Edge state tunneling in a split Hall bar model

Papa, Emiliano; MacDonald, Allan H.

doi:10.1103/physrevb.72.045324

Cited by 17 publications

(19 citation statements)

References 52 publications

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“…Similar physics has been addressed in the context of edge transport in Laughlin FQH states in Refs. 43,44. We find that in our case the Coulomb interaction drives the tunneling exponents g A and g B in the geometries 1a and 1b above the exponent g C in the geometry 1c.…”

Section: Introductionmentioning

confidence: 64%

Influence of device geometry on tunneling in theν=52quantum Hall liquid

Yang

Feldman

2013

Phys. Rev. B

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Two recent experiments [I. P. Radu et al., Science 320, 899 (2008) and X. Lin et al., Phys. Rev. B 85, 165321 (2012)] measured the temperature and voltage dependence of the quasiparticle tunneling through a quantum point contact in the ν = 5/2 quantum Hall liquid. The results led to conflicting conclusions about the nature of the quantum Hall state. In this paper, we show that the conflict can be resolved by recognizing different geometries of the devices in the experiments. We argue that in some of those geometries there is significant unscreened electrostatic interaction between the segments of the quantum Hall edge on the opposite sides of the point contact. Coulomb interaction affects the tunneling current. We compare experimental results with theoretical predictions for the Pfaffian, SU (2)2, 331 and K = 8 states and their particle-hole conjugates. After Coulomb corrections are taken into account, measurements in all geometries agree with the spin-polarized and spin-unpolarized Halperin 331 states.

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Section: Introductionmentioning

confidence: 64%

Influence of device geometry on tunneling in theν=52quantum Hall liquid

Yang

Feldman

2013

Phys. Rev. B

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“…where the = N sign means "equal modulus N," i.e., wave numbers may scatter over the edge of the Brillouin zone because of the Umklapp (flip-over) scattering (13). Using prosthaphaeresis formulas, one can show that it is impossible to find nonzero k 1 , k 2 , and k 3 satisfying Eq.…”

Section: Significancementioning

confidence: 99%

“…; N=2 , i.e., outside the Brillouin zone. The system flips back this energy into a mode contained in the domain; as mentioned, this is known as the Umklapp scattering process (13,23). These resonant modes have the following structure:…”

Section: Significancementioning

confidence: 99%

Route to thermalization in the α -Fermi–Pasta–Ulam system

Onorato

Vozella

Proment

et al. 2015

Proc. Natl. Acad. Sci. U.S.A.

137

188

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We study the original α-Fermi-Pasta-Ulam (FPU) system with N = 16, 32, and 64 masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave-wave interaction theory; i.e., we assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the α-FPU equation of motion, we find that the first nontrivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that, for small-amplitude random waves, the timescale of such interactions is extremely large and it is of the order of 1/e 8 , where e is the small parameter in the system. The wave-wave interaction theory is not based on any threshold: Equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the Umklapp (flipover) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.T he Fermi-Pasta-Ulam (FPU) chains is a simple mathematical model introduced in the 1950s to study the thermal equipartition in crystals (1). The model consists of N identical masses, each one connected by a nonlinear spring; the elastic force can be expressed as a power series in the spring deformation Δx:where γ; α, and β are elastic, spring-dependent, constants. The α-FPU chain, the system studied herein, corresponds to the case of α ≠ 0 and β = 0. Fermi, Pasta, and Ulam integrated numerically the equation of motion and conjectured that, after many iterations, the system would exhibit a thermalization, i.e., a state in which the influence of the initial modes disappears and the system becomes random, with all modes excited equally (equipartition of energy) on average. Contrary to their expectations, the system exhibited a very complicated quasiperiodic behavior. This phenomenon has been named "FPU recurrence," and this finding has spurred many great mathematical and physical discoveries such as integrability (2) and soliton physics (3). More recently, very long numerical simulations have shown clear evidence of the phenomenon of equipartition (see, for instance, ref. 4 and references therein). However, despite substantial progress on the subject (5-10), to our knowledge no complete understanding of the original problem has been achieved so far, and the numerical results of the original α-FPU system remain largely unexplained from a theoretical point of view. More precisely, the physical mechanism responsible for a first "metastable state" (4) and the observation of equipartition for very large times have not been understood.In this manuscript, we study the FPU problem using an approach based on the nonlinear interaction of weakly nonlinear dispersive waves. Our main assumption is that the irreversible transfer of energy in the spectrum in a weakly nonlinear system is achieved by exact resonant wave-w...

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“…The line junction geometry may also be relevant to a recent set of experiments using samples with an extended constriction [20]. A detailed modelling of such line junctions has been performed by Papa and McDonald [21,22].…”

Section: Introductionmentioning

confidence: 99%

Tunneling in fractional quantum Hall line junctions

2005

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We study the tunneling current between two counterpropagating edge modes described by chiral Luttinger liquids when the tunneling takes place along an extended region. We compute this current perturbatively by using a tunnel Hamiltonian. Our results apply to the case of a pair of different two-dimensional electron gases in the fractional quantum Hall regime separated by a barrier, e. g. electron tunneling. We also discuss the case of strong interactions between the edges, leading to nonuniversal exponents even in the case of integer quantum Hall edges. In addition to the expected nonlinearities due to the Luttinger properties of the edges, there are additional interference patterns due to the finite length of the barrier.

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Edge state tunneling in a split Hall bar model

Cited by 17 publications

References 52 publications

Influence of device geometry on tunneling in theν=52quantum Hall liquid

Influence of device geometry on tunneling in theν=52quantum Hall liquid

Route to thermalization in the α -Fermi–Pasta–Ulam system

Tunneling in fractional quantum Hall line junctions

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