Abstract
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^{1,2}. The resemblance of its crystallographic, magnetic and electronic structures^{1,2,3,4,5,6} to La_{2}CuO_{4}, as well as the emergence on doping of a pseudogap region^{7,8,9} and a lowtemperature dwave gap^{10,11}, has particularly strengthened analogies to cuprate highT_{c} superconductors^{12}. However, unlike the cuprate phase diagram, which features a plethora of broken symmetry phases^{13} in a pseudogap region that includes charge density wave, stripe, nematic and possibly intraunitcell loopcurrent orders, no broken symmetry phases proximate to the parent antiferromagnetic Mott insulating phase in Sr_{2}IrO_{4} have been observed so far, making the comparison of iridate to cuprate phenomenology incomplete. Using optical secondharmonic generation, we report evidence of a hidden nondipolar 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°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 magnetoelectric loopcurrent order^{14,15,16,17,18}. 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}.
Main
The crystal structure of the 5d transition metal oxide Sr_{2}IrO_{4} is tetragonal and centrosymmetric with fourfold (C_{4}) rotational symmetry about the caxis^{2,3,4,5,6}. It is composed of stacked IrO_{2} square lattices whose unit cell is doubled relative to the CuO_{2} square lattices found in highT_{c} 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, Sr_{2}IrO_{4} is a Mott insulator by virtue of strong atomic spin–orbit coupling^{1,2}. The electronic states near the Fermi level derive predominantly from the t_{2g} dorbitals of iridium ions, which are well approximated as two completely filled spin–orbital entangled J_{eff} = 3/2 bands and one halffilled J_{eff} = 1/2 band, which is split into an upper and lower Hubbard band by an onsite Coulomb interaction. Below a Néel temperature T_{N} ∼ 230 K, the spin–orbital entangled J_{eff} = 1/2 magnetic dipole moments undergo threedimensional longrange ordering into an orthorhombic antiferromagnetic structure^{2,3,4,5} that preserves global inversion symmetry but lowers the rotational symmetry of the system from C_{4} to C_{2}. No additional symmetry breaking has otherwise been observed by neutron or Xray diffraction.
Optical secondharmonic generation (SHG), a frequency doubling of light through its nonlinear interaction with a material, is strongly affected by point group symmetry changes^{19} and can be used to search for hidden phases with higherrank tensor order parameters that are difficult to detect using conventional probes^{18,20}. It is particularly sensitive to inversion symmetry breaking because the leadingorder electricdipole (ED) contribution to SHG, which is described by a thirdrank susceptibility tensor χ_{ijk}^{ED} that relates the nonlinear polarization at the secondharmonic frequency 2ω to the incident electric field via P_{i}(2ω) ∝ χ_{ijk}^{ED}E_{j}(ω)E_{k}(ω), is allowed only if the crystal lacks an inversion centre. Otherwise SHG can arise only from much weaker higherrank multipole processes, as is the case for the Sr_{2}IrO_{4} 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 roomtemperature RASHG patterns of Sr_{2}IrO_{4} measured by rotating the scattering plane about its caxis. The data measured under both P_{in}–P_{out} and P_{in}–S_{out} polarization geometries (see Supplementary Section 1 for other geometries) exhibit a clear C_{4} symmetry and are completely accounted for by the leadingorder nonlocal bulk contribution to SHG of electricquadrupole (EQ) type^{6}, which can be written as an effective nonlinear polarization P_{i}(2ω) ∝ χ_{ijkl}^{EQ}E_{j}(ω)∇_{k}E_{l}(ω). No surface ED contribution to SHG was detected within our instrument sensitivity^{6}. 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 P_{in}–S_{out} pattern clearly signify the absence of mirror symmetry across the a–c and b–c planes. Expressions for the RASHG patterns (Supplementary Section 2) calculated using the set of independent nonvanishing tensor elements of the fourthrank susceptibility tensor χ_{ijkl}^{EQ} derived from the experimentally established centrosymmetric 4/m crystallographic point group of Sr_{2}IrO_{4} fit the data extremely well (Fig. 1c).
Remarkably, the rotational symmetry of the RASHG patterns measured at low temperature (T = 170 K) is reduced from C_{4} to C_{1} (Fig. 1d). This is not caused by a structural distortion because extensive neutron and resonant Xray diffraction studies show no change in crystallographic symmetry below room temperature^{2,3,4,5}. It cannot be accounted for by the antiferromagnetic structure that develops below T_{N} ∼ 230 K because that has a centrosymmetric orthorhombic magnetic point group (2/m1′) with C_{2} 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 Sr_{2}IrO_{4} above T = 170 K (ref. 21). Instead, the ordering of a higher multipolar degree of freedom that coexists with the J_{eff} = 1/2 moment in each IrO_{6} octahedron is left as the most plausible explanation. An ordering of higherrank parityeven magnetic multipoles with the same propagation wavevector as the antiferromagnetic structure has, in fact, been proposed to occur in Sr_{2}IrO_{4} below T_{N} (refs 22, 23). Such an order would naturally be difficult to detect using diffractionbased 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 Sr_{2}IrO_{4} that do not include the twofold 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 RASHG pattern calculated using a coherent sum of the crystallographic EQ and the hiddenorderinduced ED contributions (Supplementary Section 2), with the elements of χ_{ijkl}^{EQ} derived from a 4/m point group and those of χ_{ijk}^{ED} derived from either a 2′/m or m1′ point group, fit the data extremely well (Fig. 1d). However this indicates that an ordering of higherrank parityeven magnetic multipoles, which preserves global inversion symmetry, cannot be the origin of our RASHG results.
On the other hand, our results can be explained by an ordering of higherrank parityodd 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 magnetoelectric loopcurrent order (Supplementary Section 4), which is predicted to exist in the pseudogap region of the cuprates^{14,15,16,17,18,20} but can in principle persist even at halffilling^{24}. This phase consists of a pair of countercirculating current loops in each CuO_{2} square plaquette (Fig. 1a inset) and can be described by a toroidal pseudovector^{25,26} order parameter defined as Ω = Σr_{i} × m_{i}, where r_{i} is the location of the orbital magnetic moment m_{i} inside the plaquette. Four degenerate configurations are possible because the two intraunitcell current loops can lie along either of two diagonals in the square plaquette and can have either of two timereversed 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 Sr_{2}IrO_{4} we performed widefield reflection SHG microscopy measurements. A roomtemperature P_{in}–S_{out} 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 RASHG 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 RASHG 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 Sr_{2}IrO_{4} being a magnetoelectric loopcurrent order—but other microscopic models that obey the same set of symmetries certainly cannot be ruled out.
The ordering temperature T_{Ω} of the hidden phase in Sr_{2}IrO_{4} was determined by measuring the temperature dependence of the SHG intensity in the P_{in}–S_{out} polarization geometry where the hiddenorderinduced changes are largest. The onset of change is clearly observed at T_{Ω} ∼ 232 K, as shown in Fig. 4a, which is close to but slightly higher than the value of T_{N} ∼ 230 K determined by d.c. magnetic susceptibility measurements, and evolves with an orderparameterlike 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 T_{Ω} and T_{N} should coincide. The close proximity of T_{Ω} and T_{N} observed in Sr_{2}IrO_{4} therefore suggests some microscopic mechanism by which one order can induce the other through a biquadratic coupling—an enhancement of the superexchange coupling between J_{eff} = 1/2 moments due to the presence of hidden order being one possible scenario.
To experimentally examine whether or not the Néel and hidden orders are trivially tied, we performed analogous SHG experiments on holedoped Sr_{2}Ir_{1−x}Rh_{x}O_{4} crystals (Supplementary Section 9) to track the evolution of T_{Ω} as a function of Rh concentration (x). Bulk magnetization^{27} and resonant Xray diffraction^{28} studies have shown that Néel ordering persists for x ≲ 0.17 and that T_{N} is monotonically suppressed with x. But no evidence of any broken symmetry phases beyond the Néel phase has been reported in Sr_{2}Ir_{1−x}Rh_{x}O_{4} 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 T_{Ω} is suppressed with x, the splitting between T_{Ω} and T_{N} 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 Sr_{2}IrO_{4} phase diagrams (Fig. 4d), which is further strengthened by recent observations of a pseudogap region in Sr_{2}Ir_{1−x}Rh_{x}O_{4} using angleresolved photoemission spectroscopy^{8}. Driving the hidden phase to a quantum critical point^{27} through higher doping may be a route to achieving highT_{c} (ref. 12) or parityodd^{29,30} 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 magnetoelectric loopcurrent order already suggests several interesting macroscopic responses, including a linear magnetoelectric effect^{18} and nonreciprocal optical rotation^{17,18}.
Methods
Material growth.
Single crystals of Sr_{2}IrO_{4} and Sr_{2}Ir_{1−x}Rh_{x}O_{4} were grown using a self flux technique from offstoichiometric quantities of IrO_{2}, SrCO_{3} and SrCl_{2} or RhO_{2}, IrO_{2}, SrCO_{3} and SrCl_{2} 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 energydispersive Xray spectroscopy.
RASHG 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 (KMLabsWyvern). The RASHG measurements were performed by means of a rotating scattering plane technique^{31} using an SHG wavelength λ = 400 nm that is resonant with the O 2p to J_{eff} = 1/2 upper Hubbard band transition^{32}. 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 Sr_{2}IrO_{4}. Reflected SHG light was collected using a photomultiplier tube. Samples were cleaved either in air or in a nitrogenpurged environment and immediately pumped down to a pressure <5 × 10^{−6} torr in an optical cryostat. Crystals were oriented before measurement using Xray Laue diffraction.
SHG microscopy measurements.
Widefield SHG images with a spatial resolution of ∼1 μm were collected using the same light source as that used for the RASHG 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 electronmultiplying chargecoupled device (CCD) camera.
Fitting procedure.
The low(T < T_{Ω}) and hightemperature (T > T_{Ω}) RASHG 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 χ_{ijkl}^{EQ}(ϕ) and χ_{ijk}^{ED}(ϕ) are the bulk electricquadrupole and electricdipole susceptibility tensors, respectively, transformed into the rotated frame of the scattering plane. The nonzero independent elements of the tensors in the unrotated frame of the crystal are deduced by applying the appropriate point group (4/m for χ_{ijkl}^{EQ} and 2′/m or m1′ for χ_{ijk}^{ED}) and degenerate SHG permutation symmetries. This reduces χ_{ijkl}^{EQ} to 17 nonzero 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 χ_{ijk}^{ED} to 10 nonzero independent elements (xxx; xyx = xxy; xyy; xzz; yxx; yyx = yxy; yyy; yzz; zzx = zxz; zzy = zyz).
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Acknowledgements
We thank S. Lovesey and D. Khalyavin for providing information about the magnetic point group of the Néel order in Sr_{2}IrO_{4}. We acknowledge useful discussions with P. Armitage, L. Fu, A. Kaminski, P. A. Lee, O. Motrunich, J. Orenstein, N. Perkins, S. Todadri, C. Varma and V. Yakovenko. This work was support by ARO Grant W911NF1310059. Instrumentation for the SHG measurements was partially supported by ARO DURIP Award W911NF1310293. D.H. acknowledges funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (PHY1125565) with support of the Gordon and Betty Moore Foundation through Grant GBMF1250. R.F. acknowledges the hospitality of the Aspen Center for Physics, supported by NSF Grant PHYS1066293, where some of this work was carried out. G.C. acknowledges NSF support via Grant DMR1265162. R.L. acknowledges support from the Israel Science Foundation through Grant 556/10.
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L.Z. and D.H. planned the experiment. L.Z., D.H.T., H.C. and V.I. performed the measurements. L.Z. and R.L. performed the magnetic point group symmetry analysis. R.F. performed the Landau free energy calculation. T.Q. and G.C. prepared and characterized the samples. L.Z., R.F. and D.H. analysed the data and wrote the manuscript.
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Zhao, L., Torchinsky, D., Chu, H. et al. Evidence of an oddparity hidden order in a spin–orbit coupled correlated iridate. Nature Phys 12, 32–36 (2016). https://doi.org/10.1038/nphys3517
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DOI: https://doi.org/10.1038/nphys3517
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