Abstract
Once the fractional quantum Hall (FQH) state for a finite-sized system is put on the surface of a cylinder, the distance between the two ends with open boundary conditions can be tuned by varying the aspect ratio γ. It scales linearly with increasing the system size and therefore has a larger adjustable range than that on a disk. The previous study of the quasi-hole tunneling amplitude on a disk by Hu et al (2011 New J. Phys. 13 035020) 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 behaves poorly due to the narrow range of the tunneling distance on the disk. Here we systematically study the quasiparticle tunneling amplitudes of the Laughlin state in the cylinder geometry which shows a much better scaling behavior. In particular, there are some crossover behaviors at the 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, and are found to be related to the dimension reduction and back scattering point respectively.
Highlights
The strongly correlated electron system reveals a plenty of non-trivial properties which beyond the single-particle picture
We find a richer structure in this region and two characteristic length scales appears on the quasiparticle tunneling, and in the wavefunction overlap, bipartite entanglement entropy and electron Green’s function
We confirm that the quasihole tunneling amplitude in the cylinder geometry obeys the scaling conjecture in Eq.4 and the scaling behavior is much better than that on disk
Summary
The strongly correlated electron system reveals a plenty of non-trivial properties which beyond the single-particle picture. Because of the strongly overlap of all the Landau orbitals, the Gaussian factors of each Landau wavefunction are the same and can be erased by normalization In this one-dimensional limit, the FQH wave function can be described by the Jack polynomials and all the results are the same as that we did on disk in the ring limit. Jack polynomial is a powerful method in studying the FQHE as it can construct the model wave function for Read-Rezayi series [15, 16, 17], and the low-lying excitations [18, 19] Another advantage for cylinder geometry is the computational convenient comparing with either the disk or sphere geometries which was discussed in the density matrix renormalization calculation [20].
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