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Charge modulations have been widely observed in cuprates, suggesting their centrality for understanding the high-$T_c$ superconductivity in these materials. However, the dimensionality of these modulations remains controversial, including whether their wavevector is unidirectional or bidirectional, and also whether they extend seamlessly from the surface of the material into the bulk. Material disorder presents severe challenges to understanding the charge modulations through bulk scattering techniques. We use a local technique, scanning tunneling microscopy, to image the static charge modulations on Bi$_{2-z}$Pb$_z$Sr$_{2-y}$La$_y$CuO$_{6+x}$. By comparing the phase correlation length $\xi_{\mathrm{CDW}}$ with the orientation correlation length $\xi_{\mathrm{orient}}$, we show that the charge modulations are more consistent with an underlying unidirectional wave vector. Using cluster techniques, we show that these locally 1D charge modulations are actually a bulk effect resulting from 3D criticality throughout the entire superconducting doping range.

A complicating factor in unraveling the theory of high-temperature (high-Tc) superconductivity is the presence of a "pseudogap" in the density of states, whose origin has been debated since its discovery [1]. Some believe the pseudogap is a broken symmetry state distinct from superconductivity [2-4], while others believe it arises from short-range correlations without symmetry breaking [5,6]. A number of broken symmetries have been imaged and identified with the pseudogap state [7,8], but it remains crucial to disentangle any electronic symmetry breaking from pre-existing structural symmetry of the crystal. We use scanning tunneling microscopy (STM) to observe an orthorhombic structural distortion across the cuprate superconducting Bi2Sr2Can-1CunO2n+4+x (BSCCO) family tree, which breaks two-dimensional inversion symmetry in the surface BiO layer. Although this inversion symmetry breaking structure can impact electronic measurements, we show from its insensitivity to temperature, magnetic field, and doping, that it cannot be the long-sought pseudogap state. To detect this picometer-scale variation in lattice structure, we have implemented a new algorithm which will serve as a powerful tool in the search for broken symmetry electronic states in cuprates, as well as in other materials., Comment: 4 figures