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 $κ_{xyy}$ 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.

Figures (4)

• Figure 1: Comparison of the energy gap $2G(\mathbf{k})$ and nodal lines (white dashed lines). (a) The $d$-wave model forms nodal lines along $k_{x} = \pm k_{y}$. (b) The $g$-wave model possesses a complex eightfold nodal structure. Parameters: $\Delta = 1.0$, $t_{1} = 0.5$, $\lambda = 0.3$.
• Figure 2: Distribution of the trace of the quantum metric, $\mathrm{Tr}(g_{ab})$. (a) $d$-wave and (b) $g$-wave models form geometric hotspots near nodal lines. These singular structures, shown on a logarithmic scale, act as the source for macroscopic transport. Parameters: $\Delta = 1.0$, $t_{1} = 0.5$, $\lambda = 0.3$.
• Figure 3: Contrast in the symmetry of the Berry connection polarizability $\mathcal{G}_{xy}$. (a) The asymmetric texture of $d$-wave indicates $C_{2}$ breaking. (b) The perfectly antisymmetric structure of $g$-wave visually proves the protection by $C_{2}$ symmetry. This point-symmetric structure forces the global integral to vanish. Parameters: $\Delta = 1.0$, $t_{1} = 0.5$, $\lambda = 0.3$.
• Figure 4: Numerical evaluation of the nonlinear thermal Hall conductivity $\kappa_{xyy}$. (a) The $d$-wave model displays a linear increase with the parameter $\lambda$. (b) The $g$-wave model stays exactly at zero when $C_{2}$ is preserved (dashed line). The signal recovers sharply when $C_{2}$ is broken (solid line with squares). Parameters: $\Delta = 1.0$, $\mu = -0.5$, $T = 0.05$, $\eta = 0.005$; $t_{1} = 0.5$ for $d$-wave.