Visualizing the Energy-gap Modulations of the Cuprate Pair Density Wave State

When Cooper pairs are formed with finite center-of-mass momentum, the defining characteristic is a spatially modulating superconducting energy gap {\Delta}(r). Recently, this concept has been generalized to the pair density wave (PDW) state predicted to exist in a variety of strongly correlated electronic materials such as the cuprates. Although the signature of a cuprate PDW has been detected in Cooper-pair tunnelling, the distinctive signature in single-electron tunneling of a periodic {\Delta}(r) modulation has never been observed. Here, using a new approach, we discover strong {\Delta}(r) modulations in Bi2Sr2CaCu2O8+{\delta} that have eight-unit-cell periodicity or wavevectors Q=2{\pi}/a_0(1/8,0); 2{\pi}/a_0(0,1/8). This constitutes the first energy-resolved spectroscopic evidence for the cuprate PDW state. An analysis of spatial arrangements of {\Delta}(r) modulations then reveals that this PDW is predominantly unidirectional, but with an arrangement of nanoscale domains indicative of a vestigial PDW. Simultaneous imaging of the local-density-of-states N(r,E) reveals electronic modulations with wavevectors Q and 2Q, as anticipated when the PDW coexists with superconductivity. Finally, by visualizing the topological defects in these N(r,E) density waves at 2Q, we discover them to be concentrated in areas where the PDW spatial phase changes by {\pi}, as predicted by the theory of half-vortices in a PDW state. Overall, this is a compelling demonstration, from multiple single-electron signatures, of a PDW state coexisting with superconductivity at zero magnetic field, in the canonical cuprate Bi2Sr2CaCu2O8+{\delta}.