We revisit a quantum-mechanical formula of the spin magnetic quadrupole moment (MQM) in periodic crystals. Two previous attempts were inconsistent with each other; one is gauge dependent, and the other is gauge invariant. Here we define the spin MQM by calculating the spin density in a nonuniform system. Our definition is analogous to that of the charge polarization, but the result is gauge invariant and coincides with the latter previous one. We also formulate what we call gravitomagnetoelectric (gravito-ME) effect, in which the magnetization is induced by a temperature gradient. Although the Kubo formula for the gravito-ME effect provides an unphysical divergence at zero temperature, we prove that the correct susceptibility is obtained by subtracting the spin MQM from the Kubo formula. It vanishes at zero temperature and is related to the ME susceptibility by the Mott relation. We explicitly calculate the gravito-ME susceptibility in a Rashba ferromagnet and show its experimental feasibility.
Quadrupole Order in the Frustrated Pyrochlore Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$
利用高质量单晶的比热、磁化强度和中子散射实验研究了阻挫烧绿石Tb $_{2+x}$ Ti $_{2-x}$ O $_{7+y}$与$T_{\text{c}}=0.53$ K中出现的隐序。基于赝自旋1/2哈密顿量的离子非Kramers磁偶极子的半定量分析表明，它是电四极矩的有序态。名义Tb $_2$ Ti $_2$ O $_7$的难以捉摸的自旋液体状态最有可能是U（1）量子自旋液体状态。
