标题： 绘制受挫磁体的相图：在尖晶石晶格上的简并、平带和倾斜环 显示英文标题

标题： Mapping the Phase Diagram of a Frustrated Magnet: Degeneracies, Flat Bands, and Canting Cycles on the Pyrochlore Lattice

摘要： 我们将描述稀土类尖晶石磁体的自旋哈密顿量的完整经典相图，其中包括所有允许的对称性近邻键依赖各向异性双自旋相互作用。 我们提供了一种简单的推导方法，说明自旋如何组织为描述四面体多极矩的张量自由度，其分量对应于四面体对称群的不可约表示（irreps）$T_d$。 通过直接以各个不可约表示的能量参数化哈密顿量，我们进行了所有可能的不可约表示简并性的全面搜索，这些简并性可能容纳稳定的经典自旋液体。 这样做揭示了四个单参数模型族，在这些模型族中三个相是简并的，这四个相全部在海森堡反铁磁体及其对偶体处合并，我们给出了一个完整的三维相图，展示了所有相及其交点。 一个不可约表示的两个副本的出现意味着额外的简并轨迹，该轨迹在特殊点穿过相边界，产生两个额外的孤立三重点。 我们证明单参数哈密顿量族由一个拓扑不变量表征，该不变量描述当绝热地围绕这个简并轨迹运输时基态自旋构型的缠绕方式，类似于“魔鬼点”。 我们提供了相图中所有平坦带简并性的综合目录，并讨论了可能允许或阻碍由表现出夹线奇点和相应的四重夹点的张量规范场描述的各种经典自旋液体实现的机制。 最后，我们列出了所有三个不可约表示在基态之上的简并情况，这可能导致每个有序相中的自旋波谱中出现有趣的特性。 显示英文摘要

摘要： We map the complete classical phase diagram of the spin Hamiltonian describing pyrochlore rare-earth magnets with all symmetry-allowed nearest-neighbor bond-dependent anisotropic two-spin interactions. We provide a simple derivation of the organization of spins into tensor degrees of freedom describing the multipole moments of a tetrahedron, whose components correspond to irreducible representations (irreps) of the tetrahedral symmetry group $T_d$. By parameterizing the Hamiltonian directly in terms of the energies of the individual irreps, we perform an exhaustive search of all possible irrep degeneracies which may host stable classical spin liquids. Doing so reveals four one-parameter families of models along which three phases are degenerate, all four of which merge at the Heisenberg antiferromagnet and its dual, and we give a complete three-dimensional picture of the phase diagram showing all of the phases and their intersections. The appearance of two copies of a single irrep implies an extra degenerate locus which pierces the phase boundaries at special points, yielding two additional isolated triple points. We demonstrate that one-parameter families of Hamiltonians are characterized by a topological invariant describing how the ground state spin configuration winds when adiabatically transported around this degenerate locus, analogous to a "diabolical locus". We provide a comprehensive catalog of all flat band degeneracies in the phase diagram and discuss the mechanisms that may allow for or impede the realization of a variety of classical spin liquids described by tensor gauge fields exhibiting pinch line singularities and concomitant four-fold pinch points. Lastly, we provide a list of all cases where three irreps are degenerate above the ground state, which may lead to interesting features in the spin wave spectrum within each ordered phase.