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 Bi2Sr2CaCu2O8 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.PACS numbers: 74.70.Xa, 74.62.En, Scanning tunneling microscopy (STM) methods were applied to cuprates relatively early on, but dramatic improvements in energy and spatial resolution led to a new set of classic discoveries in the early part of the last decade, giving for the first time a truly local picture of the superconducting and pseudogap states at low temperatures [1,2]. These measurements revealed gaps that were much more inhomogeneous than had previously been anticipated [3][4][5][6], exhibited localized impurity resonant states [7,8], and gave important clues to the nature of competing order [9][10][11][12][13]. More recently, STM has again been at the forefront of studies of inhomogeneities, this time as a real space probe of intra-unit cell charge ordering visible in the underdoped systems [14]. While a microscopic description of such atomic scale phenomena in superconductors is available in terms of the Bogolibuov de-Gennes equations, such calculations are always performed on a lattice with sites centered on the Cu atoms, and thus do not contain intra-unit cell information.The simplest example of a problem that can arise because of the deficiencies of theory in this regard is that of the Zn impurity substituting for Cu in Bi 2 Sr 2 CaCu 2 O 8 (BSCCO), a cuprate material which cleaves well in vacuum, leaving atomically smooth surfaces ideal for STM. The observation of a spectacularly sharp impurity resonance at the impurity site [7,8,15,16] was an important local confirmation of unconventional pairing in the cuprates. The differential conductance map near the impurity exhibits a cross-shaped real-space conductance map at resonance, as expected for a pointlike potential scatterer in a d-wave superconductor, see Fig. 1(c) [17] Upon closer examination, however, the pattern deviates from the expected theoretical one on the Cu square lat- tice in some important respects [18,19]. First, it displays a central maximum on the impurity site, unlike simple models, which have a minimum ( Fig. 1(a)). Second, the longer range intensity tails are rotated 45 degrees from the nodal directions of the ...

We study the incommensurate charge ordered states in the ¢ -t t J model using the Gutzwiller mean field theory on large systems. In particular, we explore the properties of incommensurate charge modulated states referred to as nodal pair density waves (nPDW) in the literature. nPDW states intertwine site and bond charge order with modulated d-wave pair order, and are characterized by a nonzero amplitude of uniform pairing; they also manifest a dominant intra-unit cell d-density wave form factor. To compare with a recent scanning tunneling microscopy (STM) study (Hamidian et al 2015 Nat. Phys. 12 150) of the cuprate superconductor BSCCO-2212, we compute the continuum local density of states (LDOS) at a typical STM tip height using the Wannier function based approach. By Fourier transforming Cu and O sub-lattice LDOS we also obtain bias-dependent intra-unit cell form factors and spatial phase difference. We find that in the nPDW state the behavior of form factors and spatial phase difference as a function of energy agrees remarkably well with the experiment.This is in contrast to commensurate charge modulated states, which we show do not agree with experiment. We propose that the nPDW states are good candidates for the charge density wave phase observed in the superconducting state of underdoped cuprates.

We propose a simple method of calculating inhomogeneous, atomic-scale phenomena in superconductors which makes use of the wave function information traditionally discarded in the construction of tight-binding models used in the Bogoliubov-de Gennes equations. The method uses symmetrybased first principles Wannier functions to visualize the effects of superconducting pairing on the distribution of electronic states over atoms within a crystal unit cell. Local symmetries lower than the global lattice symmetry can thus be exhibited as well, rendering theoretical comparisons with scanning tunneling spectroscopy data much more useful. As a simple example, we discuss the geometric dimer states observed near defects in superconducting FeSe.

The defining characteristic of hole-doped cuprates isd-wave high temperature superconductivity. However, intense theoretical interest is now focused on whether a pair density wave state (PDW) could coexist with cuprate superconductivity [D. F. Agterberg et al.,Annu. Rev. Condens. Matter Phys.11, 231 (2020)]. Here, we use a strong-coupling mean-field theory of cuprates, to model the atomic-scale electronic structure of an eight-unit-cell periodic,d-symmetry form factor, pair density wave (PDW) state coexisting withd-wave superconductivity (DSC). From this PDW + DSC model, the atomically resolved density of Bogoliubov quasiparticle statesNr,Eis predicted at the terminal BiO surface of Bi2Sr2CaCu2O8and compared with high-precision electronic visualization experiments using spectroscopic imaging scanning tunneling microscopy (STM). The PDW + DSC model predictions include the intraunit-cell structure and periodic modulations ofNr,E, the modulations of the coherence peak energyΔpr,and the characteristics of Bogoliubov quasiparticle interference in scattering-wavevector spaceq-space. Consistency between all these predictions and the corresponding experiments indicates that lightly hole-doped Bi2Sr2CaCu2O8does contain a PDW + DSC state. Moreover, in the model the PDW + DSC state becomes unstable to a pure DSC state at a critical hole densityp*, with empirically equivalent phenomena occurring in the experiments. All these results are consistent with a picture in which the cuprate translational symmetry-breaking state is a PDW, the observed charge modulations are its consequence, the antinodal pseudogap is that of the PDW state, and the cuprate critical point atp* ≈ 19% occurs due to disappearance of this PDW.