Correlation Effects and Hidden Spin-Orbit Entangled Electronic Order in Parent and Electron-Doped Iridates ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$

Analogs of the high-$T_c$ cuprates have been long sought after in transition metal oxides. Because of the strong spin-orbit coupling, the $5d$ perovskite iridates ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ exhibit a low-energy electronic structure remarkably similar to the cuprates. Whether a superconducting state exists as in the cuprates requires understanding the correlated spin-orbit entangled electronic states. Recent experiments discovered hidden order in the parent and electron-doped iridates, some with striking analogies to the cuprates, including Fermi surface pockets, Fermi arcs, and pseudogap. Here, we study the correlation and disorder effects in a five-orbital model derived from the band theory. We find that the experimental observations are consistent with a $d$-wave spin-orbit density wave order that breaks the symmetry of a joint twofold spin-orbital rotation followed by a lattice translation. There is a Berry phase and a plaquette spin flux due to spin procession as electrons hop between Ir atoms, akin to the intersite spin-orbit coupling in quantum spin Hall insulators. The associated staggered circulating $J_{\mathrm{eff}}=1/2$ spin current can be probed by advanced techniques of spin-current detection in spintronics. This electronic order can emerge spontaneously from the intersite Coulomb interactions between the spatially extended iridium $5d$ orbitals, turning the metallic state into an electron-doped quasi-2D Dirac semimetal with important implications on the possible superconducting state suggested by recent experiments.

Popular Summary

High-temperature superconductivity—where electricity flows with zero resistance at temperatures well above absolute zero—was first observed in copper-based materials known as cuprates. Since this discovery, researchers have sought out this behavior in other transition-metal oxides. One promising candidate is the perovskite iridate ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$. Atomically, ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ has the same structure as the cuprate ${\mathrm{La}}_2{\mathrm{CuO}}_4$. These materials also have a remarkably similar low-energy electronic structure due to strong spin-orbit coupling, raising the hope that they may share certain quantum electronic states. Understanding these states could shed light on unusual electronic behaviors observed when “high-$T_c$” superconductors are in a metallic state. Our theoretical analysis of ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ shows that these anomalous behaviors can be explained by a subtle type of spin-orbit density wave.

We used a realistic model of iridates to study how electronic interactions and disorder affect the system. We find that a hidden order (or collective behavior of the electrons) might be responsible for recent experimental observations. This order is a spin-orbit density wave, which creates a staggered pseudospin current where spin-up electrons are circulating in one direction and spin-down electrons are moving in the opposite direction.

We hope that our findings will stimulate further experimental and theoretical studies of the perovskite iridates for unconventional superconductivity.

Figure 1

Comparison of the band structures obtained using (a) LDA and (b) the five-orbital $\mathrm{TB}+\mathrm{SOC}$ model with lattice distortion caused by staggered octahedra rotation. The doubling of bands and the small band splitting in (a) is due to the doubling of unit cell along the $c$ axis. The inset in (b) shows the one-Ir Brillouin zone (solid black lines), the reduced zone (dotted black lines), and the high-symmetry points labeled by $\mathrm{\Gamma }=(0,0)$, $X=(\pi ,0)$, $Y=(0,\pi )$, $M=(\pi ,\pi )$, and $N=(\pi /2,\pi /2)$.

Figure 2

The band dispersions and the corresponding FS of (a) the noninteracting $U=0$ and (b) the nonmagnetic state in the interacting $(U,J)=(1.2,0.05)\mathrm{eV}$ system at $x=0$. The thickness of the black lines denotes the content of the $J_{\mathrm{eff}}=1/2$ doublet, while the size of the red open squares shows the content of the $J_{\mathrm{eff}}=3/2$ multiplet.

Figure 3

(a) Schematics of the in-plane canted AFM moments on the Ir square lattice with a two-dimensional rendering of the staggered ${\mathrm{IrO}}_6$ octahedral rotation. (b) QP band dispersion in the canted AFM state (red solid lines) and the nonmagnetic state (black solid lines) at $(U,J)=(1.2,0.05)\mathrm{eV}$ and $x=0$. (c) Schematics of the $d$-wave $J_{\mathrm{eff}}=1/2$ pseudospin current order and associated staggered pseudospin flux ${\mathrm{\Phi }}_{\sigma }=\pm \sigma \varphi$. (d) QP band dispersion in the canted AFM state with coexisting $d$-wave $J_{\mathrm{eff}}=1/2$ pseudospin current corresponding to ${\mathrm{\Delta }}_d=30\mathrm{meV}$. Open blue squares are data from ARPES experiments [21].

Figure 4

The PM phase at $x=0.1$ and $(U,J)=(1.4,0)\mathrm{eV}$. (a),(b) Band dispersion in the absence of $H_{\mathrm{\Delta }}$, i.e., without $d$-PSCO (black lines) and the spectral intensity of the corresponding FS. (c),(d) Band dispersion in the presence of $d$-PSCO for ${\mathrm{\Delta }}_d=30\mathrm{meV}$ showing the gapping of the BTP at $X$ (red lines) and the spectral intensity of the corresponding FS. (e) FS spectrum in the presence of disorder with ${\mathrm{\Delta }}_d=0$ and $ϵ=60\mathrm{meV}$, showing the Fermi arcs. (f) FS spectrum in the disordered $d$-PSCO with ${\mathrm{\Delta }}_d=30\mathrm{meV}$ and $ϵ=60\mathrm{meV}$. ARPES data [21] are superimposed as open blue squares.