Modular transformations through sequences of topological charge projections
Modular transformations through sequences of topological charge projections
复制标题
通过拓扑电荷投影序列进行模变换
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
M. Freedman
中科院分区：
文献类型：
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作者：
M. Barkeshli;M. Freedman
物质的拓扑相的基态子空间形成了定义状态的空间的映射类群的表示。我们证明了亏格为g的曲面的映射类群的元素可以通过沿至少三个相互交叉的不可收缩圈的拓扑荷投影序列沿着得到。我们证明了这两个通过代数理论的任意子，也通过分析的拓扑结构的时空流形。我们联合收割机将这一结果与两个观察结果相结合：（i）通过使用双层或双拓扑相，并引入适当类型的带隙边界，可以在平面几何中有效地模拟亏格g的表面;以及（ii）所需的拓扑电荷投影可以通过局部调整系统的微观参数而实现为绝热幺正变换，比如能量差。这些观察结果表明，在物理系统中有效地实现模块化转换的可能途径。特别是，他们还展示了Ising $ensuremath{igotimes}幻象{
ule{0.28em}{0 ex}}上划线{ ext{Ising}}$态，在存在断开的带隙边界的情况下，可以支持普适的拓扑量子计算.
The ground-state subspace of a topological phase of matter forms a representation of the mapping class group of the space on which the state is defined. We show that elements of the mapping class group of a surface of genus $g$ can be obtained through a sequence of topological charge projections along at least three mutually intersecting noncontractible cycles. We demonstrate this both through the algebraic theory of anyons and also through an analysis of the topology of the space-time manifold. We combine this result with two observations: (i) that surfaces of genus $g$ can be effectively simulated in planar geometries by using bilayer, or doubled, versions of the topological phase of interest, and inducing the appropriate types of gapped boundaries; and (ii) that the required topological charge projections can be implemented as adiabatic unitary transformations by locally tuning microscopic parameters of the system, such as the energy gap. These observations suggest a possible path towards effectively implementing modular transformations in physical systems. In particular, they also show how the Ising $ensuremath{igotimes}phantom{
ule{0.28em}{0ex}}overline{ ext{Ising}}$ state, in the presence of disconnected gapped boundaries, can support universal topological quantum computation.