“…For instance, the thermodynamic magnetic quadrupole moments are defined to be conjugate to the gradient of magnetic fields, and they are directly related to the magnetoelectric property (Bhowal and Spaldin, 2022;Shitade et al, 2018). It is noteworthy that the thermodynamic multipole moments contribute to other cross-correlated responses such as the nonlinear thermoelectric response , temperature-gradient-induced magnetization (gravito-magnetoelectric effect) (Shinada and Peters, 2023;Shitade et al, 2019), and spin accumulations (Shitade and Tatara, 2022). In the same spirit, the electric quadrupole moments are defined and estimated in the language of thermodynamics and quantum geometry (Daido et al, 2020;Kitamura et al, 2021).…”