Phonon Angular Momentum Multipoles and Nonlinear Thermal Responses

Chiral phonons carry angular momentum and couple lattice dynamics to magnetic and transport phenomena. In magnets, however, a systematic classification of momentum-space phonon angular momentum (PAM) textures and experimental signatures of the resulting thermal responses remain elusive. Here we classify the PAM textures for all 122 magnetic point groups (MPGs) using two equivalent approaches: directly solving the linear symmetry constraints on the multipole expansion coefficients and constructing the allowed textures from group-representation basis functions. Both methods yield the same fivefold classification according to the leading nonzero order: $s$-wave (monopole), $p$-wave (dipole), $d$-wave (quadrupole), $f$-wave (octupole), and $g$-wave (hexadecapole). We further show that a temperature gradient generates a nonequilibrium PAM density whose $n$th-order term in $\nabla T$ contains an energy-weighted PAM $2^n$-pole tensor and a kinematic remainder, thereby linking the response order and partial-wave channel through exact selection rules. Because the Berry curvature obeys the same symmetry constraints, this taxonomy is in one-to-one correspondence with Berry-curvature multipoles and their nonlinear Hall responses. We propose a cantilever torque measurement whose allowed angular harmonics are fixed by the MPG, with period $2π/n$ for a pure $n$th harmonic, providing a direct probe of the partial-wave content of PAM in both nonmagnetic and magnetic materials.