The development of sensible microscopic models is essential to elucidate the normal-state and superconducting properties of the iron-based superconductors. Because these materials are mostly metallic, a good starting point is an effective low-energy model that captures the electronic states near the Fermi level and their interactions. However, in contrast to cuprates, iron-based high-$T_{c}$ compounds are multi-orbital systems with Hubbard and Hund interactions, resulting in a rather involved 10-orbital lattice model. Here we review different minimal models that have been proposed to unveil the universal features of these systems. We first review minimal models defined solely in the orbital basis, which focus on a particular subspace of orbitals, or solely in the band basis, which rely only on the geometry of the Fermi surface. The former, while providing important qualitative insight into the role of the orbital degrees of freedom, do not distinguish between high-energy and low-energy sectors and, for this reason, generally do not go beyond mean-field. The latter allow one to go beyond mean-field and investigate the interplay between superconducting and magnetic orders as well as Ising-nematic order. However, they cannot capture orbital-dependent features like spontaneous orbital order. We then review recent proposals for a minimal model that operates in the band basis but fully incorporates the orbital composition and symmetries of the low-energy excitations. We discuss the results of the renormalization group study of such a model, particularly of the interplay between superconductivity, magnetism, and spontaneous orbital order, and compare theoretical predictions with experiments on iron pnictides and chalcogenides. We also discuss the impact of the glide-plane symmetry on the low-energy models, highlighting the key role played by the spin-orbit coupling.

Low-energy microscopic models for iron-based superconductors: a review

The development of sensible microscopic models is essential to elucidate the normal-state and superconducting properties of the iron-based superconductors. Because these materials are mostly metallic, a good starting point is an effective low-energy model that captures the electronic states near the Fermi level and their interactions. However, in contrast to cuprates, iron-based high-T c compounds are multi-orbital systems with Hubbard and Hund interactions, resulting in a rather involved 10-orbital lattice model. Here we review different minimal models that have been proposed to unveil the universal features of these systems. We first review minimal models defined solely in the orbital basis, which focus on a particular subspace of orbitals, or solely in the band basis, which rely only on the geometry of the Fermi surface. The former, while providing important qualitative insight into the role of the orbital degrees of freedom, do not distinguish between high-energy and low-energy sectors and, for this reason, generally do not go beyond mean-field. The latter allow one to go beyond mean-field and investigate the interplay between superconducting and magnetic orders as well as Ising-nematic order. However, they cannot capture orbital-dependent features like spontaneous orbital order. We then review recent proposals for a minimal model that operates in the band basis but fully incorporates the orbital composition and symmetries of the low-energy excitations. We discuss the results of the renormalization group study of such a model, particularly of the interplay between superconductivity, magnetism, and spontaneous orbital order, and compare theoretical predictions with experiments on iron pnictides and chalcogenides. We also discuss the impact of the glide-plane symmetry on the low-energy models, highlighting the key role played by the spin-orbit coupling.

https://iopscience.iop.org/article/10.1088/1361-6633/80/1/014503/pdf

Magnetism is widely considered to be a key ingredient of unconventional superconductivity. In contrast to cuprate high-temperature superconductors, antiferromagnetism in most Fe-based superconductors (FeSCs) is characterized by a pair of magnetic propagation vectors, (π,0) and (0,π). Consequently, three different types of magnetic order are possible. Of these, only stripe-type spin-density wave (SSDW) and spin-charge-density wave (SCDW) orders have been observed. A realization of the proposed spin-vortex crystal (SVC) order is noticeably absent. We report a magnetic phase consistent with the hedgehog variation of SVC order in Ni-doped and Co-doped CaKFe4As4 based on thermodynamic, transport, structural and local magnetic probes combined with symmetry analysis. The exotic SVC phase is stabilized by the reduced symmetry of the CaKFe4As4 structure. Our results suggest that the possible magnetic ground states in FeSCs have very similar energies, providing an enlarged configuration space for magnetic fluctuations to promote high-temperature superconductivity. The magnetic texture of a new superconductor adopts a in-out spin, spin-vortex crystal motif, fulfilling theoretical predictions. Many iron-based superconductors have magnetic phases arising from combining two basic magnetic structures, but only two of three possible combinations had previously been observed. A team led by Paul Canfield of Iowa State University and Ames Laboratory have synthesised a material with the third type of magnetic structure called a hedgehog spin-vortex crystal. The authors began with a compound with spatial symmetry that could help stabilise the structure, but without magnetic order. By tuning the chemical composition they induced magnetism and successfully obtained the desired phase. The sensitivity of the magnetic state to the symmetry and composition indicates that different phases are energetically close, suggesting magnetic fluctuations may play a significant role in the physics of iron-based superconductors.

/pdf/hedgehog-spin-vortex-crystal-stabilized-in-a-hole-doped-iron-2wo6dy7r2k.pdf

Hedgehog spin-vortex crystal stabilized in a hole-doped iron-based superconductor

Uneven Gap Electron pairs that are responsible for the phenomenon of superconductivity can only be broken by investing a finite amount of energy, called the energy gap. The size of the gap may depend on the position on the Fermi surface; in cuprates, the gap completely disappears at certain points. What happens in the pnictide superconductors is still a subject of debate, not least because there appear to be differences between the different pnictide families. Allan et al. (p. 563) used scanning tunneling spectroscopy to study the compound LiFeAs. The gap was mapped on three of the five bands on which the Fermi surface resides and was found to be anisotropic in momentum space. The energy needed to break up electron pairs in a pnictide superconductor depends on position on the Fermi surface. If strong electron-electron interactions between neighboring Fe atoms mediate the Cooper pairing in iron-pnictide superconductors, then specific and distinct anisotropic superconducting energy gaps Δi(k⃗) should appear on the different electronic bands i. Here, we introduce intraband Bogoliubov quasiparticle scattering interference (QPI) techniques for determination of Δik⃗. in such materials, focusing on lithium iron arsenide (LiFeAs). We identify the three hole-like bands assigned previously as γ, α2, and α1, and we determine the anisotropy, magnitude, and relative orientations of their Δik⃗. These measurements will advance quantitative theoretical analysis of the mechanism of Cooper pairing in iron-based superconductivity.

/pdf/anisotropic-energy-gaps-of-iron-based-superconductivity-from-1z88b282o2.pdf

Anisotropic Energy-Gaps of Iron-based Superconductivity from Intra-band Quasiparticle Interference in LiFeAs, Part I

Magnetism is widely considered to be a key ingredient of unconventional superconductivity. In contrast to cuprate high-temperature superconductors, antiferromagnetism in Fe-based superconductors (FeSCs) is characterized by a pair of magnetic propagation vectors. Consequently, three different types of magnetic order are possible. Of theses, only stripe-type spin-density wave (SSDW) and spin-charge-density wave (SCDW) orders have been observed. A realization of the proposed spin-vortex crystal (SVC) order is noticeably absent. We report a magnetic phase consistent with the hedgehog variation of SVC order in Ni- and Co-doped CaKFe4As4 based on thermodynamic, transport, structural and local magnetic probes combined with symmetry analysis. The exotic SVC phase is stabilized by the reduced symmetry of the CaKFe4As4 structure. Our results suggest that the possible magnetic ground states in FeSCs have very similar energies, providing an enlarged configuration space for magnetic fluctuations to promote high-temperature superconductivity.

/pdf/hedgehog-spin-vortex-crystal-stabilized-in-a-hole-doped-iron-3ocxj6e17m.pdf

Density-wave instabilities have been observed and studied in a multitude of materials. Most recently, in the context of unconventional superconductors like the iron-based superconductors, they have excited considerable interest. We analyze the fluctuation corrections to the equation of state of the density-wave order parameter for commensurate charge-density waves and spin-density waves due to perfect nesting. For XY magnets, we find that contributions due to longitudinal and transverse fluctuations cancel each other, making the mean-field analysis of the problem controlled. This is consistent with the analysis of fluctuation corrections to the BCS theory of superconductivity [\ifmmode \check{S}\else \v{S}\fi{}. Kos, A. J. Millis, and A. I. Larkin, Phys. Rev. B 70, 214531 (2004)]. However, this cancellation does not occur in density-wave systems when the order parameter is a real $N$-component object with $N\ensuremath{\ne}2$. Then, the number of transverse fluctuating modes differs from the number of longitudinal fluctuating modes. Indeed, in the case of charge-density waves as well as spin-density waves with Heisenberg symmetry, we find that fluctuation corrections are not negligible, and hence mean-field theories are not justified. These singular fluctuations originate from the intermediate length-scale regime, with wavelengths between the lattice constant and the $T=0$ correlation length. The resulting logarithmic fluctuation contributions to the gap equation originate from the derivative of the anomalous polarization function, and the crucial process is an interaction of quasiparticles through the exchange of fluctuations.

Role of fluctuations for density-wave instabilities: Failure of the mean-field description

Through the Higgs mechanism, the long-range Coulomb interaction eliminates the low-energy Goldstone phase mode in superconductors and transfers spectral weight all the way up to the plasma frequency. Here, we show that the Higgs mechanism breaks down for length scales shorter than the superconducting coherence length while it stays intact, even at high energies, in the long-wavelength limit. This effect is a consequence of the composite nature of the Higgs field of superconductivity and the broken Lorentz invariance in a solid. Most importantly, the breakdown of the Higgs mechanism inside the superconducting coherence volume is crucial to ensure the stability of the BCS mean-field theory in the weak-coupling limit. We also show that changes in the gap equation due to plasmon-induced fluctuations can lead to significant corrections to the mean-field theory and reveal that changes in the density-fluctuation spectrum of a superconductor are not limited to the vicinity of the gap.

Short-distance breakdown of the Higgs mechanism and the robustness of the BCS theory for charged superconductors

In most iron-based superconductors, the transition to the magnetically ordered state is closely linked to a lowering of structural symmetry from tetragonal $({C}_{4})$ to orthorhombic $({C}_{2})$. However, recently, a regime of ${C}_{4}$-symmetric magnetic order has been reported in certain hole-doped iron-based superconductors. This novel magnetic ground state can be understood as a double-$\mathbf{Q}$ spin density wave characterized by two order parameters ${\mathbf{M}}_{1}$ and ${\mathbf{M}}_{2}$ related to each of the two $\mathbf{Q}$ vectors. Depending on the relative orientations of the order parameters, either a noncollinear spin-vortex crystal or a nonuniform charge-spin density wave could form. Experimentally, M\"ossbauer spectroscopy, neutron scattering, and muon spin rotation established the latter as the magnetic configuration of some of these optimally hole-doped iron-based superconductors. Theoretically, low-energy itinerant models do support a transition from single-$\mathbf{Q}$ to double-$\mathbf{Q}$ magnetic order, but with nearly degenerate spin-vortex crystal and charge-spin density wave states. In fact, extensions of these low-energy models including additional electronic interactions tip the balance in favor of the spin-vortex crystal, in apparent contradiction with the recent experimental findings. In this paper we revisit the phase diagram of magnetic ground states of low-energy multiband models in the presence of weak disorder. We show that impurity scattering not only promotes the transition from ${C}_{2}$ to ${C}_{4}$-magnetic order, but it also favors the charge-spin density wave over the spin-vortex crystal phase. Additionally, in the single-$\mathbf{Q}$ phase, our analysis of the nematic coupling constant in the presence of disorder supports the experimental finding that the splitting between the structural and stripe-magnetic transition is enhanced by disorder.

Disorder-promoted C 4 -symmetric magnetic order in iron-based superconductors

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Role of fluctuations for density-wave instabilities: Failure of the mean-field description

Short-distance breakdown of the Higgs mechanism and the robustness of the BCS theory for charged superconductors

Disorder-promoted C 4 -symmetric magnetic order in iron-based superconductors