## Abstract

Measurement of the quantum-mechanical phase in quantum matter provides the most direct manifestation of the underlying abstract physics. We used resonant x-ray scattering to probe the relative phases of constituent atomic orbitals in an electronic wave function, which uncovers the unconventional Mott insulating state induced by relativistic spin-orbit coupling in the layered 5*d* transition metal oxide Sr_{2}IrO_{4}. A selection rule based on intra-atomic interference effects establishes a complex spin-orbital state represented by an effective total angular momentum = 1/2 quantum number, the phase of which can lead to a quantum topological state of matter.

Transition metal oxides (TMOs) with perovskite structure are hosts to many fascinating phenomena, including high-temperature superconductivity (*1*) and colossal magnetoresistance (*2*), in which the valence *d*-electron states are described in terms of crystal-field (CF) states: triply degenerate *t*_{2g} states (*xy, yz, zx*) and doubly degenerate *e*_{g} states (*x*^{2} – *y*^{2}, 3*z*^{2} – *r*^{2}). These CF states are all pure real functions, so that when the degeneracy is removed (e.g., by the Jahn-Teller effect), the orbital angular momentum is totally quenched. However, when the spin-orbit coupling (SOC) becomes effective, the CF states are mixed with complex phases, which may partially restore the orbital angular momentum in the *t*_{2g} manifold. This effect is particularly pronounced in TMOs with heavy 5*d* elements, where SOC is at least an order of magnitude larger than those of TMOs with 3*d* elements and can sometimes give rise to unconventional electronic states.

5*d* TMO Sr_{2}IrO_{4} is a layered perovskite with low-spin *d*^{5} configuration, in which five electrons are accommodated in almost triply degenerate *t*_{2g} orbitals. Metallic ground states are expected in 5*d* TMOs because of their characteristic wide bands and small Coulomb interactions as compared with those of 3*d* TMOs. Sr_{2}IrO_{4}, however, is known to be a magnetic insulator (*3*, *4*). A recent study has shown that the strong SOC inherent to 5*d* TMOs can induce a Mott instability even in such a weakly correlated electron system (*5*), resulting in a localized state very different from the well-known spin *S* = 1/2 state for conventional Mott insulators, proposed to be an effective total angular momentum *J*_{eff} = 1/2 state in the strong SOC limit expressed as (1) where *m* is the component of *J*_{eff} along the quantization axis and σ denotes the spin state. This state derives from the addition of *S* = 1/2 to the effective orbital angular momentum *L*_{eff} = 1, which consists of triply degenerate *t*_{2g} states but acts like the atomic *L* = 1 state with a minus sign; that is, *L*_{eff} = –*L*. As a result, *J*_{eff} = 1/2 has orbital moment parallel to spin (*6*). Note the characteristic equal mixture of *xy, yz*, and *zx* orbitals with complex number *i* involved in one of the factors and the mixed up-and-down spin states (*7*).

This realization of a Mott insulator with *J*_{eff} = 1/2 moment provides a new playground for correlated electron phenomena, because emergent physical properties that arise from it can be drastically different from those of the conventional Mott insulators. A prime example is when *J*_{eff} = 1/2 is realized in a honeycomb lattice structure where electrons hopping between *J*_{eff} = 1/2 states acquire complex phase; it generates a Berry phase leading to the recent prediction of quantum spin-Hall effect at room temperature (*8*), and it also leads to the low-energy Hamiltonian of Kitaev model relevant for quantum computing (*9*). Experimental establishment of the *J*_{eff} = 1/2 state is thus an important step toward these physics, and the direct probe of complex phase in the wave function has been awaited. However, it is usually difficult to retrieve the phase information experimentally, because it is always the intensity, the square modulus of the wave function, that is measured; and thus a reference, with which the state under measurement can interfere, is required.

The resonant x-ray scattering (RXS) technique uses resonance effects at an x-ray absorption edge to selectively enhance the signal of interest, and has become a powerful tool for investigating ordering phenomena (*10*, *11*). So far, the emphasis has been seen only in the amplification of the signal. However, the RXS signal contains important information about the phase of the wave function for valence electrons, because RXS results from quantum interference between different scattering paths via intermediate states of a single site. The RXS process is described by the second-order process of electron-photon coupling perturbation, as schematically shown in Fig. 1, and its scattering amplitude *f*_{αβ} from a single site is expressed under dipole approximation by (2) In this process, a photon with energy (ħ)ω is scattered by being virtually absorbed and emitted with polarizations α and β, respectively; and in the course of the process, an electron of mass *m*_{e} makes dipole transitions through position operators *R*_{α} and *R*_{β} from and to the initial state *i*, via all possible intermediate states *m*, collecting the phase factors associated with the intermediate states, weighted by some factors involving energy differences between the initial and intermediate states (ħ)ω_{im} and the lifetime broadening energy Γ. The interference between various scattering paths is directly reflected in the scattering intensities of the photon, and in this way the valence electronic states can be detected with phase sensitivity. This process can be contrasted with that in x-ray absorption spectroscopy (XAS), which is a first-order process and measures only the amplitudes of the individual paths, or transition probabilities to various valence states.

We have applied this technique to explore unconventional electronic states produced by the strong SOC in Sr_{2}IrO_{4}. Sr_{2}IrO_{4} is an ideal system in which to fully use this technique. The magnetic Bragg diffraction in magnetically ordered Sr_{2}IrO_{4} comes essentially from scattering by Ir *t*_{2g} electrons, to which RXS using the *L* edge (2*p*→5*d*) can be applied to examine the electronic states. The wavelength at the *L* edge of 5*d* Ir is as short as ∼1 Å, in marked contrast to >10 Å for 3*d* elements. This short wavelength makes the detection of RXS signals much easier than in 3*d* TMOs, because there exists essentially no constraint from the wavelength in detecting the magnetic Bragg signal. Moreover, the low-spin 5*d*^{5} configuration, a one-hole state, greatly reduces the number of intermediate states and makes the calculation of scattering matrix elements tractable. The excitation to the *t*_{2g} state completely fills the manifold, and the remaining degrees of freedom reside only in the 2*p* core holes. Because the intermediate states are all degenerate in this case, the denominator factors involving energies and lifetimes of the intermediate states in Eq. 2 can drop out. A careful analysis of the scattering intensity can show that the wave function given by Eq. 1 represents the ground state in Sr_{2}IrO_{4} (*4*).

Figure 2A shows the resonance enhancement of the magnetic reflection (1 0 22) at the *L* edge of a Sr_{2}IrO_{4} single crystal (*4*), overlaid with XAS spectra to show the resonant edges. Whereas there is a huge enhancement of the magnetic reflection by a factor of ∼10^{2} at the *L*_{3} edge, the resonance at *L*_{2} is small, showing less than 1% of the intensity at *L*_{3}. The constructive interference at *L*_{3} gives a large signal that allows the study of magnetic structure, whereas the destructive interference at the *L*_{2} edge hardly contributes to the resonant enhancement.

To find out the necessary conditions for the hole state leading to the destructive interference at the *L*_{2} edge, we calculate the scattering amplitudes. The most general wave function for the hole state in the *t*_{2g} manifold involves six basis states, which can be reduced by block-diagonalizing the spin-orbit Hamiltonian as (3) With its time-reversed pair, they fully span the *t*_{2g} subspace. We neglect higher-order corrections such as small residual coupling between *t*_{2g} and *e*_{g} manifolds. In the limit of the tetragonal crystal field [*Q* Ξ *E*(*d*_{xy}) – *E*(*d*_{yz,zx})] due to the elongation of octahedra much larger than SOC (λ_{SO}) (that is, *Q* ≫ λ_{SO}), the ground state will approach *c*_{1} = 1 and *c*_{2} = *c*_{3} = 0 and become a *S* = 1/2 Mott insulator, whereas in the other limit of strong SOC, *Q* ≪ λ_{SO}, *c*_{i}'s will all be equal in magnitude, with *c*_{1}, *c*_{2} pure real and *c*_{3} pure imaginary as in Eq. 1. Calculation of *f*_{αβ} for the *L*_{2} edge using Eq. 5 gives (4) Because the magnetic signal comes from the imaginary part of off-diagonal elements (*12*), the necessary condition for the vanishing intensity is (5) This condition places very stringent constraints on the allowed hole state and rules out all single-orbital *S* = 1/2 models. For the *S* = 1/2 case, the imaginary part of the off-diagonal elements does not vanish and equal resonant intensities are expected at the *L*_{2} and *L*_{3} edges (Fig. 2B). Given the constraints in Eq. 5, the scattering intensities at *L*_{3} edge are calculated to be (6) respectively. Thus, taking the phase convention *c*_{1} = 1 without loss of generality, the resonant intensity at *L*_{3} measures the imaginary part of *c*_{3} relative to the real part of *c*_{2}, which is also a measure of orbital angular momentum. The large enhancement at *L*_{3} necessarily implies that the *yz* orbital is out of phase with the *zx* orbital, the contrast between *L*_{3} and *L*_{2} being maximal when the relative phase is π/2. Taking the constraints of Eqs. 5 and 6 together, we conclude that the ground state is very close to the *J*_{eff} = 1/2 limit (*c*_{1}:*c*_{2}:*c*_{3} = 1:1:*i*), and the minute enhancement at *L*_{2} edge shows the smallness of the deviation from the *J*_{eff} = 1/2 limit coming from the factors not taken into account (*13*). The wave function in Eq. 2, representing *J*_{eff} = 1/2, indeed gives zero off-diagonal elements in *f*_{αβ} for *L*_{2} and nonzero elements for the *L*_{3} edge. This is a direct measurement of phase and provides evidence for the *J*_{eff} = 1/2 state in Sr_{2}IrO_{4}.

Having identified the nature of the local moment, we now look at the global magnetic structure using the enhanced signal due to the resonance at the *L*_{3} edge. Sr_{2}IrO_{4} shows a metamagnetic transition below 240 K and, above the metamagnetic critical field *H*_{C} (≈0.2 T well below 240 K), shows weak ferromagnetism with a saturation moment of ≈0.1 μ_{B}/Ir (*4*). The origin of this field-induced weak ferromagnetism has remained unidentified, because the neutron diffraction data did not show any detectable indication of magnetic ordering (*14*). Our RXS results indicate that the magnetic structure of Sr_{2}IrO_{4} is canted antiferromagnetic.

Figure 3A shows the crystal structure containing four IrO_{2} layers in a unit cell, enlarged by superstructure from the rotational distortion of octahedra (*14*). Figure 3B shows the magnetic ordering pattern determined from the experiment shown in Fig. 3, C to E. The arrows in Fig. 3B do not represent spins but *J*_{eff} = 1/2 moments. In zero field, the magnetic reflections are observed at (1 0 4*n*+2) and (0 1 4*n*), which implies that the moments are aligned antiferromagnetically within a layer and the symmetry changes from tetragonal to orthorhombic (Fig. 3C). The canting of the moments yields a nonzero net moment within a layer, which orders in the up-down-down-up antiferromagnetic pattern along the *c* axis. This is evidenced by the presence of (0 0 *odd*) peaks shown (Fig. 3D). The width of the peak gives an estimate of interlayer correlation length of 100 *c* or 400 IrO_{2} layers. When the magnetic field greater than *H*_{C} is applied, the peaks at (1 0 4*n*+2) disappear and new peaks show up at (1 0 *odd*) (Fig. 3E), which implies that the net moments in the planes are aligned ferromagnetically to produce a macroscopic field. The temperature dependence of the scattering intensity in the weakly ferromagnetic state above *H*_{C}, shown in Fig. 3F, scales very well with that of the magnetization and confirms again the magnetic nature of the peaks.

Our study demonstrates that x-rays can be extended to a new level to probe even finer details of magnetic structure. Until now, only the intersite interference effects were used to study ordering phenomena over a length scale of many lattice sites. A quantitative analysis on the interference effects within a single site provides phase information on the constituent wave function of the electron responsible for the magnetism. This technique should find important applications in systems where complex phases give rise to novel physics.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/323/5919/1329/DC1

Materials and Methods

Figs. S1 and S2

References