Commensurate $4a_0$ period Charge Density Modulations throughout the   $Bi_2Sr_2CaCu_2O_{8+x}$ Pseudogap Regime
  Theories based upon strong real space (r-space) electron electron interactions have long predicted that unidirectional charge density modulations (CDM) with four unit cell (4$a_0$) periodicity should occur in the hole doped cuprate Mott insulator (MI). Experimentally, however, increasing the hole density p is reported to cause the conventionally defined wavevector $Q_A$ of the CDM to evolve continuously as if driven primarily by momentum space (k-space) effects. Here we introduce phase resolved electronic structure visualization for determination of the cuprate CDM wavevector. Remarkably, this new technique reveals a virtually doping independent locking of the local CDM wavevector at $|Q_0|=2\pi/4a_0$ throughout the underdoped phase diagram of the canonical cuprate $Bi_2Sr_2CaCu_2O_8$. These observations have significant fundamental consequences because they are orthogonal to a k-space (Fermi surface) based picture of the cuprate CDM but are consistent with strong coupling r-space based theories. Our findings imply that it is the latter that provide the intrinsic organizational principle for the cuprate CDM state. 
A. Mesaros, K. Fujita, S. D. Edkins, M. H. Hamidian, H. Eisaki, S.   Uchida, J. C. S\'eamus Davis, M. J. Lawler, Eun-Ah Kim
cond-mat.str-el cond-mat.supr-con