We perform extensive classifications of ${\mathbb{Z}}_2$ quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic (fcc) lattices using a spin-rotation-invariant fermionic projective symmetry group approach. Taking into account that all three lattices share the same point group $O_h$, we apply an efficient gauge where the classification for the simple cubic lattice can be partially carried over to the other two lattices. We identify hundreds of projective representations for each of the three lattices, however, when constructing short-range mean-field models for the fermionic partons (spinons) these phases collapse to only very few relevant cases. We self-consistently calculate the corresponding mean-field parameters for frustrated Heisenberg models on all three lattices with up to third-neighbor spin interactions and discuss the spinon dispersions, ground-state energies, and dynamical spin structure factors. Our results indicate that phases with nonuniform spinon hopping or pairing amplitudes are energetically favored. An unusual situation is identified for the fcc lattice where the spinon dispersion minimizing the mean-field energy features a network of symmetry-protected linelike zero modes in reciprocal space. We further discuss characteristic fingerprints of these phases in the dynamical spin structure factor which may help to identify and distinguish them in future numerical or experimental studies.

中文翻译：

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简单立方，体心立方和面心立方晶格上的量子自旋液体的投影对称群分类

我们对...进行广泛的分类 ${\mathbb{ž}}_2$使用自旋旋转不变的铁离子投射对称群方法在简单立方，体心立方和面心立方（fcc）晶格上的量子自旋液体。考虑到所有三个晶格共享相同的点组${Ø}_H$，我们应用了一种有效的量规，其中简单立方晶格的分类可以部分地转移到其他两个晶格中。我们为这三个晶格中的每一个都确定了数百个投影表示，但是，当为铁调子（自旋子）构建短程平均场模型时，这些相仅在很少的情况下崩溃。我们自洽地计算了具有不超过第三邻域自旋相互作用的所有三个晶格上沮丧的Heisenberg模型的对应平均场参数，并讨论了旋子色散，基态能量和动力学自旋结构因子。我们的结果表明，在能量上倾向于具有不均匀的自旋跳跃或成对振幅的相位。对于fcc晶格，发现了一种不寻常的情况，在该情况下，使平均场能量最小的棘突分散在倒易空间中具有对称保护的线型零模网络。我们进一步讨论了动态自旋结构因子中这些相的特征指纹，这可能有助于在未来的数值或实验研究中识别和区分它们。