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Journal of Physics Condensed Matter

Scaling analysis of the non-Abelian quasiparticle tunneling in Z k FQH states

Qi Li, Na Jiang, Xin Wan, Zi-Xiang Hu

Published May 15, 2018• 3 citations
Publisher

Abstract

Quasiparticle tunneling between two counter propagating edges through point contacts could provide information on its statistics. Previous study of the short distance tunneling displays a scaling behavior, especially in the conformal limit with zero tunneling distance. The scaling exponents for the non-Abelian quasiparticle tunneling exhibit some non-trivial behaviors. In this work, we revisit the quasiparticle tunneling amplitudes and their scaling behavior in a full range of the tunneling distance by putting the electrons on the surface of a cylinder. The edge-edge distance can be smoothly tuned by varying the aspect ratio for a finite size cylinder. We analyze the scaling behavior of the quasiparticles for the Read-Rezayi [Formula: see text] states for [Formula: see text] and 4 both in the short and long tunneling distance region. The finite size scaling analysis automatically gives us a critical length scale where the anomalous correction appears. We demonstrate this length scale is related to the size of the quasiparticle at which the backscattering between two counter propagating edges starts to be significant.

DOI
10.1088/1361-648x/aac462

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Quasiparticle tunneling between two counter propagating edges through point contacts could provide information on its statistics. Previous study of the short distance tunneling displays a scaling behavior, especially in the conformal limit with zero tunneling distance. The scaling exponents for the non-Abelian quasiparticle tunneling exhibit some non-trivial behaviors. In this work, we revisit the quasiparticle tunneling amplitudes and their scaling behavior in a full range of the tunneling distance by putting the electrons on the surface of a cylinder. The edge-edge distance can be smoothly tuned by varying the aspect ratio for a finite size cylinder. We analyze the scaling behavior of the quasiparticles for the Read-Rezayi [Formula: see text] states for [Formula: see text] and 4 both in the short and long tunneling distance region. The finite size scaling analysis automatically gives us a critical length scale where the anomalous correction appears. We demonstrate this length scale is related to the size of the quasiparticle at which the backscattering between two counter propagating edges starts to be significant.
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.
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.
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.
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 $\ensuremath{\nu}=5/2$.
The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.
Fractional quantum Hall quasiparticles are generally characterized by two quantum numbers: electric charge QQQ and scaling dimension $\mathrm{\ensuremath{\Delta}}$. For the simplest states (such as the Laughlin series), the scaling dimension determines the anyonic statistics of the quasiparticle (the statistical phase $\ensuremath{\theta}=2\ensuremath{\pi}\mathrm{\ensuremath{\Delta}}$). For more complicated states (featuring counterpropagating modes or non-Abelian statistics), knowing the scaling dimension is not enough to extract the quasiparticle statistics. Nevertheless, even in those cases, knowing the scaling dimension facilitates distinguishing different candidate theories for describing the quantum Hall state at a particular filling (such as PH-Pfaffian and anti-Pfaffian at $\ensuremath{\nu}=\frac{5}{2}$). Here, we propose a scheme for extracting the scaling dimension of quantum Hall quasiparticles from thermal tunneling noise produced at a quantum point contact. Our scheme makes only minimal assumptions about the edge structure and features the level of robustness, simplicity, and model independence comparable with extracting the quasiparticle charge from tunneling shot noise.