Orbital Magnetization Reveals Multiband Topology

Abstract

We demonstrate that nontrivial multiband topological invariants of electronic wavefunctions can be revealed through diamagnetic orbital magnetization responses to external magnetic fields. We find that decomposing orbital magnetization into energetic and quantum-geometric contributions allows one to deduce nontrivial multiband topology, provided knowledge of the energy spectrum. We showcase our findings in general effective models with multiband Euler topology. We moreover identify such multiband topological invariants in effective models of strontium ruthenide ($\text{Sr}_2 \text{Ru} \text{O}_4$), which may in principle be verified in the state-of-the-art doping-dependent magnetization measurements. Our reconstruction scheme for multiband invariants sheds a topological perspective on the multiorbital effects in materials realizing unconventional phenomenologies of orbital currents or multiband superconductivity.

Figures (3)

• Figure 1: Orbital magnetization $(\textit{m})$ from diamagnetic loop currents $(j_{xy})$, in response to an out-of-plane magnetic field $(\textit{B})$. The diamagnetic orbital response fingerprints the nontrivial Euler band topology.
• Figure 2: Topological bands with Euler charge $e=1$ in Lieb lattice model. (a) Low-energy bands supporting the Euler invariant via a quadratic band touching. (b) The quantum geometric contribution to orbital magnetic susceptibility associated with the Euler charge ($\text{Geo)}$ dominates the band-dispersive susceptibility contribution ($\text{Eng})$. The dashed line indicates the energy of the band touching with nontrivial Euler class.
• Figure 3: Orbital magnetic susceptibility fingerprints topological Euler invariants in Sr$_2$RuO$_4$ model. (a) Band structure obtained from a tight-binding model of Sr$_2$RuO$_4$. Bands contributed by the $(\mathrm{Ru}\;d_{xz},\;\mathrm{Ru}\;d_{yz},\;\mathrm{O}1\;p_{z},\;\mathrm{O}2\;p_{z})$ orbitals (solid lines) realize the nontrivial Euler class invariants, $|e| =1$$(\text{Eu}_{1,2})$, and van Hove singularities ($\text{vH}$); dotted lines mark $(\mathrm{Ru}\;d_{xy},\;\mathrm{O}1\;p_{x},\;\mathrm{O}2\;p_{y})$ orbital bands. (b) Orbital magnetic susceptibility $\chi$ decomposed into energetic (Eng) and geometric (Geo) contributions as a function of chemical potential $\mu$, with $\chi_0= t_0(e a^2/4\pi^2\hbar c)^2$ where $a$ is the lattice constant and $t_0 = 1$ eV. Around $\mu = -0.8$ and $\mu = 0.2~\text{eV}$, the total orbital magnetic susceptibility $\chi_\text{O}$ reflects the presence of topologically-induced concentrated quantum geometries due to the Euler charges.