Short-Range Modulated Electron Lattice and d-Wave Superconductivity in Cuprates: A Phenomenological Ginzburg-Landau Framework

Abstract

A short-range charge modulation near 0.3 reciprocal lattice units along the Cu-O bond is present in every hole-doped cuprate family. Resonant x-ray scattering now shows that superconductivity does two opposite things to it at once: below Tc the modulation weakens yet becomes more phase coherent. We trace this split to symmetry: the modulation's envelope carries lattice momentum, which leaves a d-wave condensate exactly two ways to couple to it at quartic order, through the modulation's amplitude or through its phase. The first moves amplitude, coherence, and superfluid stiffness together; the second buys coherence at the expense of stiffness, so the two are separately measurable. We call this Ginzburg-Landau framework the modulated electron lattice (MEL). Classical Monte Carlo on 120x120 lattices with quenched disorder places the x-ray observation, read as a single component, at competing amplitude coupling and cooperative phase coupling, where the model gives no stiffness gain. But the measured intensity sums bond-centred and site-centred components. A two-component simulation gives the same pair of bulk signatures, intensity down and coherence up, both with a stiffness loss and with a stiffness gain, depending on the strength of the bond channel. Bulk data therefore cannot say whether this charge order stiffens the superconductor or softens it. What settles the question is the bond-channel amplitude, which form-factor-resolved scattering and phase-resolved tunnelling measure. The response follows the local pairing amplitude, so its onset need not be sharp at Tc. We also compute vortex pinning in the modulated landscape and obtain an in-plane penetration depth of about 124 nm once the transition temperature fixes the energy scale. This version corrects the first: its linear envelope coupling was symmetry-forbidden, and all numerical results are new.