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The CuO$_2$ antiferromagnetic insulator is transformed by hole-doping into anexotic quantum fluid usually referred to as the pseudogap (PG) phase. Itsdefining characteristic is a strong suppression of the electronicdensity-of-states D(E) for energies |E|<$\Delta^*$, where $\Delta^*$ is thepseudogap energy. Unanticipated broken-symmetry phases have been detected by awide variety of techniques in the PG regime, most significantly a finite Qdensity-wave (DW) state and a Q=0 nematic (NE) state. Sublattice-phase-resolvedimaging of electronic structure allows the doping and energy dependence ofthese distinct broken symmetry states to be visualized simultaneously. Usingthis approach, we show that, even though their reported ordering temperaturesT$_{DW}$ and T$_{NE}$ are unrelated to each other, both the DW and NE statesalways exhibit their maximum spectral intensity at the same energy, and usingindependent measurements that this is the pseudogap energy $\Delta^*$.Moreover, no new energy-gap opening coincides with the appearance of the DWstate (which should theoretically open an energy gap on the Fermi-surface),while the observed pseudogap opening coincides with the appearance of the NEstate (which should theoretically be incapable of opening a Fermi-surface gap).We demonstrate how this perplexing phenomenology of thermal transitions andenergy-gap opening at the breaking of two highly distinct symmetries can beunderstood as the natural consequence of a vestigial nematic state , within thepseudogap phase of Bi$_2$Sr$_2$CaCu$_2$O$_8$.

Evidence for a Vestigial Nematic State in the Cuprate Pseudogap Phase