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In a recent paper by Neupert, Santos, Chamon, and Mudry [Phys. Rev. B 86, 165133 (2012)] it is claimed that there is an elementary formula for the Hall conductivity of fractional Chern insulators. We show that the proposed formula cannot generally be correct, and we suggest one possible source of the error. Our reasoning can be generalized to show no quantity (such as Hall conductivity) expected to be constant throughout an entire phase of matter can possibly be given as the expectation of any time independent short ranged operator.
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For the fractional quantum Hall states on a finite disc, we study the thermoelectric transport properties under the influence of an edge and its reconstruction. In a recent study on a torus [Phys. Rev. B 101, 241101 (2020)], Sheng and Fu found a universal non-Fermi liquid power-law scaling of the thermoelectric conductivity $alpha_{xy} propto T^{eta}$ for the gapless composite Fermi-liquid state. The exponent $eta sim 0.5$ appears an independence of the filling factors and the details of the interactions. In the presence of an edge, we find the properties of the edge spectrum dominants the low-temperature behaviors and breaks the universal scaling law of the thermoelectric conductivity. In order to consider individually the effects of the edge states, the entanglement spectrum in real space is employed and tuned by varying the area of subsystem. In non-Abelian Moore-Read state, the Majorana neutral edge mode is found to have more significant effect than that of the charge mode in the low temperature.
It is known that the Shubnikov--de Haas oscillations can be observed in the Hall resistivity, although their amplitude is much weaker than the amplitude of the diagonal resistivity oscillations. Employing a model of two-dimensional massive Dirac fermions that exhibits anomalous Hall effect, we demonstrate that the amplitude of the Shubnikov--de Haas oscillations of the anomalous Hall conductivity is the same as that of the diagonal conductivity. We argue that the oscillations of the anomalous Hall conductivity can be observed by studying the valley Hall effect in graphene superlattices and the spin Hall effect in the low-buckled Dirac materials.
In a recent Comment, Kolf et al. (cond-mat/0503669) state that our analysis of the Fano resonance for Anderson impurity systems [Luo et al., Phys. Rev. Lett 92, 256602 (2004)] is incorrect. Here we want to point out that their comments are not based on firm physical results and their criticisms are unjustified and invalid.
Applying a unified approach, we study integer quantum Hall effect (IQHE) and fractional quantum Hall effect (FQHE) in the Hofstadter model with short range interaction between fermions. An effective field, that takes into account the interaction, is determined by both the amplitude and phase. Its amplitude is proportional to the interaction strength, the phase corresponds to the minimum energy. In fact the problem is reduced to the Harper equation with two different scales: the first is a magnetic scale (cell size corresponding to a unit quantum magnetic flux), the second scale (determines the inhomogeneity of the effective field) forms the steady fine structure of the Hofstadter spectrum and leads to the realization of fractional quantum Hall states. In a sample of finite sizes with open boundary conditions, the fine structure of the Hofstadter spectrum also includes the fine structure of the edge chiral modes. The subbands in a fine structure of the Hofstadter band (HB) are separated extremely small quasigaps. The Chern number of a topological HB is conserved during the formation of its fine structure. Edge modes are formed into HB, they connect the nearest-neighbor subbands and determine the fractional conductance for the fractional filling at the Fermi energies corresponding to these quasigaps.
We report a detailed characterization of quantum Hall effect (QHE) influence on the linear and non-linear resistivity tensor in FISDW phases of the organic conductor (TMTSF)2PF6. We show that the behavior at low electric fields, observed for nominally pure single crystals with different values of the resistivity ratio, is fully consistent with a theoretical model, which takes QHE nature of FISDW and residual quasi-particle density associated with different crystal imperfection levels into account. The non-linearity in longitudinal and diagonal resistivity tensor components observed at large electric fields reconciles preceding contradictory results. Our theoretical model offers a qualitatively good explanation of the observed features if a sliding of the density wave with the concomitant destruction of QHE, switched on above a finite electric field, is taken into account.
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