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arXiv:1505.06521 · New Journal of Physics

The length scale measurements of the fractional quantum Hall state on a cylinder

Qi Li, Na Jiang, Zheng Zhu, Zi-Xiang Hu

Published Sep 24, 2015• 3 citations
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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.

DOI
10.1088/1367-2630/17/9/095006

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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.
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Scaling analysis of the quasiparticle tunneling in the Zk{\mathbb Z}_kZk​ parafermion states
Quasiparticle tunneling between two counter-propagating edges through point contacts could provide information on the statistics of the quasiparticles. Previous study on a disk found a scaling behavior by varying the tunneling distance. It was found that in the limit with zero tunneling distance, the Abelian quasiparticles tunneling obey the scaling analysis while the non-Abelian quasiparticles exhibit some non-trivial behaviors on the scaling exponents. Because of the limitation of disk geometry, we put the fractional quantum Hall (FQH) state on the surface of a cylinder which has a larger tunable tunneling distance than that on disk by varying the aspect ratio γγγ. We analyze the scaling behavior of the quasiholes, especially the non-Abelian quasiholes in the Read-Rezayi Zk{\mathbb Z}_kZk​ parafermion states. We aim to address the existance of the anomalous correction of the scaling parameter in the long tunneling distance.
In fractional quantum Hall systems, quasiparticles of fractional charge can tunnel between the edges at a quantum point contact. Such tunneling (or backscattering) processes contribute to charge transport and provide information on both the charge and statistics of the quasiparticles involved. Here, we study quasiparticle tunneling in the Moore-Read state, in which quasiparticles of charges e/4e/4e/4 (non-Abelian) and e/2e/2e/2 (Abelian) may coexist and both contribute to edge transport. On a disk geometry, we calculate the matrix elements for e/2e/2e/2 and e/4e/4e/4 quasiholes to tunnel through the bulk of the Moore-Read state, in an attempt to understand their relative importance. We find that the tunneling amplitude for charge e/2e/2e/2 quasihole is exponentially smaller than that for charge e/4e/4e/4 quasihole, and the ratio between them can be (partially) attributed to their charge difference. We find that including long-range Coulomb interaction only has a weak effect on the ratio. We discuss briefly the relevance of these results to recent tunneling and interferometry experiments at filling factor $\ensuremath{\nu}=5/2$.
We study the scaling behavior in the tunneling amplitude when quasiparticles tunnel along a straight path between the two edges of a fractional quantum Hall annulus. Such scaling behavior originates from the propagation and tunneling of charged quasielectrons and quasiholes in an effective field analysis. In the limit when the annulus deforms continuously into a quasi-one-dimensional (1D) ring, we conjecture the exact functional form of the tunneling amplitude for several cases, which reproduces the numerical results in finite systems exactly. The results for Abelian quasiparticle tunneling is consistent with the scaling analysis; this allows for the extraction of the conformal dimensions of the quasiparticles. We analyze the scaling behavior of both Abelian and non-Abelian quasiparticles in the Read–Rezayi -parafermion states. Interestingly, the non-Abelian quasiparticle tunneling amplitudes exhibit non-trivial k -dependent corrections to the scaling exponent.
For fractional quantum Hall states on a finite disk, we study the thermoelectric transport properties under the influence of the presence of an edge and its reconstruction. In a recent study on a torus [Phys. Rev. B 101, 241101(R) (2020)], Sheng and Fu found a universal non-Fermi liquid power-law scaling of the thermoelectric conductivity ${\ensuremath{\alpha}}_{xy}\ensuremath{\propto}{T}^{\ensuremath{\eta}}$ for the gapless composite Fermi liquid state. The exponent $\ensuremath{\eta}\ensuremath{\sim}0.5$ appears an independence of the filling factors and the details of the interactions. In the presence of an edge, we find the properties of the edge spectrum dominants the low-temperature behaviors and a nonuniversal scaling behavior in low temperature is observed. To consider individually the effect of the edge states, the real space entanglement spectrum of the model wave function, which contains only the spectrum of edge excitation is employed. Its spectrum energies could be tuned by varying the radius of the subsystem. In the non-Abelian Moore-Read state, the Majorana neutral edge mode is found to have more significant effects than that of the charge mode in the low temperature.
We present a new approach for obtaining the scaling behavior of the entanglement entropy in fractional quantum Hall (FQH) states from finite-size wavefunctions. By employing the torus geometry and the fact that the torus aspect ratio can be readily varied, we can extract the entanglement entropy of a spatial block as a continuous function of the block boundary length. This approach allows us to extract the topological entanglement entropy with an accuracy superior to that possible for the spherical or disc geometry, where no natural continuously variable parameter is available. Other than the topological information, the study of entanglement scaling is also useful as an indicator of the difficulty posed by FQH states for various numerical techniques.