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The integer quantum Hall effect features a paradigmatic quantum phase transition. Despite decades of work, experimental, numerical, and analytical studies have yet to agree on a unified understanding of the critical behavior. Based on a numerical Green function approach, we consider the quantum Hall transition in a microscopic model of non-interacting disordered electrons on a simple square lattice. In a strip geometry, topologically induced edge states extend along the system rim and undergo localization-delocalization transitions as function of energy. We investigate the boundary critical behavior in the lowest Landau band and compare it with a recent tight-binding approach to the bulk critical behavior [Phys. Rev. B 99, 121301(R) (2019)] as well as other recent studies of the quantum Hall transition with both open and periodic boundary conditions.
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The integer quantum Hall transition (IQHT) is one of the most mysterious members of the family of Anderson transitions. Since the 1980s, the scaling behavior near the IQHT has been vigorously studied in experiments and numerical simulations. Despite all efforts, it is notoriously difficult to pin down the precise values of critical exponents, which seem to vary with model details and thus challenge the principle of universality. Recently, M. Zirnbauercitep{Zirnbauer2019} [Nucl. Phys. B textbf{941}, 458 (2019)] has conjectured a conformal field theory for the transition, in which linear terms in the beta-functions vanish, leading to a very slow flow in the fixed points vicinity which we term marginal scaling. In this work, we provide numerical evidence for such a scenario by using extensive simulations of various network models of the IQHT at unprecedented length scales. At criticality, we show that the finite-size scaling of the disorder averaged longitudinal Landauer conductance is consistent with its recently predicted fixed-point value and a third-order expansion of RG beta functions. In the future, our numerical findings can be checked with analytical results from the conformal field theory. Away from criticality we describe a mechanism that could account for the emergence of an emph{effective} critical exponents $ u_mathrm{eff}$, which is necessarily dependent on the parameters of the model. We further support this idea by numerical determination of $ u_mathrm{eff}$ in suitably chosen models.
Two-dimensional electron gas in the integer quantum Hall regime is investigated numerically by studying the dynamics of an electron hopping on a square lattice subject to a perpendicular magnetic field and random on-site energy with white noise distribution. Focusing on the lowest Landau band we establish an anti-levitation scenario of the extended states: As either the disorder strength $W$ increases or the magnetic field strength $B$ decreases, the energies of the extended states move below the Landau energies pertaining to a clean system. Moreover, for strong enough disorder, there is a disorder dependent critical magnetic field $B_c(W)$ below which there are no extended states at all. A general phase diagram in the $W-1/B$ plane is suggested with a line separating domains of localized and delocalized states.
Observation of interference in the quantum Hall regime may be hampered by a small edge state velocity due to finite phase coherence time. Therefore designing two quantum point contact (QPCs) interferometers having a high edge state velocity is desirable. Here, we present a new simulation method for realistically modeling edge states near QPCs in the integer quantum Hall effect (IQHE) regime. We calculate the filling fraction in the center of the QPC and the velocity of the edge states, and predict structures with high edge state velocity. The 3D Schrodinger equation is split into 1D and 2D parts. Quasi-1D Schrodinger and Poisson equations are solved self-consistently in the IQHE regime to obtain the potential profile near the edges, and quantum transport is used to solve for the edge state wavefunctions. The velocity of edge states is found to be $left< E right> / B$, where $left< E right>$ is the expectation value of the electric field for the edge state. Anisotropically etched trench gated heterostructures with double sided delta doping have the highest edge state velocity among the structures considered.
We have estimated the critical exponent describing the divergence of the localization length at the metal-quantum spin Hall insulator transition. The critical exponent for the metal-ordinary insulator transition in quantum spin Hall systems is known to be consistent with that of topologically trivial symplectic systems. However, the precise estimation of the critical exponent for the metal-quantum spin Hall insulator transition proved to be problematic because of the existence, in this case, of edge states in the localized phase. We have overcome this difficulty by analyzing the second smallest positive Lyapunov exponent instead of the smallest positive Lyapunov exponent. We find a value for the critical exponent $ u=2.73 pm 0.02$ that is consistent with that for topologically trivial symplectic systems.
Recent high-precision results for the critical exponent of the localization length at the integer quantum Hall (IQH) transition differ considerably between experimental ($ u_text{exp} approx 2.38$) and numerical ($ u_text{CC} approx 2.6$) values obtained in simulations of the Chalker-Coddington (CC) network model. We revisit the arguments leading to the CC model and consider a more general network with geometric (structural) disorder. Numerical simulations of this new model lead to the value $ u approx 2.37$ in very close agreement with experiments. We argue that in a continuum limit the geometrically disordered model maps to the free Dirac fermion coupled to various random potentials (similar to the CC model) but also to quenched two-dimensional quantum gravity. This explains the possible reason for the considerable difference between critical exponents for the CC model and the geometrically disordered model and may shed more light on the analytical theory of the IQH transition. We extend our results to network models in other symmetry classes.
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