The unclear relationship between cuprate superconductivity and the pseudogap state remains an impediment to understanding the high transition temperature (T(c)) superconducting mechanism. Here, we used magnetic field-dependent scanning tunneling microscopy to provide phase-sensitive proof that d-wave superconductivity coexists with the pseudogap on the antinodal Fermi surface of an overdoped cuprate. Furthermore, by tracking the hole-doping (p) dependence of the quasi-particle interference pattern within a single bismuth-based cuprate family, we observed a Fermi surface reconstruction slightly below optimal doping, indicating a zero-field quantum phase transition in notable proximity to the maximum superconducting T(c). Surprisingly, this major reorganization of the system's underlying electronic structure has no effect on the smoothly evolving pseudogap.

We propose a new approach to understand the origin of the pseudogap in the cuprates, in terms of bosonic entropy. The near-simultaneous softening of a large number of different q-bosons yields an extended range of short-range order, wherein the growth of magnetic correlations with decreasing temperature T is anomalously slow. These entropic effects cause the spectral weight associated with the Van Hove singularity (VHS) to shift rapidly and nearly linearly toward half filling at higher T, consistent with a picture of the VHS driving the pseudogap transition at a temperature ~T*. As a byproduct, we develop an order-parameter classification scheme that predicts supertransitions between families of order parameters. As one example, we find that by tuning the hopping parameters, it is possible to drive the cuprates across a transition between Mott and Slater physics, where a spin-frustrated state emerges at the crossover.

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