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Register a new userUniversal Conductance and Conductivity at Critical Points in Integer Quantum Hall Systems
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The sample averaged longitudinal two-terminal conductance and the respective Kubo-conductivity are calculated at quantum critical points in the integer quantum Hall regime. In the limit of large system size, both transport quantities are found to be the same within numerical uncertainty in the lowest Landau band, $0.60pm 0.02 e^2/h$ and $0.58pm 0.03 e^2/h$, respectively. In the 2nd lowest Landau band, a critical conductance $0.61pm 0.03 e^2/h$ is obtained which indeed supports the notion of universality. However, these numbers are significantly at variance with the hitherto commonly believed value $1/2 e^2/h$. We argue that this difference is due to the multifractal structure of critical wavefunctions, a property that should generically show up in the conductance at quantum critical points.
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We measure the conductance of a quantum point contact (QPC) while the biased tip of a scanning probe microscope induces a depleted region in the electron gas underneath. At finite magnetic field we find plateaus in the real-space maps of the conductance as a function of tip position at integer ( u=1,2,3,4,6,8) and fractional ( u=1/3,2/3,5/3,4/5) values of transmission. They resemble theoretically predicted compressible and incompressible stripes of quantum Hall edge states. The scanning tip allows us to shift the constriction limiting the conductance in real space over distances of many microns. The resulting stripes of integer and fractional filling factors are rugged on the micron scale, i.e. on a scale much smaller than the zero-field elastic mean free path of the electrons. Our experiments demonstrate that microscopic inhomogeneities are relevant even in high-quality samples and lead to locally strongly fluctuating widths of incompressible regions even down to their complete suppression for certain tip positions. The macroscopic quantization of the Hall resistance measured experimentally in a non-local contact configuration survives in the presence of these inhomogeneities, and the relevant local energy scale for the u=2 state turns out to be independent of tip position.
We study numerically the charge conductance distributions of disordered quantum spin-Hall (QSH) systems using a quantum network model. We have found that the conductance distribution at the metal-QSH insulator transition is clearly different from that at the metal-ordinary insulator transition. Thus the critical conductance distribution is sensitive not only to the boundary condition but also to the presence of edge states in the adjacent insulating phase. We have also calculated the point-contact conductance. Even when the two-terminal conductance is approximately quantized, we find large fluctuations in the point-contact conductance. Furthermore, we have found a semi-circular relation between the average of the point-contact conductance and its fluctuation.
The hopping ac conductance, which is realized at the transverse conductance minima in the regime of the integer Hall effect, has been measured using a combination of acoustic and microwave methods. Measurements have been made in the p-GeSi/Ge/GeSi structures with quantum wells in a wide frequency range (30-1200 MHz). The experimental frequency dependences of the real part of ac conductance $sigma_1$ have been interpreted on the basis of the model presuming hops between localized electronic states belonging to isolated clusters. At high frequencies, dominating clusters are pairs of close states; upon a decrease in frequency, large clusters that merge into an infinite percolation cluster as the frequency tends to zero become important. In this case, the frequency dependences of the ac conductance can be represented by a universal curve. The scaling parameters and their magnetic-field dependence have been determined.
We measure the longitudinal conductivity $sigma_{xx}$ at frequencies $1.246 {rm GHz} le f le 10.05$ GHz over a range of temperatures $235 {rm mK} le T le 4.2$ K with particular emphasis on the Quantum Hall plateaus. We find that $Re(sigma_{xx})$ scales linearly with frequency for a range of magnetic field around the center of the plateaus, i.e. where $sigma_{xx}(omega) gg sigma_{xx}^{DC}$. The width of this scaling region decreases with higher temperature and vanishes by 1.2 K altogether. Comparison between localization length determined from $sigma_{xx}(omega)$ and DC measurements on the same wafer show good agreement.
One dimensional tight binding models such as Aubry-Andre-Harper (AAH) model (with onsite cosine potential) and the integrable Maryland model (with onsite tangent potential) have been the subject of extensive theoretical research in localization studies. AAH can be directly mapped onto the two dimensional Hofstadter model which manifests the integer quantum Hall topology on a lattice. However, no such connection has been made for the Maryland model (MM). In this work, we describe a generalized model that contains AAH and MM as the limiting cases with the MM lying precisely at a topological quantum phase transition (TQPT) point. A remarkable feature of this critical point is that the 1D MM retains well defined energy gaps whereas the equivalent 2D model becomes gapless, signifying the 2D nature of the TQPT.
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