Spontaneous altermagnetism in multi-orbital correlated electron systems

Abstract

Altermagnets have attracted considerable attention in recent years owing to their potential technological applications in spintronics and magnonics. Recently, a new class of spontaneous altermagnets has been theoretically predicted in a correlated two orbital model, driven by the coexistence of antiferromagnetic spin and staggered orbital ordering, thus broadening the scope of altermagnetic phenomena to systems with strong correlations. It has been noted, however, that the required spin and orbital order violates the well-established Goodenough-Kanamori (GK) rules, which underlie much of our understanding of magnetism in complex systems. Here we show that materials with three active orbitals may offer a more realistic route to this exotic state. Specifically, we consider a two-dimensional system with $t_{2g}^{2}$ electrons and identify a novel microscopic mechanism that allows the formation of a spontaneous altermagnetic Mott insulator. We explain how the GK rules are circumvented and provide the stability criteria by employing unbiased mean-field and density matrix renormalization group calculations. In addition, for the first time, we uncover the presence and microscopic origin of chirally split magnons in these spontaneous altermagnets, with experimentally measurable spin conductivities. Finally, we predict that the application of a small in-plane magnetic field induces, in the presence of weak atomic spin-orbit coupling, an as-yet unreported hybrid chiral magnon-orbiton mode with a non-zero orbital polarization giving rise to finite longitudinal and transverse orbital conductivities under a thermal gradient.

Figures (9)

• Figure 1: Pictorial representation of crystallographic altermagnet and spontaneous altermagnet is depicted in panels (a) and (b), respectively. The operators $C_{4z}, {\mathcal{T}}$, and $T_{a}^{x/y}$ denote $90^ {\circ}$ rotation around the $z$-axis, time-reversal, and translation by lattice spacing along $x/y$ direction, respectively. Panel (c) shows the typical energy spectrum of a quarter-filled 2-orbital system, with GK-compliant states being most stable. The achievable spectral ordering of the 3-orbital system, specifically $t_{2g}^{2}$ under a tetragonal compressed environment, is shown in panel (d), with AFO+AFM stabilized by the additional antiferromagnetic exchange of the new half-filled third orbital.
• Figure 2: Unrestricted Hartree-Fock results: Panels (a) and (b) depict the $U$ vs $\Delta$ and $U$ vs $t_{x}/t_{z}$ phase diagrams, respectively. Panel (c) depicts the colors chosen in the phase diagrams, according to the $(\pi,\pi)$ structure factor values of spin and pseudospin. The full Brillouin zone is depicted in panel (d). The electronic bands exhibiting altermagnetic splitting are shown in panel (e). Panels (f), (g), and (h) show the pictorial representation of the main states we found in the phase diagrams. The abbreviations AFM, FM, AFO, and OD denote antiferromagnetic, ferromagnetic, antiferro-orbital, and orbital-disordered, respectively.
• Figure 3: Panels (a) and (b) depict the DMRG results for system size of $8\times4$ with cylindrical boundary conditions. Panel (c) shows the exchange parameters dependence on $J_{H}/U$, for $S$-$\tau$ model. The $J_{H}/U$ vs $t_{x}/t_{z}$ phase diagram, illustrated in panel (d), is constructed by comparing energies of AFM+AFO, AFM+FO, FM+AFO, and FM+FO states. We fix the parameters $(\Delta,t_{x},U)=(-1.5,1.0,20)t_{z}$ for the panels (a,b,c).
• Figure 4: The panels (a,b) show the specific heat, spin and pseudospin $(\pi,\pi)$ structure factors evolution with temperature, depicting the phase transitions. The temperature vs $J_{H}/U$ phase diagram is shown in panel (c). The parameters $(\Delta,t_{x},U)=(-2, 0.7,20)t_{z}$ are fixed for all the above panels.
• Figure 5: Panels (a) and (b) display the $S({\bf q}, \omega)$ and $M_{\chi}({\bf q},\omega)$, respectively, calculated using RPA, for parameters $t_{x}$=$0.7t_{z}$, $U$=$8.0t_{z}$, $J_{H}/U$=$0.2$, $\lambda$=$0.0$ and system size of $64\times 64$. The $U$-dependence of magnon energy and chiral splitting is depicted in panel (c). Panel (d) shows the $S({\bf q}, \omega)$ for the same parameters as in panel (a), but with finite $\lambda=0.04t_{z}$. The excitation spectrum calculated using LSOWT is depicted in panel (e), for parameters $J^{S}$=0.052, $J_{1}^{S\tau}$=0.054, $J^{\tau}$=0.108, $J_{\chi}$=0.005, $K$=0.0056, and $\lambda$=0.04. Panel (f) depicts the spin conductivities as a function of angle $\eta$.