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arXiv:2111.05399 · Physical review. B./Physical review. B

Extracting the scaling dimension of quantum Hall quasiparticles from current correlations

Noam Schiller, Yuval Oreg, Kyrylo Snizhko

Published Apr 27, 2022• 30 citations
arXivHTMLPDF

Abstract

Fractional quantum Hall quasiparticles are generally characterized by two quantum numbers: electric charge $Q$ 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.

DOI
10.1103/physrevb.105.165150

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Fractional quantum Hall quasiparticles are generally characterized by two quantum numbers: electric charge QQQ and scaling dimension \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ. For the simplest states (such as the Laughlin series), the scaling dimension determines the anyonic statistics of the quasiparticle (the statistical phase \ensuremathθ=2\ensuremathπ\ensuremathΔ\ensuremath{\theta}=2\ensuremath{\pi}\mathrm{\ensuremath{\Delta}}\ensuremathθ=2\ensuremathπ\ensuremathΔ). 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 \ensuremathν=52\ensuremath{\nu}=\frac{5}{2}\ensuremathν=25​). 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.
Continued improvement of heat control in mesoscopic conductors brings novel tools for probing strongly correlated electron phenomena. Motivated by these advances, we comprehensively study transport due to a temperature bias in a quantum point contact device in the fractional quantum Hall regime. We compute the charge-current noise (so-called delta- T T noise), heat-current noise, and mixed noise and elucidate how these observables can be used to infer strongly correlated properties of the device. Our main focus is the extraction of so-called scaling dimensions of the tunneling anyonic quasiparticles, of critical importance to correctly infer their anyonic exchange statistics.
We study the time-dependent dynamical phase of fractional quasiparticles in quantum Hall states using Ramsey interferometry. A Ramsey two-pulse voltage modulation generates an interference term in the current noise of a quantum point contact oscillating with the dynamical phase. We show that the interference pattern probes the Green's function of the fractional quasiparticles in the time domain and reveals the fractional charge. We address both the case of a pointlike tunnel junction and the case of momentum-resolved tunneling which also provides information on the speed of those quasiparticles propagating along the edge. For the 5/25/25/2 fractional quantum Hall case, the Ramsey signal can reveal the charge e/4e/4e/4 and differentiate between Pfaffian and anti-Pfaffian states.
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 $\ensuremath{\nu}=5/2$.
We present a comprehensive numerical study of a microscopic model of the fractional quantum Hall system at filling fraction $\ensuremath{\nu}=5∕2$, based on the disk geometry. Our model includes Coulomb interaction and a semirealistic confining potential. We also mix in a three-body interaction in some cases to help elucidate the physics. We obtain a phase diagram, discuss the conditions under which the ground state can be described by the Moore-Read state, and study its competition with neighboring stripe phases. We also study quasihole excitations and edge excitations in the Moore-Read-like state. From the evolution of the edge spectrum, we obtain the velocities of the charge and neutral edge modes, which turn out to be very different. This separation of velocities is a source of decoherence for a non-Abelian quasihole and/or quasiparticle (with charge $\ifmmode\pm\else\textpm\fi{}e∕4$) when propagating at the edge; using numbers obtained from a specific set of parameters, we estimate the decoherence length to be around $4\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$. This sets an upper bound for the separation of the two point contacts in a double point-contact interferometer, designed to detect the non-Abelian nature of such quasiparticles. We also find a state that is a potential candidate for the recently proposed anti-Pfaffian state. We find the speculated anti-Pfaffian state is favored in weak confinement (smooth edge), while the Moore-Read Pfaffian state is favored in strong confinement (sharp edge).
The quasihole states of several paired states, the Pfaffian, Haldane-Rezayi, and 331 states, which under certain conditions may describe electrons at filling factor \ensuremath{\nu}=1/2 or 5/2, are studied analytically and numerically in the spherical geometry, for the Hamiltonians for which the ground state are known exactly. We also find all the ground states (without quasiparticles) for these systems in the toroidal geometry. In each case, a complete set of linearly independent functions that are energy eigenstates of zero energy is found explicitly. For fixed positions of the quasiholes, the number of linearly independent states is ${2}^{\mathit{n}\mathrm{\ensuremath{-}}1}$ for the Pfaffian, and ${2}^{2\mathit{n}\mathrm{\ensuremath{-}}3}$ for the Haldane-Rezayi state; these degeneracies are needed if these systems are to possess non-Abelian statistics, and they agree with predictions based on conformal field theory. The dimensions of the spaces of states for each number of quasiholes agree with numerical results for moderate system sizes. The effects of tunneling and of the Zeeman term are discussed for the 331 and Haldane-Rezayi states, as well as the relation to Laughlin states of electron pairs. A model introduced by Ho, which was supposed to connect the 331 and Pfaffian states, is found to have the same degeneracies of zero-energy states as the 331 state, except at its Pfaffian point where it is much more highly degenerate than either the 331 or the Pfaffian. We introduce a modification of the model which has the degeneracies of the 331 state everywhere including the Pfaffian point; at the latter point, tunneling reduces the degeneracies to those of the Pfaffian state. An experimental difference is pointed out between the Laughlin states of electron pairs and the other paired states, in the current-voltage response when electrons tunnel into the edge. An appendix contains results for the permanent state, in which the zero modes can be occupied by composite bosons, rather than by composite fermions as in the other cases; the system is found to have an incipient instability toward a spin-polarized state. \textcopyright{} 1996 The American Physical Society.