level difference between the orbitals, the spin gap persists upon doping, whereas gapless spin excitations are generated by interactions among itinerant polarons in the pr…
In quantum materials, the electronic interaction and the electron-phonon coupling are, in general, two essential ingredients, the combined impact of which may drive exotic phases. Recently, an anomalously strong electron-electron attraction, mediated by phonons, has been unveiled in one-dimensional copper-oxide chain Ba$_{2-x}$Sr$_x$CuO$_{3+δ}$. Yet, it is unclear how this strong near-neighbor attraction $V$ influences the superconductivity pairing in the compound. Here we perform accurate many-body calculations to study the extended Hubbard model with on-site Coulomb repulsion $U>0$ and attra…
Motivated by the presence of various charge inhomogeneities in strongly correlated systems of coupled ladders, a model of spatially separated bosonic and fermionic degrees of freedom is numerically studied. In this model, bosonic chains are connected to fermionic chains by two types of generalized Andreev couplings. It is shown that for both types of couplings the long-distance pairing correlations are enhanced. Near quarter filling, this effect is much larger for the splitting of a pair in electrons which go to the two neighboring fermionic chains than for a pair hopping process. It is argued…
Based on the random phase approximation calculation in two-orbital honeycomb lattice model, we investigate the pairing symmetry of Ni-based transition-metal trichalcogenides by electron doping access to type-II van Hove singularities (vHs). We find that chiral even-parity d+id-wave (Eg) state is suppressed by odd-parity p+ip-wave (Eu) state when electron doping approaches the type-II vHs. The type-II vHs peak in density of states (DOS) enables to strengthen the ferromagnetic fluctuation, which is responsible for triplet pairing. The competition between antiferromagnetic and ferromagnetic fluct…
We study the point-contact tunneling between normal metal and ferromagnetic superconductor. In the case of magnon-induced pairing the tunneling conductance is continuous and smooth function of the applied voltage. For small values of the applied voltage the Ohm law holds. We show that one can obtain the magnetization and the superconducting order parameter from the tunneling conduc- tance. In the case of paramagnon-induced superconductivity the tunneling does not depend on the magnetization. We argue that tunneling experiment can unambiguously determine the correct pairing mechanism in the fer…
We study the sodium-iridates model on the honeycomb lattice with both BCS pairing potential and Hubbard interaction term. It is shown that this model can be exactly solved with appropriate choices of amplitude of pairing gaps, where the interacting terms are transformed to external field terms. The band structures of these exact solutions on both torus and cylinder geometry are discussed in great details. It is found that the ground state assumes an anti-ferromagnetic configuration, which breaks the time reversal symmetry spontaneously and renders the superconductor topologically trivial. On t…
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…
A general phenomenological theory is presented for the phase behavior of ferromagnetic superconductors with spin-triplet electron Cooper pairing. The theory accounts in detail for the temperature-pressure phase diagram of ZrZn$_2$, while the main features of the diagram for UGe$_2$ are also described. Quantitative criteria are deduced for the U-type (type I) and Zr-type (type II) unconventional ferromagnetic superconductors with spin-triplet Cooper electron pairing. Some basic properties of quantum phase transitions are also elucidated.
A theory of tunneling conductance in ferromagnetic metal/insulator/triplet - supercondcutor junctions is presented for unitary and non-unitary spin triplet pairing states which are promising candidates for the superconducting paring symmetry of Sr2RuO4. As the magnitude of the exchange interaction in the ferromagnetic metal is increased, the conductance for the unitary pairing state below the energy gap is reduced in contrast to the case for the non-unitary pairing state.
We perform a numerical simulation of a three-band Hubbard model with two CuO$_2$ planes and a single CuO chain layer for YBCO cuprates. The spin-fluctuation mediated pairing interaction is computed within the multiband random-phase approximation, and its pairing eigenvalues and eigenfunctions are solved as a function of chain state filling factor $n_c$. We find that for the intrinsic value of $n_c$ in YBCO samples, one obtains the usual $d$-wave pairing symmetry. However, if we dope the chain layers with holes, while keeping the plane states doping fixed, the leading pairing symmetry solution …
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 have developed the quasi-classical formalism for a self-induced vortex state in ferromagnetic superconductors. By combining the spatially averaged approximation of the superconductivity with the phenomenological form of the ferromagnetism, we demonstrate the thermodynamic properties of the p-wave triplet equal-spin-pairing state with the spontaneous vortex lattice. The occurrence condition for the self-induced vortex state is discussed within this formalism. The comparison of the calculated quantities with the observed ones indicates that the self-induced vortex state indeed occurs in UCoGe…
We study the effective interactions between Dirac fermions on the surface of a three-dimensional topological insulator due to the proximity coupling to the magnetic fluctuations in a ferromagnetic or antiferromagnetic insulator. Our results show that the magnetic fluctuations can mediate attractive interactions between Dirac fermions of both Amperean and BCS type. In the ferromagnetic case, we find pairing between fermions with parallel momenta, so-called Amperean pairing, whenever the effective Lagrangian for the magnetic fluctuations does not contain a quadratic term. The pairing interaction…
We propose a graphene-based model for realizing a new type of gapless condensate by pairing of electron-like (n) carriers of a Dirac cone conduction band with hole-like (p) carriers of a Dirac valance band. Ferromagnetic superconductivity (FS) in monolayer graphene or pairing between oppositely (n and p) doped layers of a double layer graphene allow for the formation of this p-n superconductivity. For FS in graphene, the p-n condensate dominates the zero temperature phase diagram at low levels of doping and high exchange fields. We show that p-n pairing with p+ip-wave symmetry presents a stabl…
The long-range spin-triplet supercurrent transport is an interesting phenomenon in the superconductor/ferromagnet ($S/F$) heterostructure containing noncollinear magnetic domains. Here we study the long-range superharmonic Josephson current in asymmetric $S/F_1/F_2/S$ junctions. It is demonstrated that this current is induced by spin-triplet pairs $\mid\uparrow\uparrow\rangle-$$\mid\downarrow\downarrow\rangle$ or $\mid\uparrow\uparrow\rangle+$$\mid\downarrow\downarrow\rangle$ in the thick $F_1$ layer. The magnetic rotation of the particularly thin $F_2$ layer will not only modulate the amplitu…
Majorana bound states (MBS) are well-established in the clean limit in chains of ferromagnetically aligned impurities deposited on conventional superconductors with finite spin-orbit coupling. Here we show that these MBS are very robust against disorder. By performing self-consistent calculations we find that the MBS are protected as long as the surrounding superconductor show no large signs of inhomogeneity. We find that longer chains offer more stability against disorder for the MBS, albeit the minigap decreases, as do increasing strengths of spin-orbit coupling and superconductivity.
To explore whether a flat-band system can accommodate superconductivity, we consider repulsively interacting fermions on the diamond chain, a simplest quasi-one-dimensional system that contains a flat band. Exact diagonalization and the density-matrix renormalization group (DMRG) are used to show that we have a significant binding energy of a Cooper pair with a long-tailed pair-pair correlation in real space when the total band filling is slightly below $1/3$, where the dispersive band interacts with the flat band that is empty but close to $E_F$. Pairs selectively formed across the outer site…
We propose and analyze a generalization of the Kitaev chain for fermions with long-range $p$-wave pairing, which decays with distance as a power-law with exponent $α$. Using the integrability of the model, we demonstrate the existence of two types of gapped regimes, where correlation functions decay exponentially at short range and algebraically at long range ($α> 1$) or purely algebraically ($α< 1$). Most interestingly, along the critical lines, long-range pairing is found to break conformal symmetry for sufficiently small $α$. This is accompanied by a violation of the area law for the entang…
Recent experiments have provided evidence that one-dimensional (1D) topological superconductivity can be realized experimentally by placing transition metal atoms that form a ferromagnetic chain on a superconducting substrate. We address some properties of this type of systems by using a Slater-Koster tight-binding model. We predict that topological superconductivity is nearly universal when ferromagnetic transition metal chains form straight lines on superconducting substrates and that it is possible for more complex chain structures. The proximity induced superconducting gap is $\sim ΔE_{so}…
Majorana fermions are predicted to localize at the edge of a topological superconductor, a state of matter that can form when a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin-orbit interaction. With the goal of realizing a one-dimensional topological superconductor, we have fabricated ferromagnetic iron (Fe) atomic chains on the surface of superconducting lead (Pb). Using high-resolution spectroscopic imaging techniques, we show that the onset of superconductivity, which gaps the electronic density of states in the bulk of the Fe chains, is accomp…
