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The crystal structure of the 5d transition metal oxide Sr2IrO4 is tetragonal and centrosymmetric with four-fold (C4) rotational symmetry about the c-axis2,3,4,5,6. It is composed of stacked IrO2 square lattices whose unit cell is doubled relative to the CuO2 square lattices found in high-Tc cuprates as a result of a staggered rotation of the octahedral oxygen cages (Fig. 1a). Despite the much larger spatial extent of 5d versus 3d orbitals, Sr2IrO4 is a Mott insulator by virtue of strong atomic spin–orbit coupling1,2. The electronic states near the Fermi level derive predominantly from the t2g d-orbitals of iridium ions, which are well approximated as two completely filled spin–orbital entangled Jeff = 3/2 bands and one half-filled Jeff = 1/2 band, which is split into an upper and lower Hubbard band by an on-site Coulomb interaction. Below a Néel temperature TN ∼ 230 K, the spin–orbital entangled Jeff = 1/2 magnetic dipole moments undergo three-dimensional long-range ordering into an orthorhombic antiferromagnetic structure2,3,4,5 that preserves global inversion symmetry but lowers the rotational symmetry of the system from C4 to C2. No additional symmetry breaking has otherwise been observed by neutron or X-ray diffraction.

Figure 1: Symmetry of the hidden order in Sr2IrO4.
figure 1

a, Crystal structure of a single perovskite layer in Sr2IrO4 (tetragonal point group 4/m). Inset shows the basic IrO6 unit of the magneto-electric loop-current order, with the arrows pointing in the direction of current flow. b, Schematic of the RA-SHG experiment. The electric field polarizations of the obliquely incident fundamental beam (in) and outgoing SHG beam (out) can be independently selected to lie either parallel (P) or perpendicular (S) to the light scattering plane (shaded). RA-SHG data are acquired by measuring the SHG intensity I(2ω) reflected from the (001) surface of Sr2IrO4 as a function of the angle ϕ between the scattering plane and crystal a–c-plane. c, RA-SHG data collected under Pin–Pout and Pin–Sout polarization geometries using λ = 800 nm incident light at T = 295 K. d, Analogous data to c collected at T = 170 K. All data sets are plotted on the same intensity scale, normalized to a value of 1, which corresponds to ∼20 fW. The high-temperature data are fitted to time-reversal invariant bulk electric-quadrupole-induced SHG (orange lines). The low-temperature data can only be fitted to the coherent sum of time-reversal invariant bulk electric-quadrupole-induced and time-reversal broken bulk electric-dipole-induced SHG (purple lines), as described in the text.

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Optical second-harmonic generation (SHG), a frequency doubling of light through its nonlinear interaction with a material, is strongly affected by point group symmetry changes19 and can be used to search for hidden phases with higher-rank tensor order parameters that are difficult to detect using conventional probes18,20. It is particularly sensitive to inversion symmetry breaking because the leading-order electric-dipole (ED) contribution to SHG, which is described by a third-rank susceptibility tensor χijkED that relates the nonlinear polarization at the second-harmonic frequency 2ω to the incident electric field via Pi(2ω) ∝ χijkEDEj(ω)Ek(ω), is allowed only if the crystal lacks an inversion centre. Otherwise SHG can arise only from much weaker higher-rank multipole processes, as is the case for the Sr2IrO4 crystal structure described below. The rotational symmetry of a crystal can be directly determined by performing a rotational anisotropy (RA) SHG experiment where the intensity of obliquely reflected SHG is measured as a function of the angle ϕ through which the scattering plane is rotated about the surface normal of the crystal (Fig. 1b). For our experiment, the incident light is focused onto optically flat regions of the crystal surface with a spot size d < 100 μm. The polarizations of the incident (in) and reflected (out) beams can be independently selected to be either parallel (P) or perpendicular (S) to the scattering plane, thus allowing different nonlinear susceptibility tensor elements to be probed.

Figure 1c shows room-temperature RA-SHG patterns of Sr2IrO4 measured by rotating the scattering plane about its c-axis. The data measured under both Pin–Pout and Pin–Sout polarization geometries (see Supplementary Section 1 for other geometries) exhibit a clear C4 symmetry and are completely accounted for by the leading-order non-local bulk contribution to SHG of electric-quadrupole (EQ) type6, which can be written as an effective nonlinear polarization Pi(2ω) ∝ χijklEQEj(ω)∇kEl(ω). No surface ED contribution to SHG was detected within our instrument sensitivity6. As discussed in ref. 6, the rotation of the SHG intensity maxima away from the crystallographic a and b axes and their modulated amplitude in the Pin–Sout pattern clearly signify the absence of mirror symmetry across the a–c and b–c planes. Expressions for the RA-SHG patterns (Supplementary Section 2) calculated using the set of independent non-vanishing tensor elements of the fourth-rank susceptibility tensor χijklEQ derived from the experimentally established centrosymmetric 4/m crystallographic point group of Sr2IrO4 fit the data extremely well (Fig. 1c).

Remarkably, the rotational symmetry of the RA-SHG patterns measured at low temperature (T = 170 K) is reduced from C4 to C1 (Fig. 1d). This is not caused by a structural distortion because extensive neutron and resonant X-ray diffraction studies show no change in crystallographic symmetry below room temperature2,3,4,5. It cannot be accounted for by the antiferromagnetic structure that develops below TN ∼ 230 K because that has a centrosymmetric orthorhombic magnetic point group (2/m1′) with C2 symmetry (Supplementary Section 3). Ferro- or antiferroelectric order can also be ruled out owing to the absence of any anomaly in the temperature dependence of the dielectric constant of Sr2IrO4 above T = 170 K (ref. 21). Instead, the ordering of a higher multipolar degree of freedom that coexists with the Jeff = 1/2 moment in each IrO6 octahedron is left as the most plausible explanation. An ordering of higher-rank parity-even magnetic multipoles with the same propagation wavevector as the antiferromagnetic structure has, in fact, been proposed to occur in Sr2IrO4 below TN (refs 22, 23). Such an order would naturally be difficult to detect using diffraction-based probes because it preserves the translational symmetry of the antiferromagnetic lattice and imparts no net magnetization to the crystal.

To investigate this possibility, we surveyed all magnetic subgroups of the crystallographic 4/m point group of Sr2IrO4 that do not include the two-fold rotation axis (2) as a group element. We find that our data are uniquely but equally well described by two subgroups (2′/m and m1′) of the antiferromagnetic point group. Both 2′/m and m1′ break the global inversion symmetry of the crystal and thus allow a bulk ED contribution to SHG on top of the existing EQ contribution, which is consistent with the large changes observed in the SHG amplitude. Moreover, expressions for the RA-SHG pattern calculated using a coherent sum of the crystallographic EQ and the hidden-order-induced ED contributions (Supplementary Section 2), with the elements of χijklEQ derived from a 4/m point group and those of χijkED derived from either a 2′/m or m1′ point group, fit the data extremely well (Fig. 1d). However this indicates that an ordering of higher-rank parity-even magnetic multipoles, which preserves global inversion symmetry, cannot be the origin of our RA-SHG results.

On the other hand, our results can be explained by an ordering of higher-rank parity-odd magnetic multipoles that preserves the translational symmetry of either the antiferromagnetic structure or the crystallographic structure. A microscopic model that satisfies this condition as well as the 2′/m or m1′ point group symmetries is the magneto-electric loop-current order (Supplementary Section 4), which is predicted to exist in the pseudogap region of the cuprates14,15,16,17,18,20 but can in principle persist even at half-filling24. This phase consists of a pair of counter-circulating current loops in each CuO2 square plaquette (Fig. 1a inset) and can be described by a toroidal pseudovector25,26 order parameter defined as Ω = Σri × mi, where ri is the location of the orbital magnetic moment mi inside the plaquette. Four degenerate configurations are possible because the two intra-unit-cell current loops can lie along either of two diagonals in the square plaquette and can have either of two time-reversed configurations, which correspond to four 90°-rotated directions of the pseudovector. In a real material, one therefore expects domains of all four types to be populated.

To search for domains of the hidden order in Sr2IrO4 we performed wide-field reflection SHG microscopy measurements. A room-temperature Pin–Sout SHG image collected on a clean ∼500 μm × 500 μm region parallel to the a–b plane is shown in Fig. 2a. This region produces a uniform SHG response consistent with the behaviour of a single crystallographic domain. On cooling to T = 175 K, brighter and darker patches separated by boundaries that are straight over a length scale of tens of micrometres become visible (Fig. 2b). The distribution and shapes of these patches can be rearranged on thermal cycling (Supplementary Section 5), which suggests that they are not pinned to structural defects in the crystal. To examine whether the different patches observed in Fig. 2b correspond to domains with different Ω orientation, we performed local RA-SHG measurements within each of the patches. An exhaustive study over the entire crystal area in Fig. 2b reveals a total of only four types of patches, characterized by the four distinct RA-SHG patterns (Supplementary Section 6) shown in Fig. 3, which are exactly 0°-, 90°-, 180°- and 270°-rotated copies of those shown in Fig. 1. These results are consistent with the hidden phase in Sr2IrO4 being a magneto-electric loop-current order—but other microscopic models that obey the same set of symmetries certainly cannot be ruled out.

Figure 2: Spatial mapping of hidden order domains.
figure 2

a,b, Wide-field reflection SHG microscopy images of the cleaved (001) plane of Sr2IrO4 measured under Pin–Sout polarization geometry at ϕ = 78° at T = 295 K (a) and at T = 175 K (b). A patchwork of light and dark regions present in the low-temperature image arises from domains of the hidden order. The dark curves at the sample edge and dark micrometre-sized spots near the sample centre that are visible in both the high- and low-temperature images are structural defects.

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Figure 3: Degenerate ground-state configurations of the hidden order.
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Four different types of RA-SHG patterns found by performing local measurements within all of the domains mapped in Fig. 2b. Red lines are fits to the expressions described in the text. The large lobes are shaded pink to emphasize the orientation of each pattern. Schematics of the four degenerate magneto-electric loop-current order configurations are shown below each pattern to illustrate the possible correspondence. The red arrows denote the direction of the toroidal moment Ω in each plaquette.

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The ordering temperature TΩ of the hidden phase in Sr2IrO4 was determined by measuring the temperature dependence of the SHG intensity in the Pin–Sout polarization geometry where the hidden-order-induced changes are largest. The onset of change is clearly observed at TΩ ∼ 232 K, as shown in Fig. 4a, which is close to but slightly higher than the value of TN ∼ 230 K determined by d.c. magnetic susceptibility measurements, and evolves with an order-parameter-like behaviour on further cooling (Supplementary Section 7). In a Landau free energy expansion (Supplementary Section 8), a bilinear coupling between the antiferromagnetic Néel and hidden order parameters is forbidden by symmetry, so there is no a priori reason that TΩ and TN should coincide. The close proximity of TΩ and TN observed in Sr2IrO4 therefore suggests some microscopic mechanism by which one order can induce the other through a biquadratic coupling—an enhancement of the super-exchange coupling between Jeff = 1/2 moments due to the presence of hidden order being one possible scenario.

Figure 4: Temperature and hole-doping dependence of the hidden order.
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a-c, Change in SHG intensity from Sr2Ir1−xRhxO4 at x = 0 (a),x = 0.04 (b) and x = 0.11 (c) measured relative to their room-temperature values as a function of temperature. Data were taken under Pin–Sout polarization geometry at ϕ = 78°. The error bars represent the standard deviation over 60 independent measurements. No difference was observed between curves measured on cooling and heating. Lines are guides to the eye. The dashed red lines mark the transition temperature TΩ of the hidden order phase deduced from our SHG data and the dashed black lines mark the Néel temperature TN determined from d.c. magnetic susceptibility measurements. d, Temperature versus doping phase diagram of Sr2Ir1−xRhxO4 showing the boundaries of the hidden order and the long-range (LRO) and short-range (SRO) Néel ordered regions. Points where a pseudogap is present are also marked, although a pseudogap phase boundary is not yet experimentally known.

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To experimentally examine whether or not the Néel and hidden orders are trivially tied, we performed analogous SHG experiments on hole-doped Sr2Ir1−xRhxO4 crystals (Supplementary Section 9) to track the evolution of TΩ as a function of Rh concentration (x). Bulk magnetization27 and resonant X-ray diffraction28 studies have shown that Néel ordering persists for x ≲ 0.17 and that TN is monotonically suppressed with x. But no evidence of any broken symmetry phases beyond the Néel phase has been reported in Sr2Ir1−xRhxO4 so far. Remarkably, our SHG measurements show that the hidden phase transition observed in the parent compound also persists on hole doping (Fig. 4b, c) and that, although TΩ is suppressed with x, the splitting between TΩ and TN grows monotonically from approximately 2 K to 75 K between x = 0 and x ∼ 0.11. This provides strong evidence that the Néel and hidden orders are not trivially tied, but are independent and distinct electronic phases.

Our finding of a hidden broken symmetry phase in proximity to an antiferromagnetic Mott insulator reveals a striking parallel between the cuprate and Sr2IrO4 phase diagrams (Fig. 4d), which is further strengthened by recent observations of a pseudogap region in Sr2Ir1−xRhxO4 using angle-resolved photoemission spectroscopy8. Driving the hidden phase to a quantum critical point27 through higher doping may be a route to achieving high-Tc (ref. 12) or parity-odd29,30 superconductivity in the iridates. Although further theoretical studies are required to establish the microscopic origin of the hidden phase, the fact that it bears the symmetries of a magneto-electric loop-current order already suggests several interesting macroscopic responses, including a linear magneto-electric effect18 and non-reciprocal optical rotation17,18.

Methods

Material growth.

Single crystals of Sr2IrO4 and Sr2Ir1−xRhxO4 were grown using a self flux technique from off-stoichiometric quantities of IrO2, SrCO3 and SrCl2 or RhO2, IrO2, SrCO3 and SrCl2 respectively. The ground mixtures of powders were melted at 1,470 °C in partially capped platinum crucibles. The soaking phase of the synthesis lasted for >20 h and was followed by a slow cooling at 2 °C h−1 to reach 1,400 °C. From this point the crucible is brought to room temperature through a rapid cooling at a rate of 100 °C h−1. The Rh concentration was determined by energy-dispersive X-ray spectroscopy.

RA-SHG measurements.

Ultrashort optical pulses with 35 fs duration and 800 nm centre wavelength were produced at a 10 kHz repetition rate from a regeneratively amplified Ti:sapphire laser (KMLabs-Wyvern). The RA-SHG measurements were performed by means of a rotating scattering plane technique31 using an SHG wavelength λ = 400 nm that is resonant with the O 2p to Jeff = 1/2 upper Hubbard band transition32. Light was obliquely incident onto the sample at a 30° angle of incidence with a fluence that was maintained below 1 mJ cm−2, which is well below the damage threshold of Sr2IrO4. Reflected SHG light was collected using a photomultiplier tube. Samples were cleaved either in air or in a nitrogen-purged environment and immediately pumped down to a pressure <5 × 10−6 torr in an optical cryostat. Crystals were oriented before measurement using X-ray Laue diffraction.

SHG microscopy measurements.

Wide-field SHG images with a spatial resolution of ∼1 μm were collected using the same light source as that used for the RA-SHG measurements. A 40° oblique angle of incidence was used and the incident light fluence was maintained below 1 mJ cm−2. The reflected SHG light was collected by an objective lens and focused onto an electron-multiplying charge-coupled device (CCD) camera.

Fitting procedure.

The low-(T < TΩ) and high-temperature (T > TΩ) RA-SHG data were fitted to the expressions and respectively, where A is a constant determined by the experimental geometry, I(ω) is the intensity of the incident beam, is the polarization of the incoming fundamental or outgoing SHG light and χijklEQ(ϕ) and χijkED(ϕ) are the bulk electric-quadrupole and electric-dipole susceptibility tensors, respectively, transformed into the rotated frame of the scattering plane. The non-zero independent elements of the tensors in the unrotated frame of the crystal are deduced by applying the appropriate point group (4/m for χijklEQ and 2′/m or m1′ for χijkED) and degenerate SHG permutation symmetries. This reduces χijklEQ to 17 non-zero independent elements (xxxx = yyyy; zzzz; zzxx = zzyy = zxxz = zyyz; xyzz = −yxzz; xxyy = yyxx = xyyx = yxxy; xxxy = −yyyx = xyxx = −yxyy; xxzz = yyzz; zzxy = −zzyx = −zxyz = zyxz; xyxy = yxyx; xxyx = −yyxy; zxzx = zyzy; xzyz = −yzxz; xzxz = yzyz; zxzy = −zyzx; yxxx = −xyyy; xzzx = yzzy; xzzy = −yzzx) and reduces χijkED to 10 non-zero independent elements (xxx; xyx = xxy; xyy; xzz; yxx; yyx = yxy; yyy; yzz; zzx = zxz; zzy = zyz).