where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Proceedings of the National Academy of Sciences

Annals of physics

Materials with flat electronic bands often exhibit exotic quantum phenomena owing to strong correlations. An isolated low-energy flat band can be induced in bilayer graphene by simply rotating the layers by 1.1°, resulting in the appearance of gate-tunable superconducting and correlated insulating phases. In this study, we demonstrate that in addition to the twist angle, the interlayer coupling can be varied to precisely tune these phases. We induce superconductivity at a twist angle larger than 1.1°—in which correlated phases are otherwise absent—by varying the interlayer spacing with hydrostatic pressure. Our low-disorder devices reveal details about the superconducting phase diagram and its relationship to the nearby insulator. Our results demonstrate twisted bilayer graphene to be a distinctively tunable platform for exploring correlated states.

https://science.sciencemag.org/content/sci/early/2019/01/23/science.aav1910.full.pdf

Tuning superconductivity in twisted bilayer graphene.

Twisted bilayer graphene (TBG) was recently shown to host superconductivity when tuned to special "magic angles" at which isolated and relatively flat bands appear. However, until now the origin of the magic angles and their irregular pattern have remained a mystery. Here we report on a fundamental continuum model for TBG which features not just the vanishing of the Fermi velocity, but also the perfect flattening of the entire lowest band. When parametrized in terms of α∼1/θ, the magic angles recur with a remarkable periodicity of Δα≃3/2. We show analytically that the exactly flat band wave functions can be constructed from the doubly periodic functions composed of ratios of theta functions-reminiscent of quantum Hall wave functions on the torus. We further report on the unusual robustness of the experimentally relevant first magic angle, address its properties analytically, and discuss how lattice relaxation effects help justify our model parameters.

Origin of Magic Angles in Twisted Bilayer Graphene.

Intermetallic compounds containing $f$-electron elements display a wealth of superconducting phases, which are prime candidates for unconventional pairing with complex order parameter symmetries. For instance, superconductivity has been found at the border of magnetic order as well as deep within ferromagnetically and antiferromagnetically ordered states, suggesting that magnetism may promote rather than destroy superconductivity. Superconducting phases near valence transitions or in the vicinity of magnetopolar order are candidates for new superconductive pairing interactions such as fluctuations of the conduction electron density or the crystal electric field, respectively. The experimental status of the study of the superconducting phases of $f$-electron compounds is reviewed.

Superconducting phases of f -electron compounds

We consider the superconducting and Mott-insulating states for twisted bilayer graphene, modeled as a two-narrow-band system of electrons with appreciable intra-atomic Coulomb interactions The interaction induces kinetic exchange which leads to real space, either triplet- or singlet-spin pairing, in direct analogy to heavy fermions and high-temperature superconductors By employing the statistically consistent Gutzwiller method, we construct explicitly the phase diagram as a function of electron concentration for the spin-triplet ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$ paired case, as well as determine the topological edge states The model reproduces principal features observed experimentally in a semiquantitative manner The essential role of electronic correlations in driving both the Mott-insulating and superconducting transitions is emphasized The transformation of the spin-triplet state into its spin-singlet analog is also analyzed, as well as the appearance of the phase-separated superconducting+Mott-insulating state close to the half-filling

Unconventional topological superconductivity and phase diagram for an effective two-orbital model as applied to twisted bilayer graphene

The NiS{sub 2{minus}x}Se{sub x} system represents one of the best examples of a Mott-Hubbard system, i.e., a system in which, under appropriate conditions of concentration, temperature, or pressure, a metal-insulator transition driven by electron-electron interaction takes place. Here, the metallic phase is either antiferromagnetic (for 0.44 {le} x {le} 1) or paramagnetic (for x {ge} 1), whereas the insulating phase is as a rule antiferromagnetic (including the spin-canted phases). In this paper the authors review both the physical properties and outline the basic features of the theoretical approach to those correlated electron systems. Emphasis is placed on a qualitative understanding of the observed transformation of the system from semiconductor (or magnetic insulator) to metal.

Electronic Properties of NiS2-xSex Single Crystals: From Magnetic Mott−Hubbard Insulators to Normal Metals

In this overview I sketch briefly the path to the so-called {\em t-J model} derived for the first time 30 years ago and provide its original meaning within the theory of strongly correlated magnetic metals with a non-Fermi (non-Landau) liquid ground state. An emergence of the concept of {\em real space pairing}, is discussed in a historical prospective. A generalization of this model to the many-orbital situation is briefly discussed. The emphasis is put on didactical exposition of ideas, as they were transformed into mathematical language. The concept of {\em hybrid pairing} is introduced in the same context at the end.

/pdf/t-j-model-then-and-now-a-personal-perspective-from-the-3mb1sfnckm.pdf

t-J model then and now: A personal perspective from the pioneering times

A systematic diagrammatic expansion for Gutzwiller-wave functions (DE-GWF) is formulated and used for the description of superconducting (SC) ground state in the two-dimensional Hubbard model with electron-transfer amplitudes $t$ (and ${t}^{\ensuremath{'}}$) between nearest (and next-nearest) neighbors. The method is numerically very efficient and allows for a detailed analysis of the phase diagram as a function of all relevant parameters ($U$, $\ensuremath{\delta}$, ${t}^{\ensuremath{'}}$) and a determination of the kinetic-energy driven pairing region. SC states appear only for substantial interactions, $U/t\ensuremath{\gtrsim}3$, and for not too large hole doping, $\ensuremath{\delta}\ensuremath{\lesssim}0.32$ for ${t}^{\ensuremath{'}}=0.25t$; this upper critical doping value agrees well with experiment for the cuprate high-temperature superconductors. We also obtain other important features of the SC state: (i) the SC gap at the Fermi surface resembles ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave only around the optimal doping and the corrections to this state are shown to arise from the longer range of the pairing; (ii) the nodal Fermi velocity is almost constant as a function of doping and agrees quantitatively with the experimental results; (iii) the SC transition is driven by the kinetic-energy lowering for low doping and strong interactions.

Superconductivity in the two-dimensional Hubbard model: Gutzwiller wave function solution

Selected universal experimental properties of high-temperature superconducting (HTS) cuprates have been singled out in the last decade. One of the pivotal challenges in this field is the designation of a consistent interpretation framework within which we can describe quantitatively the universal features of those systems. Here we analyze in a detailed manner the principal experimental data and compare them quantitatively with the approach based on a single-band model of strongly correlated electrons supplemented with strong antiferromagnetic (super)exchange interaction (the so-called $t\ensuremath{-}J\ensuremath{-}U$ model). The model rationale is provided by estimating its microscopic parameters on the basis of the three-band approach for the Cu-O plane. We use our original full Gutzwiller wave-function solution by going beyond the renormalized mean-field theory (RMFT) in a systematic manner. Our approach reproduces very well the observed hole doping ($\ensuremath{\delta}$) dependence of the kinetic-energy gain in the superconducting phase, one of the principal non-Bardeen-Cooper-Schrieffer features of the cuprates. The calculated Fermi velocity in the nodal direction is practically $\ensuremath{\delta}$-independent and its universal value agrees very well with that determined experimentally. Also, a weak doping dependence of the Fermi wave vector leads to an almost constant value of the effective mass in a pure superconducting phase which is both observed in experiment and reproduced within our approach. An assessment of the currently used models ($t\ensuremath{-}J$, Hubbard) is carried out and the results of the canonical RMFT as a zeroth-order solution are provided for comparison to illustrate the necessity of the introduced higher-order contributions.

Jozef Spałek

Papers

Unconventional topological superconductivity and phase diagram for an effective two-orbital model as applied to twisted bilayer graphene

Electronic Properties of NiS2-xSex Single Crystals: From Magnetic Mott−Hubbard Insulators to Normal Metals

t-J model then and now: A personal perspective from the pioneering times

Superconductivity in the two-dimensional Hubbard model: Gutzwiller wave function solution

Universal properties of high-temperature superconductors from real-space pairing: t − J − U model and its quantitative comparison with experiment