(a) Longitudinal, Txx (red), and transverse, Txy (blue), transmission coefficients of a disordered nanowire between the second and first Hall plateaus as functions of φ. The respective transmission increment series are shown in the inset. (b) LDOS in the transition between the second and first Landau levels for increasing values of the magnetic flux φ = 0.50, 0.55, 0.56, 0.64 (from top to bottom). The probes are shown in grey. The LDOS increases as the color changes from red to blue.

Contexts in source publication

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... [66,67]. The four-terminal transmission coefficients are calculated via the Landauer-Büttiker formula, Note that, the coefficients satisfy the relation T 11 (φ) + T xx (φ) + T xy (φ) = N , where N is the number of propagating wave modes in the terminals, which is tuned by the Fermi energy. The time series of T xy (φ) and T xx (φ) shown in Fig. 2 were obtained for one realization of the disorder potential with 10 4 time steps. One sees that both longitudinal and transverse transmission coefficients fluctuate in a seemingly random fashion as the system is driven from one plateau to the next, as reported previously, by e.g., [32,34,36]. The corresponding series of transmission ...

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... both longitudinal and transverse transmission coefficients fluctuate in a seemingly random fashion as the system is driven from one plateau to the next, as reported previously, by e.g., [32,34,36]. The corresponding series of transmission increments, ∆T (φ) = T (φ + ∆φ) − T (φ), where T stands for either T xx or T xy , are shown in the inset of Fig. 2(a), where one sees that the transmission increments fluctuate rather intermittently. Figure 2(b) shows colour-coded plots of the local density of states (LDOS) for increasing values of φ (from top to bottom). Note that in the Hall plateaus (top and bottom panels of Fig. 2(b)) the LDOS is localized near the upper and lower edges of the ...

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... corresponding series of transmission increments, ∆T (φ) = T (φ + ∆φ) − T (φ), where T stands for either T xx or T xy , are shown in the inset of Fig. 2(a), where one sees that the transmission increments fluctuate rather intermittently. Figure 2(b) shows colour-coded plots of the local density of states (LDOS) for increasing values of φ (from top to bottom). Note that in the Hall plateaus (top and bottom panels of Fig. 2(b)) the LDOS is localized near the upper and lower edges of the device, as expected, since in such cases the only extended states that connect contacts are edge states. ...

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... − T (φ), where T stands for either T xx or T xy , are shown in the inset of Fig. 2(a), where one sees that the transmission increments fluctuate rather intermittently. Figure 2(b) shows colour-coded plots of the local density of states (LDOS) for increasing values of φ (from top to bottom). Note that in the Hall plateaus (top and bottom panels of Fig. 2(b)) the LDOS is localized near the upper and lower edges of the device, as expected, since in such cases the only extended states that connect contacts are edge states. As the system is driven away from a plateau by varying φ, the LDOS penetrate into the bulk and a complex spatial pattern develops (middle pannels of Fig. 2(b)), leading to ...

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... and bottom panels of Fig. 2(b)) the LDOS is localized near the upper and lower edges of the device, as expected, since in such cases the only extended states that connect contacts are edge states. As the system is driven away from a plateau by varying φ, the LDOS penetrate into the bulk and a complex spatial pattern develops (middle pannels of Fig. 2(b)), leading to the formation of coherent structures of different sizes inside the device. This process is somewhat similar to a laminar-to-turbulence transition, in the sense that near a Hall plateau the LDOS are rather laminar (albeit restricted to the edges), whereas it becomes very irregularly distributed in space as the magnetic ...

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... of length scales in the shown patterns. Such a multiscale dynamics will be examined below from the viewpoint of both a multifractal analysis and a turbulence-like cascade model. In order to perform our analysis, we conveniently chose the range of fluxes 0.545 < φ < 0.615, which includes only transmission fluctuations values between 0 and 1, see Fig. ...

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... Analysis. Here we shall employ the Multifractal Detrended Fluctuation Analysis (MF-DFA) [68] for time series of transmission coefficients T (φ), Fig. 2(a). For a brief description of the main steps of the MF-DFA algorithm, see the Supplemental Material (SM) [69]. Fig. 3(a) shows the generalized Hurst exponent h(q), which is obtained through the scaling relation of the q-th order fluctuation function F q (τ ) ∼ τ h(q) , as computed for both T xy (φ) and T xx (φ) [69]. The strong ...

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... to have the same functional form as the large-scale distribution; and ε is a random variable representing the slowly fluctuating background. For instance, in turbulence ε represents the energy flux from the adjacent larger scale eddies; whereas in the IQHT context, the background is provided by the large structures in the energy density of Fig. 2(b), under which the 'electron flow' evolves over short (fictitious) time scales. As the last and crucial ingredient of the H-theory formalism, the probability density f (ε) of the background variable is obtained explicitly from a hierarchical intermittency model in terms of special functions, namely the Meijer G-functions (see SM ...

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... with the theoretical distri- , superimposed with the respective empirical distribution for the transmission increments. Note that in both cases we observe an excellent agreement between the theoretical predictions and the empirical distributions for N = 5, which can be understood as the number of levels in the hierarchy of length scales seen in Fig. 2(b). This confirms that the plateau transition in the quantum Hall effect (in the mesoscopic regime) displays a turbulence-like hierarchical ...

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... [66,67]. The four-terminal transmission coefficients are calculated via the Landauer-Büttiker formula, Note that, the coefficients satisfy the relation T 11 (φ) + T xx (φ) + T xy (φ) = N , where N is the number of propagating wave modes in the terminals, which is tuned by the Fermi energy. The time series of T xy (φ) and T xx (φ) shown in Fig. 2 were obtained for one realization of the disorder potential with 10 4 time steps. One sees that both longitudinal and transverse transmission coefficients fluctuate in a seemingly random fashion as the system is driven from one plateau to the next, as reported previously, by e.g., [32,34,36]. The corresponding series of transmission ...

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... both longitudinal and transverse transmission coefficients fluctuate in a seemingly random fashion as the system is driven from one plateau to the next, as reported previously, by e.g., [32,34,36]. The corresponding series of transmission increments, ∆T (φ) = T (φ + ∆φ) − T (φ), where T stands for either T xx or T xy , are shown in the inset of Fig. 2(a), where one sees that the transmission increments fluctuate rather intermittently. Figure 2(b) shows colour-coded plots of the local density of states (LDOS) for increasing values of φ (from top to bottom). Note that in the Hall plateaus (top and bottom panels of Fig. 2(b)) the LDOS is localized near the upper and lower edges of the ...

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... corresponding series of transmission increments, ∆T (φ) = T (φ + ∆φ) − T (φ), where T stands for either T xx or T xy , are shown in the inset of Fig. 2(a), where one sees that the transmission increments fluctuate rather intermittently. Figure 2(b) shows colour-coded plots of the local density of states (LDOS) for increasing values of φ (from top to bottom). Note that in the Hall plateaus (top and bottom panels of Fig. 2(b)) the LDOS is localized near the upper and lower edges of the device, as expected, since in such cases the only extended states that connect contacts are edge states. ...

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... − T (φ), where T stands for either T xx or T xy , are shown in the inset of Fig. 2(a), where one sees that the transmission increments fluctuate rather intermittently. Figure 2(b) shows colour-coded plots of the local density of states (LDOS) for increasing values of φ (from top to bottom). Note that in the Hall plateaus (top and bottom panels of Fig. 2(b)) the LDOS is localized near the upper and lower edges of the device, as expected, since in such cases the only extended states that connect contacts are edge states. As the system is driven away from a plateau by varying φ, the LDOS penetrate into the bulk and a complex spatial pattern develops (middle pannels of Fig. 2(b)), leading to ...

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... and bottom panels of Fig. 2(b)) the LDOS is localized near the upper and lower edges of the device, as expected, since in such cases the only extended states that connect contacts are edge states. As the system is driven away from a plateau by varying φ, the LDOS penetrate into the bulk and a complex spatial pattern develops (middle pannels of Fig. 2(b)), leading to the formation of coherent structures of different sizes inside the device. This process is somewhat similar to a laminar-to-turbulence transition, in the sense that near a Hall plateau the LDOS are rather laminar (albeit restricted to the edges), whereas it becomes very irregularly distributed in space as the magnetic ...

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... of length scales in the shown patterns. Such a multiscale dynamics will be examined below from the viewpoint of both a multifractal analysis and a turbulence-like cascade model. In order to perform our analysis, we conveniently chose the range of fluxes 0.545 < φ < 0.615, which includes only transmission fluctuations values between 0 and 1, see Fig. ...

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... Analysis. Here we shall employ the Multifractal Detrended Fluctuation Analysis (MF-DFA) [68] for time series of transmission coefficients T (φ), Fig. 2(a). For a brief description of the main steps of the MF-DFA algorithm, see the Supplemental Material (SM) [69]. Fig. 3(a) shows the generalized Hurst exponent h(q), which is obtained through the scaling relation of the q-th order fluctuation function F q (τ ) ∼ τ h(q) , as computed for both T xy (φ) and T xx (φ) [69]. The strong ...

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... to have the same functional form as the large-scale distribution; and ε is a random variable representing the slowly fluctuating background. For instance, in turbulence ε represents the energy flux from the adjacent larger scale eddies; whereas in the IQHT context, the background is provided by the large structures in the energy density of Fig. 2(b), under which the 'electron flow' evolves over short (fictitious) time scales. As the last and crucial ingredient of the H-theory formalism, the probability density f (ε) of the background variable is obtained explicitly from a hierarchical intermittency model in terms of special functions, namely the Meijer G-functions (see SM ...

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... with the theoretical distri- , superimposed with the respective empirical distribution for the transmission increments. Note that in both cases we observe an excellent agreement between the theoretical predictions and the empirical distributions for N = 5, which can be understood as the number of levels in the hierarchy of length scales seen in Fig. 2(b). This confirms that the plateau transition in the quantum Hall effect (in the mesoscopic regime) displays a turbulence-like hierarchical ...