Non-Abelian statistics in condensed matter systems

September 21, 2020

The particles in our world can be divided into two classes: those with Bose statistics (such as photons, Hydrogen atom, etc) and those with Fermi statistics (such as electron, quark, etc). The possibility of third kind of particles with fractional statistics in 2-dimensional space was pointed out by Leinaas and Myrheim [1] and Wilczek [2], and was named as anyon. Wu [3] has setup a general theory for quantum statistics in two dimensions in terms of braid group, and Goldin et al. [4] pointed out that such a setup contains higher dimensional representations of braid group, which correspond to non-abelian statistics. A full description of non-abelian statistics in topological quantum field theory was given by Witten [5].

Braiding of particles is described by braid group

The most striking property of non-Abelian anyons is that they carry non-integer degrees of freedom. An electron, with spin-up and spin-down, has 2 degrees of freedom. An SU(2)2 non-Abelian anyon carries 21/2 = 1.414 degrees of freedom. An SU(3)2 non-Abelian anyon carries (1+51/2)/2 = 1.618 degrees of freedom. Such a property is so strange, it seems impossible to be realized by an electron system. But one should not underestimate the richness of many-body systems. Non-Abelian statistics actually can appear in condensed matter systems (in fractional quantum Hall (FQH) systems), as pointed out by Wen [6] and Moore-Read [7] in 1991.

Wen proposed to make non-Abelain FQH state using degenerate Landau levels (or multi-layers) and short range repulsion. If the first three Landau level are degenerate, the following wave function χ1(zi)[χ2(zi)]2 will realize one version of SU(2)2 non-Abelian topological order (i.e. Ising non-Abelian topological order or Z2 parafermion non-Abelian topological order). Here χn(zi) is the electron wave function of n-filled Landau level. If the first four Landau level are degenerate, the following wave function [χ2(zi)]3 will realize one version of SU(3)2 non-Abelian topological order (also known as Z3 parafermion non-Abelian topological order, or SU(2)3 non-Abelian topological order, or Fibonacci topological order).

Moore-Read proposed a many-body wave function using a correlation function of simple current operators in Ising conformal field theory, which realize a version of Ising (i.e. SU(2)2 or Z2 parafermion) non-Abelian topological order. Such a wave function can be realized by a 3-body interaction (or by electrons in the second Landau level with core-softened Coulomb interaction).

The Ising non-Abelian FQH state has a very special property: its edge exacitions are described by chiral Majorana fermion [8]. It was predicted that the appearance of chiral Majorana fermion on the edge implies the apprearance of Ising non-Abelian statistics in the bulk [8]. The chiral Majorana fermion on the edge can be measured by half-integer quantized chiral central charge (i.e. thermal Hall effect [9]). Recently, Heiblum's group discovered half-integer quantized thermal Hall effect [10] in FQH system, indirectly discovered Ising non-Abelian statistics. In a more recent work, a superconductor-FQH heterostructure was realized, that may produce non-Abelian statistics [11].