We present a microscopic theory of the orbital magnetic quadrupole moment density ${\mathcal{Q}}_{ij}$ in periodic crystals with combined time-reversal ($\mathbf{T}$) and inversion ($\mathbf{I}$) symmetry. We obtain a gauge-invariant expression with a clear physical interpretation and demonstrate the typical behavior of ${\mathcal{Q}}_{ij}$ in a minimal two-band model that hosts a tilted Dirac cone. We then show that ${\mathcal{Q}}_{ij}$ leads to an intrinsic nonlinear anomalous thermoelectric current. As an example, we calculate the nonlinear Nernst and Hall current in the loop-current model for cuprate superconductors, and demonstrate their unique behavior and capability of indicating $\mathbf{TI}$ invariance.

• Received 19 March 2018

DOI:https://doi.org/10.1103/PhysRevB.98.060402