Universality of commensurate 4$a$-period charge density modulations throughout the cuprate pseudogap regime

Theories for the hole-doped Mott insulator, representing underdoped cuprates, are based upon the strong real space ($r$-space) interactions, and have long predicted a modulation of charge that is commensurate with the underlying lattice. Such a charge density modulation (CDM) state is unrelated to any momentum space ($k$-space) features such as the nesting of regions on a Fermi surface. Experimentally, with increasing hole density, the reported wavevector $Q$ of the CDM diminishes continuously with increasing hole-density as if driven by $k$-space phenomena. Using a novel technique based upon phase-sensitive electronic structure visualization, we demonstrate that the cuprate CDM actually exhibits a commensurate 4$a$-period throughout the entire underdoped region of the $Bi_2Sr_2CaCu_2O_8$ phase diagram. Our technique is designed for extracting $Q$ from inhomogeneous, short-ranged CDM, as the ones observed in experiments. Thus, a strong-interaction $r$-space perspective appears to be relevant to achieving a predictive theory for the cuprate pseudogap regime.