Mapping the Braiding Properties of Non-Abelian Fractional Quantum Hall Liquids

Non-Abelian FQH states have elementary excitations that cannot be individually locally-created. When widely separated, the elementary excitations (quasiparticles) give rise to topological (quasi-)degeneracy of the quantum states; braiding of such non-Abelian quasiparticles implements unitary transformations among the degenerate states that may be useful for "topological quantum computing".

In this talk I will focus on the non-Abelian Moore-Read 1/2 state. In the first part, I will discuss the elementary excitations of this quantum Hall state, which are quasiparticles carrying half flux and a quarter of the elementary charge. The quasiparticles are paired like in a p-wave super-conducting state.

In the second part, I will discuss a new technique that allows me to pin the quasiparticles to any desired configuration in space. The resulting configurations and states can be visualized by plotting the non-homogeneous particle density and pair amplitude. I will also discuss the possibility of splitting the paired quasiparticles by using external 1-body potentials, a fact that is essential for "topological computation".

In the third part, I will discuss the non-Abelian braiding properties. For systems of small numbers of quasiparticles, I have computed the non-Abelian Berry curvature and Pontrjagin density as one quasiparticle is adiabatically moved relative to a fixed configuration of the others, showing how the topological and non-Abelian properties develop as the system size increases. I will also describe the geometry of the quantum states and conclude with a discussion of the non-Abelian statistics of the quasiparticles.