The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter θ. θ = 0, π correspond to bosons and fermions, respectively; at intermediate values, we say that we have fractional statistics. In two dimensions, θ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group realize this behavior. More complex charge-flux constructions can involve nonabelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of nonabelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Anyons of all kinds are described by quantum field theories that include Chern–Simons terms. The crossings of one-dimensional anyons on a ring are unidirectional, such that a fractional phase θ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently, the anyon behavior predicted for quasiparticles in the ν = 1/3 fractional quantum Hall state has been observed in both scattering and interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.

中文翻译：

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 分数统计


运动局限于两个（或一个）空间维度的粒子集合体的量子力学描述提供了许多不同于玻色子和费米子的可能性。我们称这种粒子为任意子。最简单的任意子由角相位参数 θ 参数化。θ = 0，π分别对应于玻色子和费米子;在中间值处，我们说我们有分数统计量。在二维中，θ 将波函数获得的相位描述为两个任意子逆时针相互缠绕。它会在相对角动量的允许值中生成偏移。与阿贝尔 U（1） 规范群相关的局域电荷和磁通量的复合材料实现了这种行为。更复杂的电荷-磁通量结构可能涉及作用于允许电荷和磁通量范围的 nonabelian 和乘积基团，从而产生 nonabelian 和 mutual 统计。非阿贝尔任意子的交换在内部状态的紧急空间内实现波函数的幺正变换。各种 Anyons 都由包括 Chern-Simons 项的量子场论描述。环上一维任意子的交叉是单向的，因此在互换时获得的分数相位 θ 会导致任意子之间相对动量的分数偏移。分数量子霍尔态的准粒子激发长期以来一直被预测为包括任意子。最近，在散射和干涉实验中都观察到了 ν = 1/3 分数量子霍尔态准粒子预测的任意子行为。设计系统（尤其是包括超导电路）中的激励可以表现出任意子行为。 正在开发此类系统，以可能用于量子信息处理。