Nonequilibrium superfield and lattice Weyl transform transport approach to quantum Hall effect

Abstract

Using Buot's superfield and lattice Weyl transform nonequilibirum quantum transport formalism, we derive the topological Chern number of the integer quantum Hall effect in electrical conductivity. The method is naturally simple and direct, and employs neither the conventional use of Kubo formula nor the retarded Green's function in linear response theory. We have identified the topological invariant in ( p, q; E, t)-phase space nonequilibrium quantum transport equation, generally not to first-oder in electric field but to first-order in the gradient expansion. We have also derive the Kubo current-current correlation for the Hall current as a by-product of our new approach. The Berry curvatur related to orbital magnetic moment is also calculated.

On Fractional Quantum Hall Effect (FQHE): A Chern-Simons and nonequilibrium quantum transport Weyl transform approach

“…In the present paper, we show that the idea of flux attachment is inherent in fully occupied as well as in partially occupied Landau levels, which results in scaling factor closely related to the scaling k-factor in Chern-Simons gauge theory. This is shown to be crucial in giving a simple explanation of fractional quantum Hall effect (FQHE), using nonequilibrium quantum transport in the lattice Weyl transform formalism [5,7] used in previous papers [1].…”

“…The following transport analysis give a simple account of the experimental results [23,24] with the fully occupied lowest Landau level as a reference point, i.e., the point v = 1 moving towards successive v's < 1 with increasing magnetic fields, or moving backward of successive v's > 1 with decreasing magnetic fields. In particular, we will focus on v = 1 3 or k = 3 as a particular case since this looks like a very defined state in the experiments, although the following analysis holds for any values of k.…”

“…In previous papers [1][2][3][4], we make use of the gapped energy-band structure of solids under external electric field to derive the integer quantum Hall effect (IQHE) of Chern insulator. We employ the real-time superfield and lattice Weyl transform nonequilibrium Green's function (SFLWT-NEGF) [5] quantum transport formalism [6,7] to the first-order gradient expansion to derive the topological Chern number of the IQHE for two-dimensional systems, as an integral multiple of quantum conductance, also known as the minimal contact conductance in mesoscopic physics [5].…”

“…In Ref. [1], we have shown that the (p.q; E.t) phase-space is renormalized to that of K, E phase-space, where in the absence of magnetic fields,…”

On Hofstadter butterfly spectrum: Chern-Simons theory, subband gap mapping, IQHE and FQHE labelling

“…One most notable characteristics of the Hofstadter butterfly is the reconstruction of the topmost subband as well as that of the lowest subband. Although the left and right portion of the spectrum can be characterize as regions of IQHE regions [6,7,8], the top and bottom regions of the spectrum can be identified with the phenomenon of FQHE, since obviously this involves the reconstruction of a single subband [3] at high magnetic fields . This reconstruction is evidenced by the measurements [9,10,11] and Chern-Simons theoretical explanation of the FQHE given by Buot [3].…”