Abstract
Recent studies revealed that the electric multipole moments of insulators result in fractional electric charges localized to the hinges and corners of the sample. We here explore the magnetic analog of this relation. We show that a collinear antiferromagnet with spin $S$ defined on a $d$-dimensional cubic lattice features fractionally quantized magnetization ${M}_{\text{c}}^{z}=S/{2}^{d}$ at the corners. We find that the quantization is robust even in the presence of gapless excitations originating from the spontaneous formation of the N\'eel order, although the localization length diverges, suggesting a power-law localization of the corner magnetization. When the spin rotational symmetry about the $z$ axis is explicitly broken, the corner magnetization is no longer sharply quantized. Even in this case, we numerically find that the deviation from the quantized value is negligibly small based on quantum Monte Carlo simulations.
Highlights
Multipole insulators feature fractional electric charges bound to hinges and corners of the system [1, 2]
Given that fractional corner charges can appear in ionic crystals, it is natural to expect fractional “corner magnetizations” in the magnetic analog of ionic crystals
Two fundamental problems inherit to spin systems that are absent in the electric counterparts have not been addressed in the earlier works: (i) The U(1) symmetry underlying the charge conservation may be explicitly broken because of the crystal anisotropy and spin-orbit coupling, and (ii) excitations may not be gapped because the spontaneous formation of the Neel order results in gapless Nambu-Goldstone excitations
Summary
Multipole insulators feature fractional electric charges bound to hinges and corners of the system [1, 2]. Two fundamental problems inherit to spin systems that are absent in the electric counterparts have not been addressed in the earlier works: (i) The U(1) symmetry underlying the charge conservation (i.e., the spin rotational symmetry about the z axis) may be explicitly broken because of the crystal anisotropy and spin-orbit coupling, and (ii) excitations may not be gapped because the spontaneous formation of the Neel order results in gapless Nambu-Goldstone excitations For these reasons, the results for electric multiple moments established in the previous works [2,3,4,5,6, 8] cannot be directly applied to spin systems.
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