The length scale measurements of the Fractional quantum Hall state on cylinder

Once the fractional quantum Hall (FQH) state for a finite size system is put on the surface of a cylinder, the distance between the two ends with open boundary conditions can be tuned as varying the aspect ratio $\gamma$. It scales linearly as increasing the system size and therefore has a larger adjustable range than that on disk. The previous study of the quasi-hole tunneling amplitude on disk in Ref.~\cite{Zk2011} indicates that the tunneling amplitudes have a scaling behavior as a function of the tunneling distance and the scaling exponents are related to the scaling dimension and the charge of the transported quasiparticles. However, the scaling behaviors poorly due to the narrow range of the tunneling distance on disk. Here we systematically study the quasiparticle tunneling amplitudes of the Laughlin state in the cylinder geometry which shows a much better scaling behavior. Especially, there are some corssover behaviors at two length scales when the two open edges are close to each other. These lengths are also reflected in the bipartite entanglement and the electron Green's function as either a singularity or a crossover. These two critical length scales of the edge-edge distance, $L_x^{c_1}$ and $L_x^{c_2}$, are found to be related to the dimension reduction and back scattering point respectively.
Comment: 8 pages, 7 figures