Density Waves Cause Sub-Gap Structures but no Pseudogap in Superconducting Cuprates S. Verret, M. Charlebois, D. S´ en´ echal, and A.-M. S. Tremblay ∗ D´ epartement de physique, Universit´ e de Sherbrooke, Qu´ ebec, Canada J1K 2R1 (Dated: August 11, 2018) In scanning tunneling microscopy (STM) conductance curves, the superconducting gap of cuprates is sometimes accompanied by small sub-gap structures at very low energy. This was documented early on near vortex cores and later at zero magnetic field. Using mean-field toy models of coexisting d-wave superconductivity (dSC), d -form factor density wave (dFF-DW), and extended s-wave pair density wave (s ′ PDW), we find agreement with this phenomenon, with s ′ PDW playing a critical role. We explore the high variability of the gap structure with changes in band structure and density wave (DW) wave vector, thus explaining why sub-gap structures may not be a universal feature in cuprates. In the absence of nesting, non-superconducting results never show signs of pseudogap, even for large density waves magnitudes, therefore reinforcing the idea of a distinct origin for the pseudogap, beyond mean-field theory. Therefore, we also briefly consider the effect of DWs on a pre-existing pseudogap. PACS numbers: 74.81.-g, 74.55.+v, 74.72.-h, 71.10.Fd I. INTRODUCTION The presence of density waves (DWs) in high tempera- ture superconducting cuprates is now well-established 1,2 , and growing evidence suggest that DWs and the pseudo- gap (PG) are related 3 but distinct phenomena 4–6 . How- ever, it is not yet clear how this distinction between DWs and the PG appears in the tunneling density of state (DOS) of cuprates 7,8 . Scanning tunneling microscopy played a key role in the discovery of DWs in cuprates. The early find- ing of a checkerboard DW in vortex cores 9 , where d- wave superconductivity (dSC) is weakened by a mag- netic field, suggested a competition between dSC and DWs. However, finding DWs was the culmination of much work previously focusing on the presence of low- energy sub-gap structures in the local DOS surrounding vortex cores 10 . Sub-gap structures were typically found between ±5 meV and ±10 meV in the conductance spec- tra of optimally doped YbBa 2 Cu 3 O 7−δ (YBCO) 11 and Bi 2 Sr 2 CaCu 2 O 8 (BSCCO) 12,13 . The occurrence of sub-gap structures (SGS) exactly where charge order was found indicates a likely relation between these phenomena 14 . Moreover, sub-gap struc- tures are also found at zero-field, so vortices and mag- netic fields are not the key to explain them. Bru` er et al. recently reported those structures in the averaged zero- field spectra of YBCO, and suggested: “it is tempting to link the SGS with the static charge density wave discov- ered recently in Y123 ” 15 . Equivalent zero-field structures had been extensively studied in underdoped samples of BSCCO 8,16,17 and mainly occur in locally resolved spec- tra where density waves are enhanced. Fig. 1 reproduces typical examples of sub-gap structures at zero-field in BSCCO and YBCO. Along with proposed scenarios of spin density waves 19 and staggered flux 20,21 , early theoretical work firmly stated the likeliness of pair density waves (PDWs) 22–25 Gap-map 70 mV 20 mV -80 -40 0 40 80 dI/dV (arb.) Sample Bias (mV) (a) sub-gap structure ∆ SC dI/dV (nS) -50 0 50 Bias V (mV) Averaged (b) BSCCO YBCO 0.2 0.8 FIG. 1. Sub-gap structures seen at zero magnetic field, a) in the inhomogeneous conductance spectra in BSCCO (adapted from Ref. 18, preprint version of Ref. 16) b) and in the aver- aged spectra for YBCO (adapted from 15 ) to explain the checkerboard patterns found in STM. Re- cent work by Agterberg and Garaud further showed that for competing dSC and PDW, a vortex core will favor PDW through the suppression of dSC, and also drive complementary charge modulation, as those seen in ex- periments 26 . After the observation by Lee that some pho- toemission results in cuprates agree better with a PDW scenario than with one based on charge density waves 27 , and after much work stating how many theoretical mod- els are prone to several forms of PDWs 28–32 , experimen- tal evidence of pair density wave was finally obtained, at zero-field, for an underdoped sample of BSCCO, through scanning Josephson tunneling spectroscopy 33 . It would be hard, at this point, to exclude PDWs from the cuprate puzzle. In this work, we consider phenomenological mean-field Hamiltonians for coexisting bond-centered density wave and pair density waves (similar to those experimentally reported in Ref. 33 and 34) combined to d-wave super- conductivity, and we find qualitative agreement with ob- arXiv:1610.01109v2 [cond-mat.supr-con] 23 Dec 2016