基于模糊超几何的量子反铁磁体拓扑多体态
Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry
摘要：
近年来对拓扑序的研究不仅发现了新的物质拓扑态，而且对“已知”的拓扑态也有了新的认识。拓扑有序的一个既定例子是量子反铁磁体中的价键固体（VBS）态。VBS态是无序的自旋液体，没有自发对称性破缺，但最典型的表现为一维链上的拓扑有序，称为隐弦序。有趣的是，VBS模型基于类似于模糊几何的数学。我们回顾了模糊超几何数学在构建VBS（SVBS）态的超对称版本中的应用，并给出了SVBS模型及其性质的教学介绍[arXiv：0809.4885，1105.3529，1210.0299]。作为具体例子，我们详细分析了SU（2）和SO（5）VBS态的超对称形式，即UOSp（N| 2）和UOSp（N| 4）数学上与模糊二超球和模糊四超球密切相关的SVBS状态。SVBS态在物理上被解释为具有超导性质的空穴掺杂VBS态，其根据空穴掺杂参数的值插入各种VBS态。明确地构造了SVBS态的父哈密顿量，并在一维SVBS链的单模近似下导出了它们的带隙激发。突出的特点SVBS链的详细讨论，如一个广义的字符串序参量和纠缠谱。结果表明，不管体（超）自旋的奇偶性如何，纠缠谱至少是双简并的。讨论了具有超对称性的拓扑相的稳定性，重点讨论了它与特定边（超）自旋态的关系。
英文摘要：
Recent vigorous investigations of topological order have not only discovered new topological states of matter but also shed new light to "already known" topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking but most typically manifest topological order known as hidden string order on 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy super-geometry in the construction of supersymmetric versions of VBS (SVBS) states, and give a pedagogical introduction of SVBS models and their properties [arXiv:0809.4885, 1105.3529, 1210.0299]. As concrete examples, we present detail analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e. UOSp(N|2) and UOSp(N|4) SVBS states whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with superconducting property that interpolate various VBS states depending on value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate regardless of the parity of bulk (super)spins. Stability of topological phase with supersymmetry is discussed with emphasis on its relation to particular edge (super)spin states.
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