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When very high magnetic fields suppress the superconductivity in underdopedcuprates, an exceptional new electronic phase appears. It supports remarkableand unexplained quantum oscillations and exhibits an unidentified density wave(DW) state. Although generally referred to as a "charge" density wave (CDW)because of the observed charge density modulations, theory indicates that thiscould actually be the far more elusive electron-pair density wave state (PDW).To search for evidence of a field-induced PDW in cuprates, we visualize themodulations in the density of electronic states $N(\bf{r})$ within the halosurrounding Bi$_2$Sr$_2$CaCu$_2$O$_8$ vortex cores. This reveals multiplesignatures of a field-induced PDW, including two sets of $N(\bf{r})$modulations occurring at wavevectors $\bf{Q}_P$ and $2\bf{Q}_P$, both havingpredominantly $s$-symmetry form factors, the amplitude of the latter decayingtwice as rapidly as the former, along with induced energy-gap modulations at$\bf{Q}_P$ . Such a microscopic phenomenology is in detailed agreement withtheory for a field-induced primary PDW that generates secondary CDWs within thevortex halo. These data indicate that the fundamental state generated byincreasing magnetic fields from the underdoped cuprate superconducting phase isactually a PDW with approximately eight CuO$_2$ unit-cell periodicity ($\lambda= 8a_0$) and predominantly $d$-symmetry form factor.

Magnetic-field Induced Pair Density Wave State in the Cuprate Vortex Halo