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Register a new userGeometry of random potentials: Induction of 2D gravity in Quantum Hall plateau transitions
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In the context of the Integer Quantum Hall plateau transitions, we formulate a specific map from random landscape potentials onto 2D discrete random surfaces. Critical points of the potential, namely maxima, minima and saddle points uniquely define a discrete surface $S$ and its dual $S^*$ made of quadrangular and $n-$gonal faces, respectively, thereby linking the geometry of the potential with the geometry of discrete surfaces. The map is parameter-dependent on the Fermi level. Edge states of Fermi lakes moving along equipotential contours between neighbour saddle points form a network of scatterings, which define the geometric basis, in the fermionic model, for the plateau transitions. The replacement probability characterizing the network model with geometric disorder recently proposed by Gruzberg, Klumper, Nuding and Sedrakyan, is physically interpreted within the current framework as a parameter connected with the Fermi level.
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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.
This paper investigates quantum diffusion of matter waves in two-dimensional random potentials, focussing on expanding Bose-Einstein condensates in spatially correlated optical speckle potentials. Special care is taken to describe the effect of dephasing, finite system size, and an initial momentum distribution. We derive general expressions for the interference-renormalized diffusion constant, the disorder-averaged probability density distribution, the variance of the expanding atomic cloud, and the localized fraction of atoms. These quantities are studied in detail for the special case of an inverted-parabola momentum distribution as obtained from an expanding condensate in the Thomas-Fermi regime. Lastly, we derive quantitative criteria for the unambiguous observation of localization effects in a possible 2D experiment.
Effects of backward scattering between fractional quantum Hall (FQH) edge modes are studied. Based on the edge-state picture for hierarchical FQH liquids, we discuss the possibility of the transitions between different plateaux of the tunneling conductance $G$. We find a selection rule for the sequence which begins with a conductance $G=m/(mppm 1)$ ($m$: integer, $p$: even integer) in units of $e^2/h$. The shot-noise spectrum as well as the scaling behavior of the tunneling current is calculated explicitly.
We show how to model the transition between distinct quantum Hall plateaus in terms of D-branes in string theory. A low energy theory of 2+1 dimensional fermions is obtained by considering the D3-D7 system, and the plateau transition corresponds to moving the branes through one another. We study the transition at strong coupling using gauge/gravity duality and the probe approximation. Strong coupling leads to a novel kind of plateau transition: at low temperatures the transition remains discontinuous due to the effects of dynamical symmetry breaking and mass generation, and at high temperatures is only partially smoothed out.
We study a c=-2 conformal field theory coupled to two-dimensional quantum gravity by means of dynamical triangulations. We define the geodesic distance r on the triangulated surface with N triangles, and show that dim[r^{d_H}]= dim[N], where the fractal dimension d_H = 3.58 +/- 0.04. This result lends support to the conjecture d_H = -2alpha_1/alpha_{-1}, where alpha_{-n} is the gravitational dressing exponent of a spin-less primary field of conformal weight (n+1,n+1), and it disfavors the alternative prediction d_H = -2/gamma_{str}. On the other hand, we find dim[l] = dim[r^2] with good accuracy, where l is the length of one of the boundaries of a circle with (geodesic) radius r, i.e. the length l has an anomalous dimension relative to the area of the surface. It is further shown that the spectral dimension d_s = 1.980 +/- 0.014 for the ensemble of (triangulated) manifolds used. The results are derived using finite size scaling and a very efficient recursive sampling technique known previously to work well for c=-2.
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