Evidence of an odd-parity hidden order in a spin-orbit coupled correlated iridate

A rare combination of strong spin-orbit coupling and electron-electron correlations makes the iridate Mott insulator Sr$_2$IrO$_4$ a promising host for novel electronic phases of matter. The resemblance of its crystallographic, magnetic and electronic structures to La$_2$CuO$_4$, as well as the emergence upon doping of a pseudogap region and a low temperature $d$-wave gap, has particularly strengthened analogies to cuprate high-$T_c$ superconductors. However, unlike the cuprate phase diagram that features a plethora of broken symmetry phases in a pseudogap region that include charge density wave, stripe, nematic and possibly intra-unit cell loop-current orders, no broken symmetry phases proximate to the parent antiferromagnetic Mott insulating phase in Sr$_2$IrO$_4$ have been observed to date, making the comparison of iridate to cuprate phenomenology incomplete. Using optical second harmonic generation, we report evidence of a hidden non-dipolar magnetic order in Sr$_2$IrO$_4$ that breaks both the spatial inversion and rotational symmetries of the underlying tetragonal lattice. Four distinct domain types corresponding to discrete 90$^{\circ}$ rotated orientations of a pseudovector order parameter are identified using nonlinear optical microscopy, which is expected from an electronic phase that possesses the symmetries of a magneto-electric loop-current order. The onset temperature of this phase is monotonically suppressed with bulk hole doping, albeit much more weakly than the Néel temperature, revealing an extended region of the phase diagram with purely hidden order. Driving this hidden phase to its quantum critical point may be a path to realizing superconductivity in Sr$_2$IrO$_4$.