Higher order topological insulators (HOTIs) are a novel form of insulating quantum matter, which are characterized by having gapped boundaries that are separated by gapless corner or hinge states. Recently, it has been proposed that the essential features of a large class of HOTIs are captured by topological multipolar response theories. In this work, we show that these multipolar responses can be realized in interacting lattice models, which conserve both charge and dipole. In this work we study several models in both the strongly interacting and mean-field limits. In $2$D we consider a ring-exchange model which exhibits a quadrupole response, and can be tuned to a $C_4$ symmetric higher order topological phase with half-integer quadrupole moment, as well as half-integer corner charges. We then extend this model to develop an analytic description of adiabatic dipole pumping in an interacting lattice model. The quadrupole moment changes during this pumping process, and if the process is periodic, we show the total change in the quadrupole moment is quantized as an integer. We also consider two interacting $3$D lattice models with chiral hinge modes. We show that the chiral hinge modes are heralds of a recently proposed "dipolar Chern-Simons" response, which is related to the quadrupole response by dimensional reduction. Interestingly, we find that in the mean field limit, both the $2$D and $3$D interacting models we consider here are equivalent to known models of non-interacting HOTIs (or boundary obstructed versions). The self-consistent mean-field theory treatment provides insight into the connection between free-fermion (mean-field) theories having vanishing polarization and interacting models where dipole moments are microscopically conserved.
Kosterlitz--Thouless Criticality in a Dipole-Conserving XY Model
Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, State Key Laboratory of Optoelectronic Materials and Technologies, and School of Physics, Sun Yat-sen University（中山大学物理学院）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Department of Physics, School of Science and Research Center for Industries of the Future, Westlake University（西湖大学理学院）；Institute of Natural Sciences, Westlake Institute for Advanced Study（西湖高等研究院自然科学研究中心）
Reply to "Comment on: 'Case for a U(1)$_π$ Quantum Spin Liquid Ground State in the Dipole-Octupole Pyrochlore $\mathrm{Ce}_2\mathrm{Zr}_2\mathrm{O}_7$' "
在他的评论[arXiv：2209.03235]，S. W. Lovesey认为我们对Ce $_2$ Zr $_2$ O $_7$的中子散射实验的分析是无效的。Lovesey认为，我们没有正确地考虑高阶多极对磁散射的贡献，我们使用赝自旋- $1/2$算符来描述散射是不合适的。在这个答复中，我们表明，多极校正Lovesey讨论只有在散射波矢量超过我们的实验中访问的显着。这与我们的工作并不矛盾，也没有破坏我们的工作，因为我们的工作从未声称直接观察到高阶多极子的散射。我们进一步表明，Lovesey的反对我们使用的pseudospins是没有根据的，和pseudospin运营商能够描述所有的磁散射过程中的能量尺度，我们的实验，远远低于晶场间隙。最后，我们评论某些假设Lovesey的计算的散射振幅是不符合实验。
Invariant Form of Hyperfine Interaction with Multipolar Moments - Observation of Octupolar Moments in NpO$_{2}$ and CeB$_{6}$ by NMR -
导出了多极矩与核自旋之间超精细相互作用的不变量形式，并讨论了利用NMR实验识别反铁八极矩的可能性.详细研究了NpO $_{2}$的有序相和Ce $_{1-x}$ La $_{x}$ B $_{6}$的IV相。从理论上讨论了NpO多晶样品的$^{17}$ O NMR。观察到的$^{17}$ O NMR谱分裂为尖线和宽线的特征、它们的强度比以及位移和宽度的磁场依赖性可以根据Paixao提出的三重$\bq$ AFO有序模型（\it et. al.}因此，本理论表明，$^{17}$ O NMR谱给出了强有力的支持模型。在fcc NpO $_2$中的4个O位置由于AF四极子的次级三重$\bq$排序而变得不等价：一个立方和三个非立方位置。事实证明，由于磁场引起的反铁偶极子和AFO矩的超精细场，以及非立方位置处的四极场是理解所观察到的光谱的关键成分。对Ce $_{1-x}$ La $_{x}$ B $_{6}$中存在争议的IV相的性质问题也进行了研究。本文指出，如果Ce $_{1-x}$ La $_{x}$ B $_{6}$中实现了$Γ_{5}$（$T^β_{x}=T^β_{y}=T^β_{z}$）AFO有序化，则其NMR谱具有独特的特征。也就是说，当磁场围绕$[001]$轴旋转时，$({1/2},{1/2},\pm u)$位点上的B原子对的超精细分裂在$(1\bar{1}0)$平面上穿过零。
