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

其运动被限制在两个（或一个）空间维度的粒子集合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分别对应玻色子和费米子；在中间值，我们说我们有分数统计。在二维中，θ 描述了当两个任意子逆时针相互缠绕时波函数获得的相位。它会产生相对角动量允许值的变化。与阿贝尔 U(1) 规范组相关的局部电荷和磁通量的复合材料实现了这种行为。更复杂的电荷通量结构可能涉及作用于一系列允许的电荷和通量的非阿贝尔和乘积群，从而产生非阿贝尔统计和互统计。非阿贝尔任意子的交换在内部态的涌现空间内实现了波函数的酉变换。所有类型的任意子都可以通过包括陈-西蒙斯项的量子场论来描述。一维任意子在环上的交叉是单向的，使得交换时获得的分数相位θ引起任意子之间的相对动量的分数位移。长期以来，人们一直预测分数量子霍尔态的准粒子激发包括任意子。最近，在散射和干涉实验中都观察到了 ν = 1/3 分数量子霍尔态中准粒子的任意子行为。设计系统（特别是超导电路）内的激励可以表现出任意子行为。此类系统正在开发中，以可能用于量子信息处理。