Pseudo-spin rotation symmetry breaking by
Coulomb interaction terms in spin-orbit coupled systems

Arising from a novel interplay between crystal field, spin-orbit coupling (SOC) and intermediate-strength Coulomb interactions, the emergent quantum states which essentially determine the electronic and magnetic properties of the iridium based transition-metal oxides involve correlated motion of electrons in spin-orbital entangled states.krempa_AR_2014 ; rau_AR_2016 ; bertinshaw_AR_2018 In the spin-orbit Mott insulator $\rm Sr_{2}IrO_{4}$ with $d^{5}$ configuration, electronic states near the Fermi energy have dominantly $J$=1/2 character, and important magnetic properties such as in-plane canted antiferromagnetic (AFM) order and magnon excitations have been extensively discussed in terms of the effectively single pseudo orbital ($J$=1/2) picture.kim_PRL_2008 ; kim_Science_2009 ; watanabe_PRL_2010 ; kim1_PRL_2012 Finite-interaction and finite-SOC effects are responsible for the strong zone-boundary magnon dispersion measured in resonant inelastic X-ray scattering (RIXS) studies, highlighting the observable effect of mixing between $J$=1/2 and 3/2 sectors.iridate_one

The Dzyaloshinskii-Moriya (DM) and pseudo-dipolar (PD) anisotropic interactions in the $J=1/2$ sector, although weakly affected by the tetragonal splitting,kim_PRL_2012 are not the source of true anisotropy in $\rm Sr_{2}IrO_{4}$, as they yield pseudo-spin canting with no magnon gap due to compensation. True anisotropy has been ascribed to the Hund’s coupling term ($J_{\rm H}$) using strong-coupling expansion (including virtual excitations to $J$=3/2 states) and numerical self-consistent calculation.jackeli_PRL_2009 ; igarashi_PRB_2013 ; perkins_PRB_2014 ; vale_PRB_2015 ; igarashi_JPSJ_2014 While Coulomb interactions were considered within the pure-spin-orbital basis ($t_{\rm 2g}$ orbitals, pure spins) in above approaches, their treatment within a pseudo-spin-orbital basis, and the role of weak magnetism in the other two pseudo orbitals ($J$=3/2 sector) on the $J_{\rm H}$-induced easy-plane magnetic anisotropy and magnon gap ($\sim 40$ meV), as measured in recent resonant inelastic X-ray scattering (RIXS) studies,kim_NATCOMM_2014 ; pincini_PRB_2017 ; porras_PRB_2019 have not been elucidated. Furthermore, the pseudo-spin-orbital based approach can allow for a unified study of both intra-orbital (magnon) and inter-orbital (spin-orbit exciton) excitations within a single formalism.

Magnetic anisotropy is generally associated with spin rotation symmetry breaking. Therefore, a general pseudo-spin rotation symmetry analysis of the different Coulomb interaction terms, treating all three pseudo orbitals on the same footing, can provide additional insight into the origin of true magnetic anisotropy in $\rm Sr_{2}IrO_{4}$ arising from the interplay of spin-orbital entanglement and Coulomb interaction. Due to the spin-orbital entanglement, the same pseudo-spin rotation for all three pseudo orbitals ($l=1,2,3$) corresponds to different pure-spin rotations for the three pure ($t_{\rm 2g}$) orbitals. This follows directly from the relation $\psi_{\mu}=\sigma_{\mu}\sum_{l}c_{\mu l}\psi_{l}$ between the fermionic field operators in the pure-spin-orbital basis ($\mu=yz,xz,xy$) and pseudo-spin-orbital basis ($l=1,2,3$), where the Pauli matrices $\sigma_{\mu}=\sigma_{x},\sigma_{y},\sigma_{z}$ corresponding to the three pure orbitals $\mu=yz,xz,xy$. The same SU(2) transformation $\psi_{l}\rightarrow\psi_{l}^{\prime}=[U]\psi_{l}$ for all three pseudo orbitals corresponds to different SU(2) transformations $\psi_{\mu}\rightarrow\psi_{\mu}^{\prime}=[U_{\mu}]\psi_{\mu}$ for the three pure orbitals due to the orbital dependence of $[U_{\mu}]=\sigma_{\mu}[U]\sigma_{\mu}$.

Therefore, the question of how the different Coulomb interaction terms transform under same pseudo-spin rotation for all three pseudo orbitals assumes importance. In other words, while all Coulomb interaction terms are invariant under same pure-spin rotation for all three pure orbitals, does this invariance hold under same pseudo-spin rotation in the pseudo-spin-orbital basis? Pseudo-spin rotation symmetry breaking by any Coulomb interaction term would imply true magnetic anisotropy and gapped magnon spectrum.

In this paper, we will show that while the Hubbard $(U)$ and density $(U^{\prime})$ interaction terms do preserve pseudo-spin rotation symmetry, the Hund’s coupling ($J_{\rm H}$) and pair-hopping ($J_{\rm P}$) interaction terms explicitly break this symmetry systematically. This symmetry breaking results in (on-site) anisotropic interactions between moments in the $J$=1/2 and $J$=3/2 sectors, highlighting the importance of the weak magnetism in the nominally filled $J$=3/2 sector as well as the mixing between the two sectors. Magnetic anisotropy will not survive in the large SOC limit when the two sectors become effectively decoupled. The physically transparent approach for investigating magnetic ordering and anisotropy effects will be illustrated for the perovskite compound $\rm Sr_{2}IrO_{4}$, where the sign of $J$=3/2 sector moments directly yields easy $x$-$y$ plane anisotropy.

The structure of this paper is as below. After introducing the transformation between the pure-spin-orbital and pseudo-spin-orbital bases in Sec. II, a symmetry analysis of the individual Coulomb interaction terms is carried out in Sec. III, where the Hund’s coupling and pair hopping interaction terms are shown to explicitly break pseudo-spin rotation symmetry. The transformed Coulomb interaction terms are obtained in Sec. IV, explicitly showing that the symmetry breaking terms account for the easy $x$-$y$ plane anisotropy in $\rm Sr_{2}IrO_{4}$, as illustrated for the ground state energy in Sec. V, and for magnon excitations and anisotropy gap in Sec. VI. Comparison with an independent $t_{\rm 2g}$ orbital based approach is discussed in Sec. VII, which provides confirmation of the above symmetry analysis and also illustrates the significant simplification achieved in the pseudo-spin-orbital basis. After a critical comparison (Sec. VIII) of different but equivalent approaches for studying magnetic anisotropy in $\rm Sr_{2}IrO_{4}$, some conclusions are finally presented in Sec. IX.

Due to large crystal-field splitting ($\sim$3 eV) in the $\rm IrO_{6}$ octahedra, low-energy physics in $d^{5}$ iridates is effectively described by projecting out the empty ${\rm e_{g}}$ levels which are well above the $t_{\rm 2g}$ levels. Spin-orbit coupling (SOC) further splits the $t_{\rm 2g}$ states into $J$=1/2 doublet ($m_{J}=\pm 1/2$) and $J$=3/2 quartet ($m_{J}=\pm 1/2,\pm 3/2$), with an energy gap of $3\lambda/2$ (Fig. 1). Nominally, four of the five electrons fill the $J$=3/2 states, leaving one electron for the $J$=1/2 sector, rendering it magnetically active in the ground state.

In the ($yz$$\downarrow$,$xz$$\downarrow$,$xy$$\uparrow||\;yz$$\uparrow$,$xz$$\uparrow$,$xy$$\downarrow$) basis, the SOC Hamiltonian becomes block diagonal with two [$3\times 3$] blocks. There is no mixing between the ($\downarrow,\downarrow,\uparrow$) and ($\uparrow,\uparrow,\downarrow$) sectors, which can therefore be treated as pseudo spins $\uparrow$ and $\downarrow$ for the three doubly degenerate eigenstates (pseudo orbitals).watanabe_PRL_2010 Corresponding to the three Kramers pairs $|J,m_{j}\rangle$ above, the pseudo-spin-orbital basis states $|l,\tau\rangle$ for the three pseudo orbitals ($l=1,2,3$), with pseudo spins ($\tau=\uparrow,\downarrow$) each, have the form:

where $\Ket{yz,\sigma}$, $\Ket{xz,\sigma}$, $\Ket{xy,\sigma}$ are the t_2g basis states and the signs $\pm$ correspond to spins $\sigma=\uparrow/\downarrow$. The coherent superposition of different-symmetry $t_{\rm 2g}$ orbitals, with opposite spin polarization between $xz$/$yz$ and $xy$ levels implies spin-orbital entanglement, and also imparts unique extended 3D shape to the pseudo-orbitals $l=1,2,3$, as shown in Fig 1. The pseudo-spin dynamics in iridate heterostructures are gaining interest as their magnetic properties are much more sensitive to structural distortion compared to pure spin systems due to spin-orbital entanglement.meyers_SREP_2019 ; mohapatra_PRB_2019

Taking the conjugate to express the above basis transformation in terms of the $\langle l,\tau|$ and $\langle\mu,\sigma|$ states, and rewriting in terms of the corresponding fermionic field operators:

involving the annihilation operators for the pseudo orbitals ($l=1,2,3$, $\tau=\uparrow,\downarrow$) and the $t_{2g}$ orbitals ($\mu=yz,xz,xy$, $\sigma=\uparrow,\downarrow$), we obtain (using Pauli matrices):

Inverting the above transformation yields the $t_{2g}$ basis states represented in terms of the pseudo-spin-orbital basis states:

The above equations are convenient for transforming the hopping and Coulomb interaction terms to the pseudo-spin-orbital basis, and can be expressed in the compact form:

where $\sigma_{\mu}=\sigma_{x},\sigma_{y},\sigma_{z}$ for the three orbitals $\mu=yz,xz,xy$, respectively, and the (real) transformation coefficients $c_{\mu l}$ are explicitly shown in Eq. (4).

We consider the on-site Coulomb interaction terms in the $t_{2g}$ basis ($\mu,\nu=yz,xz,xy$):

including the intra-orbital $(U)$ and inter-orbital $(U^{\prime})$ density interaction terms, the Hund’s coupling term $(J_{\rm H})$, and the pair hopping interaction term $(J_{\rm P}$=$J_{\rm H})$. Here $a_{i\mu\sigma}^{\dagger}$ and $a_{i\mu\sigma}$ are the creation and annihilation operators for site $i$, orbital $\mu$, spin $\sigma=\uparrow,\downarrow$, the density operator $n_{i\mu\sigma}=a_{i\mu\sigma}^{\dagger}a_{i\mu\sigma}$, the total density operator $n_{i\mu}=n_{i\mu\uparrow}+n_{i\mu\downarrow}=\psi_{i\mu}^{\dagger}\psi_{i\mu}$, and $U^{\prime\prime}=U^{\prime}-J_{\rm H}/2$. All interaction terms above are SU(2) invariant and thus possess pure-spin rotation symmetry. In the following, we consider the transformation of individual Coulomb interaction terms to the pseudo-spin-orbital basis using Eq. (5), and examine their SU(2) transformation behavior in pseudo-spin space.

In the following, we illustrate the transformation of the different Coulomb interaction terms to the pseudo-spin-orbital basis, starting with $l$=1 ($J$=1/2) sector of the intra-pseudo-orbital interaction terms. Similar transformation to the $J$ basis has been discussed recently, focussing only on the density interaction terms.martins_JPCM_2017 Considering first the pair-hopping interaction term (Eq. 9), and retaining only the $l$=$m$=$l^{\prime}$=$m^{\prime}$=1 terms (indicated by $\leadsto$ below), we obtain:

Similarly, for the Hund’s coupling term in Eq. (6), we obtain:

as the product $c_{\mu 1}^{2}c_{\nu 1}^{2}$ is identical for all three orbital pairs, and for the rotation matrices (Eq. 16), we have $R_{x}R_{y}+R_{y}R_{z}+R_{z}R_{x}=-{\bf 1}$. Although the pair-hopping and Hund’s coupling interaction terms generally break pseudo-spin rotation symmetry, within the magnetically active $l$=1 ($J$=1/2) sector, they individually yield classically isotropic terms of the form ${\bf S}_{1}\cdot{\bf S}_{1}$. This follows from the special symmetry within this sector, as reflected by the identical coefficients $c_{\mu l}$ for all three orbitals $\mu=yz,xz,xy$ for $l=1$ (Eq. 4).

Finally, from the remaining $J_{\rm H}$ term (in the $U^{\prime\prime}$ term of Eq. 6), again retaining only the $l=m=l^{\prime}=m^{\prime}=1$ term, we obtain (using Eq. 7 for site $i$):

where we have used $n_{1\sigma}^{2}=n_{1\sigma}$ and $2n_{1\uparrow}n_{1\downarrow}=n_{1}-(4/3){\bf S}_{1}\cdot{\bf S}_{1}$. Collecting all the ${\bf S}_{1}\cdot{\bf S}_{1}$ interaction terms resulting from the pair-hopping, Hund’s coupling, and density interaction terms corresponding to $J_{\rm H}$, as obtained in Eqs. 19, 23, 24, yields an exact cancellation.

Similarly, considering the other two intra-pseudo-orbital cases ($l=m=l^{\prime}=m^{\prime}=2,3$) in the $J=3/2$ sector, we obtain from the three explicitly $J_{\rm H}$ interaction terms:

explicitly showing the symmetry breaking contributions $(S_{lz}S_{lz})$ from the pair-hopping and Hund’s coupling interaction terms, as expected from the SU(2) transformation analysis in Sec. III, in the absence of the special symmetry in the $J=3/2$ sector. However, since $S_{\alpha}S_{\alpha}=(1/4)[(n_{\uparrow}+n_{\downarrow})-2n_{\uparrow}n_{\downarrow}]=(1/3){\bf S}\cdot{\bf S}$ for all three components $\alpha=x,y,z$ of $S=1/2$ quantum spin operators $S_{\alpha}=(1/2)\psi^{\dagger}\sigma_{\alpha}\psi$, there is no true magnetic anisotropy even if classically anisotropic terms such as $S_{lz}S_{lz}$ are present, as in Eq. (25).

Substituting $S_{lz}S_{lz}=(1/3){\bf S}_{l}\cdot{\bf S}_{l}$ in Eq. (25) yields an exact cancellation of the three explicitly $J_{\rm H}$ interaction terms in the $J=3/2$ sector also. Therefore, the Hubbard like (or equivalently ${\bf S}_{l}\cdot{\bf S}_{l}$) intra-pseudo-orbital interaction terms in all three sectors result only from the $U$ and $U^{\prime}$ terms in Eq. (6). Using similar analysis as above, and dropping one-particle density terms as in Eq. (24), one obtains (for site $i$):

From the above discussion it follows that true magnetic anisotropy results only from the inter-orbital anisotropic interaction terms such as $S_{lz}S_{l^{\prime}z}$ with $l^{\prime}\neq l$. Including inter-pseudo-orbital cases such as $l$=$m$, $l^{\prime}$=$m^{\prime}$ and $l$=$m^{\prime}$, $l^{\prime}$=$m$ in Eqs. (19) and (23), and keeping all interaction terms relevant for the present study (Hubbard, Hund’s coupling, and density), we obtain (for site $i$):

where ${\bf S}_{m}=\psi_{m}^{\dagger}\frac{\mbox{\boldmath$\tau$}}{2}\psi_{m}$ and $n_{m}=\psi_{m}^{\dagger}{\bf 1}\psi_{m}=n_{m\uparrow}+n_{m\downarrow}$ are the spin and charge density operators. Using the spherical symmetry condition ($U^{\prime}$=$U$-$2J_{\mathrm{H}}$), the transformed interaction Hamiltonian (27) simplifies to:

In the above equation, the Hubbard-like terms ${\cal U}_{m}n_{m\uparrow}n_{m\downarrow}\sim-{\cal U}_{m}{\bf S}_{m}\cdot{\bf S}_{m}$, the Hund’s-coupling-like term ${\bf S}_{1}\cdot{\bf S}_{2}$, and the density terms $n_{l}n_{m}$, are all invariant under pseudo-spin rotation and the corresponding SU(2) transformation $\psi_{m}\rightarrow\psi_{m}^{\prime}=[U]\psi_{m}$. Therefore, only the interaction terms $S_{1}^{z}S_{2}^{z}$ and $S_{1}^{z}S_{3}^{z}$ between moments in the $J$=1/2 and $J$=3/2 sectors are responsible for the magneto-crystalline anisotropy in $\rm Sr_{2}IrO_{4}$, highlighting the importance of the weak magnetism in the nominally filled $J$=3/2 sector due to the mixing between the two sectors. Magnetic anisotropy will not survive in the large SOC limit when the two sectors become effectively decoupled. As shown below, an easy $x$-$y$ plane anisotropy is obtained from the dominant term ($-2J_{\rm H}S_{1}^{z}S_{3}^{z}$) due to opposite sign of $J$=3/2 sector moment.

We consider the various interaction terms in Eq. (28) in the Hartree-Fock (HF) approximation, focussing on the staggered field terms corresponding to ($\pi,\pi$) ordered AF state on the square lattice. For general ordering direction with components $\Delta_{l}$= $(\Delta_{l}^{x},\Delta_{l}^{y},\Delta_{l}^{z})$, the staggered field term for sector $l$ in the pseudo-orbital basis is given by:

where $\psi_{{\bf k}ls}^{\dagger}=(a_{{\bf k}ls\uparrow}^{\dagger}\;\;a_{{\bf k}ls\downarrow}^{\dagger})$, $s=\pm 1$ for the two sublattices A/B, and the staggered field components $\Delta_{l=1,2,3}^{\alpha=x,y,z}$ are self-consistently determined from:

in terms of the staggered pseudo-spin magnetization components $m_{l=1,2,3}^{\alpha=x,y,z}$. In practice, it is easier to choose set of ${\bf\Delta}_{l=1,2,3}$ and self-consistently determine the Hubbard-like interaction strengths ${\mathcal{U}}_{l=1,2,3}$ such that ${\mathcal{U}}_{1}=U-\frac{4}{3}J_{\rm H}$ and ${\mathcal{U}}_{2}={\mathcal{U}}_{3}=U-J_{\rm H}$ using Eq. (30). The interaction strengths are related by ${\mathcal{U}}_{l=2,3}={\mathcal{U}}_{l=1}+J_{\rm H}/3$.

Transforming the staggered-field term back to the three-orbital basis $(yz\sigma,xz\sigma,xy\bar{\sigma})$, and including the crystal field, SOC and band terms,watanabe_PRL_2010 we have considered $\mathcal{H}_{\rm HF}=\mathcal{H}_{\rm SO}+\mathcal{H}_{\rm band+cf}+\mathcal{H}_{\rm sf}$ in our band structure and spin fluctuation analysis, where

in the composite three-orbital, two-sublattice basis. The crystal field induced tetragonal splitting is included as the $xy$ orbital energy offset $\epsilon_{xy}$ from the degenerate $yz/xz$ orbitals. The different hopping terms in Eq. (31) connecting the same ($s=s^{\prime}$) and opposite ($s\neq s^{\prime}$) sublattice(s) are given by: