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Emergent electronic phenomena in iron-based superconductors have been at the forefront of condensed matter physics for more than a decade. Much has been learned about the origins and intertwined roles of ordered phases, including nematicity, magnetism, and superconductivity, in this fascinating class of materials. In recent years, focus has been centered on the peculiar and highly unusual properties of FeSe and its close cousins. This family of materials has attracted considerable attention due to the discovery of unexpected superconducting gap structures, a wide range of superconducting critical temperatures, and evidence for nontrivial band topology, including associated spin-helical surface states and vortex-induced Majorana bound states. Here, we review superconductivity in iron chalcogenide superconductors, including bulk FeSe, doped bulk FeSe, FeTe1−xSex, intercalated FeSe materials, and monolayer FeSe and FeTe1−xSex on SrTiO3. We focus on the superconducting properties, including a survey of the relevant experimental studies, and a discussion of the different proposed theoretical pairing scenarios. In the last part of the paper, we review the growing recent evidence for nontrivial topological effects in FeSe-related materials, focusing again on interesting implications for superconductivity.

Iron pnictides are the only known family of unconventional high-temperature superconductors besides cuprates. Until recently, it was widely accepted that superconductivity is driven by spin fluctuations and intimately related to the fermiology, specifically, hole and electron pockets separated by the same wavevector that characterizes the dominant spin fluctuations, and supporting order parameters (OP) of opposite signs. This picture was questioned after the discovery of intercalated or monolayer form of FeSe-based systems without hole pockets, which seemingly undermines the basis for spin-fluctuation theory and the idea of a sign-changing OP. Using the recently proposed phase-sensitive quasiparticle interference technique, here we show that in LiOH-intercalated FeSe compound the OP does change sign, albeit within the electronic pockets. This result unifies the pairing mechanism of iron-based superconductors with or without the hole Fermi pockets and supports the conclusion that spin fluctuations play the key role in electron pairing.

Visualizing atomic-orbital degrees of freedom is a frontier challenge in scanned microscopy. Some types of orbital order are virtually imperceptible to normal scattering techniques because they do not reduce the overall crystal lattice symmetry. A good example is d xz / d yz (π,π) orbital order in tetragonal lattices. For enhanced detectability, here we consider the quasiparticle scattering interference (QPI) signature of such (π,π) orbital order in both normal and superconducting phases. The theory reveals that sublattice-specific QPI signatures generated by the orbital order should emerge strongly in the superconducting phase. Sublattice-resolved QPI visualization in superconducting CeCoIn 5 then reveals two orthogonal QPI patterns at lattice-substitutional impurity atoms. We analyze the energy dependence of these two orthogonal QPI patterns and find the intensity peaked near E  = 0, as predicted when such (π,π) orbital order is intertwined with d -wave superconductivity. Sublattice-resolved superconductive QPI techniques thus represent a new approach for study of hidden orbital order. Orbital order that does not break the overall crystal lattice symmetry is difficult to observe. Here, the authors use scanning tunneling microscopy on the superconductor CeCoIn 5 to detect a signature of the orbital order in quasiparticle interference which is enhanced in the superconducting state, as predicted theoretically.

An unidentified quantum fluid designated the pseudogap (PG) phase is produced by electron-density depletion in the CuO 2 antiferromagnetic insulator. Current theories suggest that the PG phase may be a pair density wave (PDW) state characterized by a spatially modulating density of electron pairs. Such a state should exhibit a periodically modulating energy gap Δ P ( r ) in real-space, and a characteristic quasiparticle scattering interference (QPI) signature Λ P ( q ) in wavevector space. By studying strongly underdoped Bi 2 Sr 2 CaDyCu 2 O 8 at hole-density ~0.08 in the superconductive phase, we detect the 8 a 0 -periodic Δ P ( r ) modulations signifying a PDW coexisting with superconductivity. Then, by visualizing the temperature dependence of this electronic structure from the superconducting into the pseudogap phase, we find the evolution of the scattering interference signature Λ ( q ) that is predicted specifically for the temperature dependence of an 8 a 0 -periodic PDW. These observations are consistent with theory for the transition from a PDW state coexisting with d -wave superconductivity to a pure PDW state in the Bi 2 Sr 2 CaDyCu 2 O 8 pseudogap phase. The pseudogap phase in cuprate superconductors is predicted to be a pair density wave state (PDW) but experimental evidence has been lacking. Here, the authors detect the temperature evolution of energy gap modulations and scattering interference signature suggesting the Bi 2 Sr 2 CaDyCu 2 O 8 pseudogap phase contains a PDW.

This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.

Resolving the microscopic pairing mechanism and its experimental identification in unconventional superconductors is among the most vexing problems of contemporary condensed matter physics. We show that Raman spectroscopy provides an avenue towards this aim by probing the structure of the pairing interaction at play in an unconventional superconductor. As we study the spectra of the prototypical Fe-based superconductor Ba 1−x K x Fe 2 As 2 for 0.22 ≤  x  ≤ 0.70 in all symmetry channels, Raman spectroscopy allows us to distill the leading s -wave state. In addition, the spectra collected in the B 1 g symmetry channel reveal the existence of two collective modes which are indicative of the presence of two competing, yet sub-dominant, pairing tendencies of d x 2 - y 2 symmetry type. A comprehensive functional Renormalization Group and random-phase approximation study on this compound confirms the presence of the two sub-leading channels, and consistently matches the experimental doping dependence of the related modes. The consistency between the experimental observations and the theoretical modeling suggests that spin fluctuations play a significant role in superconducting pairing. Iron-based superconductors: competing pairing interactions Two collective Raman modes are observed in an iron-based superconductor, indicative of the presence of two competing pairing tendencies alongside the dominant s-wave state. An international team led by R. Hackl from the Walther Meissner Institut perform Raman spectroscopy measurements to probe the structure of pairing interactions in Ba 1− x K x Fe 2 As 2 for 0.22 ≤  x  ≤ 0.70 for all symmetry channels. The Raman spectra not only shows the dominant peak marking the dominant s-wave superconducting pairing state, but also reveals the existence of two collective modes in the B 1 g symmetry channel, indicative of two competing, sub-dominant, paring tendencies of d x 2 - y 2 symmetry type. Numerical calculations confirm the finding and consistently match the doping dependencies of the related modes. The results suggest a significant role of spin-fluctuations in superconducting pairing.

Integrating deep learning with the search for new electron-phonon superconductors represents a burgeoning field of research, where the primary challenge lies in the computational intensity of calculating the electron-phonon spectral function, α 2 F ( ω ), the essential ingredient of Midgal-Eliashberg theory of superconductivity. To overcome this challenge, we adopt a two-step approach. First, we compute α 2 F ( ω ) for 818 dynamically stable materials. We then train a deep-learning model to predict α 2 F ( ω ), using a training strategy tailored for limited data to temper the model’s overfitting, enhancing predictions. Specifically, we train a Bootstrapped Ensemble of Tempered Equivariant graph neural NETworks (BETE-NET), obtaining an MAE of 0.21, 45 K, and 43 K for the moments derived from α 2 F ( ω ): λ , ω log , and ω 2 , respectively, yielding an MAE of 2.5 K for the critical temperature, T c . Further, we incorporate domain knowledge of the site-projected phonon density of states to impose inductive bias into the model’s node attributes and enhance predictions. This methodological innovation decreases the MAE to 0.18, 29 K, and 28 K, respectively, yielding an MAE of 2.1 K for T c . We illustrate the practical application of our model in high-throughput screening for high- T c materials. The model demonstrates an average precision nearly five times higher than random screening, highlighting the potential of ML in accelerating superconductor discovery. BETE-NET accelerates the search for high- T c superconductors while setting a precedent for applying ML in materials discovery, particularly when data is limited.

The single-layered ruthenate Sr2RuO4 is one of the most enigmatic unconventional superconductors. While for many years it was thought to be the best candidate for a chiral p-wave superconducting ground state, desirable for topological quantum computations, recent experiments suggest a singlet state, ruling out the original p-wave scenario. The superconductivity as well as the properties of the multi-layered compounds of the ruthenate perovskites are strongly influenced by a van Hove singularity in proximity of the Fermi energy. Tiny structural distortions move the van Hove singularity across the Fermi energy with dramatic consequences for the physical properties. Here, we determine the electronic structure of the van Hove singularity in the surface layer of Sr2RuO4 by quasi-particle interference imaging. We trace its dispersion and demonstrate from a model calculation accounting for the full vacuum overlap of the wave functions that its detection is facilitated through the octahedral rotations in the surface layer.

The quest to identify new superconducting materials with enhanced properties is hindered by the prohibitive cost of computing electron-phonon spectral functions, severely limiting the materials space that can be explored. Here, we introduce a Bootstrapped Ensemble of Equivariant Graph Neural Networks (BEE-NET), a machine-learning model trained to predict the Eliashberg spectral function and superconducting critical temperature with a mean-absolute-error of 0.87 K relative to DFT-based Allen-Dynes calculations. Intriguingly, BEE-NET achieves a true-negative-rate of 99.4%, enabling highly efficient screening for the rare property of superconductivity. Integrated into a multi-stage, AI-accelerated discovery pipeline that incorporates elemental-substitution strategies and machine-learned interatomic potentials, our workflow reduced over 1.3 million candidate structures to 741 dynamically and thermodynamically stable compounds with DFT-confirmed T c > 5 K. We report the successful synthesis and experimental confirmation of superconductivity in two of these previously unreported compounds. This study establishes a data-driven framework that integrates machine learning, quantum calculations, and experiments to systematically accelerate superconductor discovery.