Symmetry selection rules for the intrinsic nonlinear thermal Hall effect in altermagnets: Role of quantum metric and $C_{2}$ rotational symmetry

Abstract

We establish symmetry-based selection rules for the intrinsic nonlinear thermal Hall effect driven by the quantum metric in altermagnets. We show that a nonvanishing nonlinear thermal Hall conductivity $\kappa_{xyy}$ requires three conditions: (i) a nontrivial quantum metric, (ii) breaking of mirror symmetry $M_{x}$, and (iii) breaking of twofold rotational symmetry $C_{2}$. Using tight-binding models on a square lattice, we demonstrate that $d$-wave altermagnets naturally break $C_{2}$ through parity-mixing orbital hybridizations, while $g$-wave systems preserve $C_{2}$, forcing the response to vanish identically. Step-by-step Taylor expansions and explicit unitary matrix proofs establish these results. Our framework provides predictive power for material selection and lays the groundwork for nonlinear spin-caloritronic devices.