Symmetry, Maximally Localized Wannier States, and a Low-Energy Model for Twisted Bilayer Graphene Narrow Bands

We discuss a method for determining the optimally-localized set of generalized Wannier functions associated with a set of Bloch bands in a crystalline solid. By ``generalized Wannier functions'' we mean a set of localized orthonormal orbitals spanning the same space as the specified set of Bloch bands. Although we minimize a functional that represents the total spread sum_n [ _n - _n^2 ] of the Wannier functions in real space, our method proceeds directly from the Bloch functions as represented on a mesh of k-points, and carries out the minimization in a space of unitary matrices U_mn^k describing the rotation among the Bloch bands at each k-point. The method is thus suitable for use in connection with conventional electronic-structure codes. The procedure also returns the total electric polarization as well as the location of each Wannier center. Sample results for Si, GaAs, molecular C2H4, and LiCl will be presented.

The electronic ground state of a periodic system is usually described in terms of extended Bloch orbitals, but an alternative representation in terms of localized "Wannier functions" was introduced by Gregory Wannier in 1937. The connection between the Bloch and Wannier representations is realized by families of transformations in a continuous space of unitary matrices, carrying a large degree of arbitrariness. Since 1997, methods have been developed that allow one to iteratively transform the extended Bloch orbitals of a first-principles calculation into a unique set of maximally localized Wannier functions, accomplishing the solid-state equivalent of constructing localized molecular orbitals, or "Boys orbitals" as previously known from the chemistry literature. These developments are reviewed here, and a survey of the applications of these methods is presented. This latter includes a description of their use in analyzing the nature of chemical bonding, or as a local probe of phenomena related to electric polarization and orbital magnetization. Wannier interpolation schemes are also reviewed, by which quantities computed on a coarse reciprocal-space mesh can be used to interpolate onto much finer meshes at low cost, and applications in which Wannier functions are used as efficient basis functions are discussed. Finally the construction and use of Wannier functions outside the context of electronic-structure theory is presented, for cases that include phonon excitations, photonic crystals, and cold-atom optical lattices.

The understanding of strongly-correlated materials, and in particular unconventional superconductors, has puzzled physicists for decades. Such difficulties have stimulated new research paradigms, such as ultra-cold atom lattices for simulating quantum materials. Here we report on the realization of intrinsic unconventional superconductivity in a 2D superlattice created by stacking two graphene sheets with a small twist angle. For angles near \(1.1^\circ\), the first `magic' angle, twisted bilayer graphene (TBG) exhibits ultra-flat bands near charge neutrality, which lead to correlated insulating states at half-filling. Upon electrostatic doping away from these correlated insulating states, we observe tunable zero-resistance states with a critical temperature \(T_c\) up to 1.7 K. The temperature-density phase diagram shows similarities with that of the cuprates, including superconducting domes. Moreover, quantum oscillations indicate small Fermi surfaces near the correlated insulating phase, in analogy with under-doped cuprates. The relative high \(T_c\), given such small Fermi surface (corresponding to a record-low 2D carrier density of \(10^{11} \textrm{cm}^{-2}\) , renders TBG among the strongest coupling superconductors, in a regime close to the BCS-BEC crossover. These novel results establish TBG as the first purely carbon-based 2D superconductor and as a highly tunable platform to investigate strongly-correlated phenomena, which could lead to insights into the physics of high-\(T_c\) superconductors and quantum spin liquids.