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.
The superconductor iron selenide (FeSe) is of intense interest owing to its unusual nonmagnetic nematic state and potential for high-temperature superconductivity. But its Cooper pairing mechanism ...has not been determined. We used Bogoliubov quasiparticle interference imaging to determine the Fermi surface geometry of the electronic bands surrounding the G = (0, 0) and X = (p/aFe, 0) points of FeSe and to measure the corresponding superconducting energy gaps. We show that both gaps are extremely anisotropic but nodeless and that they exhibit gap maxima oriented orthogonally in momentum space. Moreover, by implementing a novel technique, we demonstrate that these gaps have opposite sign with respect to each other. This complex gap configuration reveals the existence of orbital-selective Cooper pairing that, in FeSe, is based preferentially on electrons from the dyz orbitals of the iron atoms.
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In weakly coupled BCS superconductors, only electrons within a tiny energy window around the Fermi energy, E
, form Cooper pairs. This may not be the case in strong coupling superconductors such as ...cuprates, FeSe, SrTiO
or cold atom condensates where the pairing scale, E
, becomes comparable or even larger than E
. In cuprates, for example, a plausible candidate for the pseudogap state at low doping is a fluctuating pair density wave, but no microscopic model has yet been found which supports such a state. In this work, we write an analytically solvable model to examine pairing phases in the strongly coupled regime and in the presence of anisotropic interactions. Already for moderate coupling we find an unusual finite temperature phase, below an instability temperature T
, where local pair correlations have non-zero center-of-mass momentum but lack long-range order. At low temperature, this fluctuating pair density wave can condense either to a uniform d-wave superconductor or the widely postulated pair-density wave phase depending on the interaction strength. Our minimal model offers a unified framework to understand the emergence of both fluctuating and long range pair density waves in realistic systems.
The recently discovered Fe-pnictide and chalcogenide superconductors display low-temperature properties suggesting superconducting gap structures which appear to vary substantially from family to ...family, and even within families as a function of doping or pressure. We propose that this apparent nonuniversality can actually be understood by considering the predictions of spin fluctuation theory and accounting for the peculiar electronic structure of these systems, coupled with the likely 'sign-changing s-wave' (s sub(+ or -)) symmetry. We review theoretical aspects, materials properties and experimental evidence relevant to this suggestion, and discuss which further measurements would be useful to settle these issues.
Superconductivity is a remarkably widespread phenomenon that is observed in most metals cooled to very low temperatures. The ubiquity of such conventional superconductors, and the wide range of ...associated critical temperatures, is readily understood in terms of the well-known Bardeen-Cooper-Schrieffer theory. Occasionally, however, unconventional superconductors are found, such as the iron-based materials, which extend and defy this understanding in unexpected ways. In the case of the iron-based superconductors, this includes the different ways in which the presence of multiple atomic orbitals can manifest in unconventional superconductivity, giving rise to a rich landscape of gap structures that share the same dominant pairing mechanism. In addition, these materials have also led to insights into the unusual metallic state governed by the Hund's interaction, the control and mechanisms of electronic nematicity, the impact of magnetic fluctuations and quantum criticality, and the importance of topology in correlated states. Over the fourteen years since their discovery, iron-based superconductors have proven to be a testing ground for the development of novel experimental tools and theoretical approaches, both of which have extensively influenced the wider field of quantum materials.
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We provide a band structure with low-energy properties consistent with recent photoemission and quantum oscillation measurements on FeSe, assuming mean-field-like site- and/or bond-centered ...ferro-orbital ordering at the structural transition. We show how the resulting model provides a consistent explanation of the temperature dependence of the measured Knight shift and the spin-relaxation rate. Furthermore, the superconducting gap structure obtained from spin-fluctuation theory exhibits nodes on the electron pockets, consistent with the V-shaped density of states obtained by tunneling spectroscopy on this material, and the temperature dependence of the London penetration depth.
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We apply a recently developed method combining first principles based Wannier functions with solutions to the Bogoliubov-de Gennes equations to the problem of interpreting STM data in cuprate ...superconductors. We show that the observed images of Zn on the surface of Bi_{2}Sr_{2}CaCu_{2}O_{8} can only be understood by accounting for the tails of the Cu Wannier functions, which include significant weight on apical O sites in neighboring unit cells. This calculation thus puts earlier crude "filter" theories on a microscopic foundation and solves a long-standing puzzle. We then study quasiparticle interference phenomena induced by out-of-plane weak potential scatterers, and show how patterns long observed in cuprates can be understood in terms of the interference of Wannier functions above the surface. Our results show excellent agreement with experiment and enable a better understanding of novel phenomena in the cuprates via STM imaging.
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Strong electronic correlations, emerging from the parent Mott insulator phase, are key to copper-based high-temperature superconductivity. By contrast, the parent phase of an iron-based ...high-temperature superconductor is never a correlated insulator. However, this distinction may be deceptive because Fe has five actived d orbitals while Cu has only one. In theory, such orbital multiplicity can generate a Hund's metal state, in which alignment of the Fe spins suppresses inter-orbital fluctuations, producing orbitally selective strong correlations. The spectral weights Z
of quasiparticles associated with different Fe orbitals m should then be radically different. Here we use quasiparticle scattering interference resolved by orbital content to explore these predictions in FeSe. Signatures of strong, orbitally selective differences of quasiparticle Z
appear on all detectable bands over a wide energy range. Further, the quasiparticle interference amplitudes reveal that Formula: see text, consistent with earlier orbital-selective Cooper pairing studies. Thus, orbital-selective strong correlations dominate the parent state of iron-based high-temperature superconductivity in FeSe.
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Over the past few years iron chalcogenides have been intensively studied as part of the wider family of iron-based superconductors, with many intriguing results reported so far on intercalated and ...monolayer FeSe. Nevertheless, bulk FeSe itself remains an unusual case when compared with pnictogen-based iron superconductors, and may hold clues to understanding the more exotic derivatives of the FeSe system. The FeSe phase diagram is distinct from the pnictides: the orthorhombic distortion, which is likely to be of a 'spin-nematic' nature in numerous pnictides, is not accompanied by magnetic order in FeSe, and the superconducting transition temperature Tc rises significantly with pressure before decreasing. Here we show that the magnetic interactions in FeSe, as opposed to most pnictides, demonstrate an unusual and unanticipated frustration, which suppresses magnetic (but not nematic) order, triggers ferro-orbital order in the nematic phase and can naturally explain the non-monotonic pressure dependence of the superconducting critical temperature Tc (P).
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