Triplet pairing, orbital selectivity and correlations in iron-based superconductors

A major open question in the theory of strongly correlated electronic systems is the origin of the superconductivity in iron-based superconductors (FeSC) [1]. Since their discovery more than a decade ago, a large body of data has been collected which has largely been interpreted in the framework of $s_{\pm}$ singlet superconductivity [2, 3]. A widely accepted paradigm for the pairing in these materials, based on itinerant electrons in the presence of an attractive pairing glue [4, 5] that binds them into Cooper pairs, has been successful in explaining a subset of this data. There is, however, less consensus on the origin of the pairing glue. Various exchange bosons have been suggested, from spin, nematic and orbital fluctuations to a combination thereof [6, 7]. At the same time, as we expand upon below, a variety of new experimental data has shown signatures of an interplay between complex orbital and local moment physics [8] with superconductivity. The consistency of these findings with the paradigm of electrons subject to a fluctuation-mediated pairing glue is currently unclear.

Considering the complexity of this multi-orbital system and presence of many competing orders and putative critical points, the pursuit of a single origin for the superconductivity may be fraught with difficulty. A more modest goal is to seek the minimal ingredients for superconductivity in FeSC. In a recent paper [9] we suggested that Hund’s coupling is a the driver [10] of the superconductivity, giving rise to an orbitally-odd spin-triplet superconductor. Although itinerant triplet pairing in iron-based SC was discussed early on by Wen and Lee [11] this idea was quickly dismissed due to the observation of a Knight shift and the absence of nodes on the Fermi surface (FS). Triplet pairing in iron-based superconductors has recently been re-visited [12, 13, 14, 15] in the context of a possible interplay between pairing and the topological bandstructure.

Given the pre-dominant support for an $s_{\pm}$ singlet pairing scenario, it is instructive to re-iterate the motivation for a triplet state. Dynamical mean-field theory (DMFT) [16] has long shown that the Hund’s interaction plays a dominant role in the correlations of the normal state [17, 18, 19, 20, 21, 22], a picture that is also confirmed in slave-spin mean-field studies [21]. The basic idea is that coherent metallicity in DMFT can be understood within a self-consistent impurity framework, where the underlying physics is akin to Kondo screening of an equivalent impurity problem. Hund’s interactions tend to force electrons into high-spin configurations which are hard to screen, thus effectively suppressing the Kondo coherence to low temperatures [23, 24]. At the same time, the orbital degrees of freedom can be quenched at higher temperature creating a spin-orbital separation (SOS) regime [25, 26]. A related phenomenon is the spin freezing effect [27], where under orbital symmetry the spin correlation functions do not decay in time. Therefore, unhindered by the hybridization and down to lowest temperatures, the Hund’s interaction is the dominant interaction whose pair-hopping effects can give rise to superconductivity. Indeed, the authors in [28] proposed spatially isotropic triplet pairing for SrRuO₃ in the spin-freezing regime.

An important experimental development is the realization that FeSCs appear to display a universal yet non-BCS ratio of maximal zero-temperature gap and transition temperature [29] $2\Delta/T_{C}\sim 7.2$. Given the wide variety of different Fermi surface topologies in the large FeSC family, this apparent universality suggests that superconductivity may be intimately linked to the only common feature they share - namely, the local crystalline environment of the Fe ion. Indeed the universal $2\Delta/T_{C}$ ratio can be explained [30] semi-phenomenologically by local critical spin-fluctuations of the form $-\chi^{\prime\prime}/\omega\sim\omega^{-\gamma}$ and $\gamma=1.2$. Pairing by power-law-spin fluctuations is a problem of substantial present-day interest [31], partly due to its interconnection to Sachdev-Ye-Kitaev models [32]. The value of $\gamma=1.2$ relevant to the observed $\Delta/T_{c}$ in FeSC occurs in DMFT solutions of Hund’s metals [20, 25], and (approximately) in the aforementioned SOS regime of multi-orbital [33] and mixed-valent Hund’s impurities.[34] Combining experimental observations with the above-mentioned Hund’s phenomenology, naturally leads to the question of investigating the possibility of triplet superconductivity in FeSC.

Another pertinent aspect of these materials concerns the role of local atomic orbitals in the development of orbital selective Mott phases (OSMP) [35], orbital selective pairing [36], orbital loop currents [37] and the pairing enhanced by orbital fluctuations [38]. In particular, the angle-resolved photo-emission spectroscopy (ARPES) has enabled orbitally resolved study of the band-structure. The strong orbital selectivity, first predicted by DMFT [20] has been confirmed by ARPES [39, 40]. Certainly, by focusing on the Fermi surface the entirely itinerant approach picture misses the orbital degrees of freedom of the local physics.

A further question of importance in the context of superconductivity is the local Coulomb repulsion which tends to enforce the anomalous Green’s function to vanish at equal points in space and time. Phonon-mediated superconductivity satisfies this Coulomb constraint by exploiting the strong retardation of the electron phonon interaction, to build in nodes in frequency space [41, 42]. By contrast, non-BCS pairing of strongly correlated fermions typically involves a finite angular momentum of Cooper pair, which is symmetry protected against onsite repulsion [43, 44]. The theory of electron-interaction-mediated $s_{\pm}$ in FeSC is criticized [45] for realizing neither of these established escape-routes.

A conceptually different paradigm which may be easier to reconcile with the orbital complexity of this material is the picture of pre-formed pairs [46, 47, 48] as in the theory of resonating valence bonds in a doped Mott insulator [49]. The idea of Hund’s driven spin-triplet superconductivity in strongly correlated materials was first discussed by Anderson [50, 51, 52] who, in the context of heavy-fermions, argued that an odd-parity of the Cooper pair requires at least two atoms per unit cell with a center of inversion between the two, an ingredient that is satisfied in FeSC. Recently, the idea of triplet pairing resurfaced in the context of ferromagnetic systems, with the observation of strange metal phase in a pure heavy-fermion CeRh₆Ge₄ [53]. In this material, a magnetic easy-plane anisotropy was argued to lead to presence of triplet resonating valence bonds (tRVB) in the ground state and a highly entangled ordered phase. It was subsequently shown that doping such a tRVB host can lead to odd-parity spin-triplet superconductivity [54, 55].

Recently, we proposed [9] that in FeSC the Hund’s induced electron triplets resonate between various orbitals of each atom. To stabilize of a non-zero superconducting parameter order, Anderson’s above-mentioned argument about the two atoms per unit cell proves critical. We argued that a tRVB state is consistent with the body of available experimental data, from Knight shift [56], quasi-particle interference [57, 58] and neutron spin-resonance measurements [9]. Moreover, a tRVB order parameter is relatively robust against disorder and predicts staggered component to the pair wavefunction that, we argued, will be discernible using scanning Josephson spectroscopy. The goal of the present paper is to extend these early studies, contrasting the orbital complexity and the effects of strong intra-orbital Coulomb repulsion in the tRVB and $s_{\pm}$ states within these two perspectives.

To this end, we study the (orbital selective) band renormalization and superconducting instability of the normal state of FeSC under strong on-site Coulomb interaction. In the trade-off between complexity and tractability, we choose the simplest possible model, i.e. a two-dimensional three-band model of layered FeSC, which seem to capture the main physics. We also assume absent nematicity and use slave-bosons [59, 60] to represent the Hubbard operators in the limit of infinite intra-orbital repulsion, which are treated via mean-field theory. Our approach connects previous slave-boson works of [61, 62] and more recent slave-spin approaches [63].

The content of the paper is as follows: in section II we introduce the model and the decoupling of the Hund’s interaction. Section III describes how the slave-bosons are used to treat the intra-orbital Coulomb repulsion and the Hund’s induced renormalization of the inter-orbital Coulomb interaction. Section IV contains the mean-field analysis of the interacting Hamiltonian and an analysis of band-renormalization, orbital selectivity and pair susceptibility within the mean-field theory.

The model is described by the Hamiltonian

where the non-interacting Hamiltonian matrix is

Here, $\alpha,\beta=\uparrow,\downarrow$ are spin, $i,j$ site, and $\mu,\nu=xy,xz,yz$ are the orbital indices (the ladder within the d-shell of Fe). Furthermore, $\sigma^{a}_{\alpha\beta}$ are Pauli matrices in spin space and $L^{a}_{\mu\nu}=-i\epsilon_{a\mu\nu}$ are three totally anti-symmetric matrices in orbital space. Throughout this paper, the upper/lower position of indices is equivalent. We use a three band model [64] expressed in the original two atom per unit cell basis (cf. Appendix A). The atomic spin-orbit coupling $\lambda_{S}$ can be regarded as a spin-dependent inter-orbital hopping. The interaction is on-site and using the notation $\bar{\delta}_{\mu\nu}=1-\delta_{\mu\nu}$, can be written as

in terms of $n_{j\mu\sigma}=c^{\dagger}_{j\mu\sigma}c^{\vphantom{\dagger}}_{j\mu\sigma}$ and $\vec{S}_{j\mu}=\frac{1}{2}c^{\dagger}_{j\mu\alpha}\vec{\sigma}_{\alpha\beta}c^{\vphantom{\dagger}}_{j\mu\beta}$. The interaction $H_{\rm int}$ contains an intra-orbital Coulomb interaction $U$, an inter-orbital part $U^{\prime}$ as well as a local Hund’s interaction $J_{H}>0$, which is an intra-atomic spin-spin interaction which favors higher-spin states.

The largest energy scale in $H_{\rm int}$ is the intra-orbital Coulomb repulsion $U\sim$1-5eV. In terms of $U$, the remaining parameters have the typical hierarchy $U^{\prime}\sim U/4$, $t\sim J_{H}\sim U/10$ and $\lambda_{S}\sim U/100$, to be compared with a typical iron-based superconducting transition temperature $T_{c}\sim U/1000\sim$ 10-50K. Our strategy is to study a simplified limit of the problem in which the intra-orbital $U$ is sent to infinity.

After the onsite Coulomb interaction, the most important interaction term is the Hund’s interaction. We now re-write this term in terms of the inter-orbital triplet interactions. By using a modified Fierz identity (Appendix B):

we can decouple the Hund’s interaction in the triplet channel. Restoring the fermions by multiplying this identity by $c^{\dagger}_{j\mu\alpha}c^{\dagger}_{j\nu\alpha^{\prime}}$ on the left and $c_{j\nu\beta^{\prime}}c_{j\mu\beta}$ on the right, the Hund’s interaction can be written as

Here we use the notation $c^{*}_{j\nu}\equiv(c^{\dagger}_{j\nu})^{T}$ to denote the transpose of the creation operator and we have rewritten the Fermion bi-linears in terms of the triplet pair and particle-hole operators

We note that the anti-commutation properties of the fermion operators enforce an orbital antisymmetry on the local triplet pair operators $\vec{\Psi}_{\mu\nu}(j)=-\vec{\Psi}_{\nu\mu}(j)$, whereas the particle-hole operators are Hermitian, $(\vec{\Phi}_{\mu\nu})^{\dagger}=\vec{\Phi}_{\nu\mu}$. Both interaction channels in Eq. (II) are attractive and can acquire an expectation value for ferromagnetic Hund’s coupling. In a bulk system, thee triplet operators can condense into a ferromagnetic ground state [54] similar to the way short-range singlet RVBs can exhibit long-range AFM order [65].

It is convenient to decompose the triplet pair and particle-hole operators as follows:

Here the $\lambda^{A}_{\mu\nu}$ are the eight Gell-Mann matrices. The orbital-antisymmetry of the pair operators means that only the three antisymmetric Gell-Mann matrices appear in the triplet pair operators $\vec{\Psi}_{\mu\nu}$, denoted by the angular momentum operators $L^{a}_{\mu\nu}=-i\epsilon_{a\mu\nu}\equiv(\lambda^{7},-\lambda^{5},\lambda^{2})$. We can rewrite

where the magnetic vectors $\vec{\Phi}_{A}=\vec{\Phi}^{\dagger}_{A}$ are real. If we combine the three vectors $\vec{\Psi}_{a}$ into a three-dimensional matrix $\Psi_{ab}=(\vec{\Psi}_{a})_{b}$, and similarly, denote $(\vec{\Phi}_{A})_{b}=\Phi_{Ab}$, then the Hund’s interaction can be written as

Carrying out a Hubbard Stratonovich transformation, we can decouple the Hund’s interaction in terms of magnetic order parameters $\Lambda_{Ab}=(\vec{\Lambda}_{A})_{b}$ and a triplet gap matrix $\Delta_{ab}=(\vec{\Delta}_{a})_{b}$ as follows

where $g=J_{H}/2$ is the bare coupling constant.

Under renormalization, the above interaction is expected to develop anisotropies. Moreover, the magnetic and the triplet channels will behave differently, since the particle-hole triplet channel associated with $\Phi$ couples to the spin-orbit interaction (SOI) and the particle-hole channels are un-nested, whereas the presence of center of symmetry between the two iron-atoms per unit cell in the iron-based superconductors means that the triplet Cooper channel will couple to the Fermi surface, and will undergo a logarithmic renormalization. In [9], it was shown that the effects of spin-orbit coupling at an atomic level create anisotropies in the above pairing interactions, favoring diagonal gap functions, $\Gamma_{ab}\sim{\rm diag}(1,1,1)$, and $\Delta_{ab}\sim{\rm diag}(1,1,-2)$. The latter is an example of a triplet resonating valence bond (tRVB) state. In the next section, we introduce slave-bosons to study this problem in the $U\to\infty$ limit, but first we comment on the use of the reduced three band model.

Since the reduced band in the parent compound has the occupation of $n_{e}=4$ among three orbitals, at least one orbital is doubly occupied. The electron annihilation operator is represented by $c_{j\mu\sigma}$ where $\mu=xz,yz,xy$ is the orbital index, $\sigma=\uparrow,\downarrow$ is the spin index and $j$ is the site index. In order to study doping of the parent compound in an infinite-$U$ problem, we choose to preserve the doubly occupied states of the $\mu=xz,yz$ orbitals (“doublons”) and the empty state (“holon”) of the $xy$ orbital. In other words, we discard empty $xz/yz$ orbitals and doubly occupied $xy$ orbitals, so that the fermionic Hubbard operators are represented as

where $\tilde{\sigma}={\rm sign}{(}\sigma)$. These representations are subject to the constraints

where $n_{j\mu}^{b}=b_{j\mu}^{\dagger}b^{\vphantom{\dagger}}_{j\mu}$ and $n^{f}_{j\mu}=\sum_{\sigma}f^{\dagger}_{j\mu\sigma}f^{\vphantom{\dagger}}_{j\mu\sigma}$. The number of physical electrons is $n_{\mu}=n^{f}_{\mu}+2n^{b}_{\mu}$ for $\mu=xz,yz$ and $n_{\mu}=n^{f}_{\mu}$ for $\mu=xy$, so that the total number of electrons per site is then

where we have imposed the constraint (13) in the last step. In the mean-field theory, we adjust the overall chemical potential so that the average number of electrons per site is $n_{e}=4$, i.e. by (14),

The constraints (13-16) are imposed via separate Lagrange multipliers. A consequence of these constraints is that $n_{xy}^{f}=n_{xz}^{f}+n_{yz}^{f}$. Assuming that xy is dominated by electrons and $xz/yz$ by holes, this would indicate fully compensated electron-hole pockets at the FS.

The inter-orbital Coulomb interaction remains as in Eq. (3) but we may use the constraint to express it entirely in terms of $n^{b}_{j\mu}$. The $U^{\prime}$ terms drive various phenomena [64] including the nematic phase, which is not considered in this paper. Within mean-field theory

and the $U^{\prime}$ is completely absorbed by shifting the chemical potential and the Lagrange multiplier used to impose Eq. (16). In other words within mean-field theory, physical quantities expressed in terms of densities are insensitive to the value of the $U^{\prime}$ interaction. Beyond mean-field theory, we will argue in the next section that in the Hund’s dominated regime, inter-orbital repulsion is renormalized to smaller values by spin-fluctuations.

For future convenience we collect the bosons into $\tilde{b}$ and spinons into $\tilde{f}$ such that $c_{j\mu\sigma}=\tilde{b}^{\vphantom{\dagger}}_{j\mu}\tilde{f}^{\dagger}_{j\mu\sigma}$ for all $\mu$.

A mean-field decoupling of the slave-boson Hamiltonian leads to

The coefficients ${\cal H}^{b}$ and ${\cal H}^{f}$ are chosen so that

leading to a set of self-consistent equations. Limiting ourselves to the normal state and assuming absence of nematicity, we have self-consistently solved these equations in momentum space, imposing the constraints (13,16).

This enables us to study the mean-field Hamiltonian beyond the single-site approximation used in earlier slave-spin [63, 67, 68] or DMFT approaches, allowing us to address the possibility of orbitally selective Mott transitions in the presence of inter-orbital hopping. Our results for the band renormalizations and pair susceptibility are summarized in the next two sections, while the technical details of the calculation are discussed in Appendix G.

Both tRVB and $s_{\pm}$ states show a superconducting dome around the doping $n_{xy}\approx 0.3$ where electrons in the $xy$ orbital are delocalized. In the presence of realistic spin-orbit coupling, the mean-field pair susceptibilities of the tRVB state are enhanced by a factor of about five. Moreover, our mean field theory demonstrates a clear correlation between increasing $xy$-orbital localization and decreasing $T_{c}$ for both superconducting states, a feature which is consistent with experimental observations.

Such correlations between $xy$ occupation and $T_{c}$ is observed in FeTe_1-xSe_x [71] which exhibits an anti-ferromagnetic (AFM) order at $x=0$. As $x$ is varied [71] from FeTe to FeSe, the antiferromagnetism disappears at about $x=0.1$ and superconductivity develops at $x>0.25$ with $T_{c}\sim 10K$ which depends only weakly on doping in the range $x\in(0.3,0.5)$. A particularly fascinating feature, is that the renormalization of the $xy$ band, as determined by the effective mass, diverges as $x\to 0.2$, so that the $T_{c}$ and bandwidth of the $xy$ orbital are correlated at low doping, the latter strongly depending on the temperature. It is plausible that as the $xy$ electrons localize, they produce the AFM order [72].

In summary, we have studied the pair susceptibility of the iron-based superconductors, taking into account the effects of correlations, orbital selectivity and Hund’s interaction. The influence on band renormalization, evolution of orbitals and Fermi surface reconstruction close to the orbital selective Mott transition is captured. Away from the OSMT, a mostly electron-hole compensated Fermi surface develops. We have argued the importance of including charge projectors in the decoupling of the interaction which enable a study of pair susceptibility in terms of physical electrons. Furthermore, this provides a Hund’s driven mechanism for renormalizing down the inter-orbital charge repulsion via spin-fluctuation.

We have performed this calculation for the tRVB state to study the effects of orbital localization and compare it to the s_± state. We employed a three-band tight-binding model for this calculation taking into account both intra-band and inter-band contributions to the susceptibility. Both states show a superconducting dome around $n_{xy}\sim 0.5$, but the tRVB state also includes a much weaker superconducting dome close to the orbitally selective Mott phase at $n_{xy}\sim 1$ due to inter-band contributions.