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Register a new userInterplay of Short-Range Interactions and Quantum Interference Near the Integer Quantum Hall Transition
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Short-range electron-electron interactions are incorporated into the network model of the integer quantum Hall effect. In the presence of interactions, the electrons, propagating along one link, experience exchange scattering off the Friedel oscillations of the density matrix of electrons on the neighboring links. As a result, the energy dependence of the transmission, ${cal T}(epsilon)$, of the node, connecting the two links, develops an anomaly at the Fermi level, $epsilon=epsilon_F$. We show that this interaction-induced anomaly in ${cal T}(epsilon)$ translates into the anomalous behavior of the Hall conductivity, $sigma_{xy}( u)$, where $ u$ is the filling factor (we assume that the electrons are {em spinless}). At low temperatures, $T to 0$, the evolution of the quantized $sigma_{xy}$ with decreasing $ u$ proceeds as $1to 2 to 0$, in apparent violation of the semicircle relation. The anomaly in ${cal T}(epsilon)$ also affects the temperature dependence of the peak in the diagonal conductivity, $sigma_{xx}( u, T)$. In particular, unlike the case of noninteracting electrons,the maximum value of $sigma_{xx}$ stays at $sigma_{xx} = 0.5$ within a wide temperature interval.
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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.
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.
We present a calculation for the second moment of the local density of states in a model of a two-dimensional quantum dot array near the quantum Hall transition. The quantum dot array model is a realistic adaptation of the lattice model for the quantum Hall transition in the two-dimensional electron gas in an external magnetic field proposed by Ludwig, Fisher, Shankar and Grinstein. We make use of a Dirac fermion representation for the Green functions in the presence of fluctuations for the quantum dot energy levels. A saddle-point approximation yields non-perturbative results for the first and second moments of the local density of states, showing interesting fluctuation behaviour near the quantum Hall transition. To our knowledge we discuss here one of the first analytic characterizations of chaotic behaviour for a two-dimensional mesoscopic structure. The connection with possible experimental investigations of the local density of states in the quantum dot array structures (by means of NMR Knight-shift or single-electron-tunneling techniques) and our work is also established.
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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