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We study the spectral properties of infinite rectangular quantum graphs in the presence of a magnetic field. We study how these properties are affected when three-dimensionality is considered, in particular, the chaological properties. We then establish the quantization of the Hall transverse conductivity for these systems. This quantization is obtained by relating the transverse conductivity to topological invariants. The different integer values of the Hall conductivity are explicitly computed for an anisotropic diffusion system which leads to fractal phase diagrams.
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The analog of two seminal quantum optics experiments are considered in a condensed matter setting with single electron sources injecting electronic wave packets on edge states coupled through a quantum point contact. When only one electron is injected, the measurement of noise correlations at the output of the quantum point contact corresponds to the Hanbury-Brown and Twiss setup. When two electrons are injected on opposite edges, the equivalent of the Hong-Ou-Mandel collision is achieved, exhibiting a dip as in the coincidence measurements of quantum optics. The Landauer-Buttiker scattering theory is used to first review these phenomena in the integer quantum Hall effect, next, to focus on two more exotic systems: edge states of two dimensional topological insulators, where new physics emerges from time reversal symmetry and three electron collisions can be achieved; and edges states of a hybrid Hall/superconducting device, which allow to perform electron quantum optics experiments with Bogoliubov quasiparticles.
We report on numerical studies into the interplay of disorder and electron-electron interactions within the integer quantum Hall regime, where the presence of a strong magnetic field and two-dimensional confinement of the electronic system profoundly affects thermodynamic and transport properties. We emphasise the behaviour of the electronic compressibility, the local density of states, and the Kubo conductivity. Our treatment of the electron-electron interactions relies on the Hartree-Fock approximation so as to achieve system sizes comparable to experimental situations. Our results clearly exhibit manifestations of various interaction-mediated features, such as non-linear screening, local charging, and g-factor enhancement, implying the inadequacy of independent-particle models for comparison with experimental results.
Electron pairing is a rare phenomenon appearing only in a few unique physical systems; e.g., superconductors and Kondo-correlated quantum dots. Here, we report on an unexpected, but robust, electron pairing in the integer quantum Hall effect (IQHE) regime. The pairing takes place within an interfering edge channel circulating in an electronic Fabry-Perot interferometer at a wide range of bulk filling factors, $2<{ u} _B<5$. The main observations are: (a) High visibility Aharonov-Bohm conductance oscillations with magnetic flux periodicity ${Delta}{phi}={varphi}_0/2=h/2e$ (instead of the ubiquitous $h/e$), with $e$ the electron charge and $h$ the Planck constant; (b) An interfering quasiparticle charge $e ^* {sim} 2e$ - revealed by quantum shot noise measurements; and (c) Full dephasing of the $h/2e$ periodicity by induced dephasing of the adjacent edge channel (while keeping the interfering edge channel intact) : a clear realization of inter-channel entanglement. While this pairing phenomenon clearly results from inter-channel interaction, the exact mechanism that leads to e-e attraction within a single edge channel is not clear.
In recent interference experiments with an electronic Fabry-Perot interferometer (FPI), implemented in the integer quantum Hall effect regime, a flux periodicity of $h/2e$ was observed at bulk fillings $ u_B>2.5$. The halved periodicity was accompanied by an interfering charge $e^*=2e$, determined by shot noise measurements. Here, we present measurements, demonstrating that, counterintuitively, the coherence and the interference periodicity of the interfering chiral edge channel are solely determined by the coherence and the enclosed flux of the adjacent edge channel. Our results elucidate the important role of the latter and suggest that a neutral chiral edge mode plays a crucial role in the pairing phenomenon. Our findings reveal that the observed pairing of electrons is not a curious isolated phenomenon, but one of many manifestations of unexpected edge physics in the quantum Hall effect regime.
Conductivity of Integer Quantum Hall Effect (IQHE) may be expressed as the topological invariant composed of the two - point Green function. Such a topological invariant is known both for the case of homogeneous systems with intrinsic Anomalous Quantum Hall Effect (AQHE) and for the case of IQHE in the inhomogeneous systems. In the latter case we may speak, for example, of the AQHE in the presence of elastic deformations and of the IQHE in presence of magnetic field. The topological invariant for the general case of inhomogeneous systems is expressed through the Wigner transformed Green functions and contains Moyal product. When it is reduced to the expression for the IQHE in the homogeneous systems the Moyal product is reduced to the ordinary one while the Wigner transformed Green function (defined in phase space) is reduced to the Green function in momentum space. Originally the mentioned above topological representation has been derived for the non - interacting systems. We demonstrate that in a wide range of different cases in the presence of interactions the Hall conductivity is given by the same expression, in which the noninteracting two - point Green function is substituted by the complete two - point Green function with the interactions taken into account. Several types of interactions are considered including the contact four - fermion interactions, Yukawa and Coulomb interactions. We present the complete proof of this statement up to the two loops, and argue that the similar result remains to all orders of perturbation theory. It is based on the incorporation of Wigner - Weyl calculus to the perturbation theory. We, therefore, formulate Feynmann rules of diagram technique in terms of the Wigner transformed propagators.
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