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Register a new userBoundary multifractality at the integer quantum Hall plateau transition: implications for the critical theory
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We study multifractal spectra of critical wave functions at the integer quantum Hall plateau transition using the Chalker-Coddington network model. Our numerical results provide important new constraints which any critical theory for the transition will have to satisfy. We find a non-parabolic multifractal spectrum and we further determine the ratio of boundary to bulk multifractal exponents. Our results rule out an exactly parabolic spectrum that has been the centerpiece in a number of proposals for critical field theories of the transition. In addition, we demonstrate analytically exact parabolicity of related boundary spectra in the 2D chiral orthogonal `Gade-Wegner symmetry class.
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Statistical properties of critical wave functions at the spin quantum Hall transition are studied both numerically and analytically (via mapping onto the classical percolation). It is shown that the index $eta$ characterizing the decay of wave function correlations is equal to 1/4, at variance with the $r^{-1/2}$ decay of the diffusion propagator. The multifractality spectra of eigenfunctions and of two-point conductances are found to be close-to-parabolic, $Delta_qsimeq q(1-q)/8$ and $X_qsimeq q(3-q)/4$.
We present an ultra-high-precision numerical study of the spectrum of multifractal exponents $Delta_q$ characterizing anomalous scaling of wave function moments $<|psi|^{2q}>$ at the quantum Hall transition. The result reads $Delta_q = 2q(1-q)[b_0 + b_1(q-1/2)^2 + ...]$, with $b_0 = 0.1291pm 0.0002$ and $b_1 = 0.0029pm 0.0003$. The central finding is that the spectrum is not exactly parabolic, $b_1 e 0$. This rules out a class of theories of Wess-Zumino-Witten type proposed recently as possible conformal field theories of the quantum Hall critical point.
The statistical properties of wave functions at the critical point of the spin quantum Hall transition are studied. The main emphasis is put onto determination of the spectrum of multifractal exponents $Delta_q$ governing the scaling of moments $<|psi|^{2q}>sim L^{-qd-Delta_q}$ with the system size $L$ and the spatial decay of wave function correlations. Two- and three-point correlation functions are calculated analytically by means of mapping onto the classical percolation, yielding the values $Delta_2=-1/4$ and $Delta_3=-3/4$. The multifractality spectrum obtained from numerical simulations is given with a good accuracy by the parabolic approximation $Delta_qsimeq q(1-q)/8$ but shows detectable deviations. We also study statistics of the two-point conductance $g$, in particular, the spectrum of exponents $X_q$ characterizing the scaling of the moments $<g^q >$. Relations between the spectra of critical exponents of wave functions ($Delta_q$), conductances ($X_q$), and Green functions at the localization transition with a critical density of states are discussed.
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.
The temperature dependence of the magneto-conductivity in graphene shows that the widths of the longitudinal conductivity peaks, for the N=1 Landau level of electrons and holes, display a power-law behavior following $Delta u propto T^{kappa}$ with a scaling exponent $kappa = 0.37pm0.05$. Similarly the maximum derivative of the quantum Hall plateau transitions $(dsigma_{xy}/d u)^{max}$ scales as $T^{-kappa}$ with a scaling exponent $kappa = 0.41pm0.04$ for both the first and second electron and hole Landau level. These results confirm the universality of a critical scaling exponent. In the zeroth Landau level, however, the width and derivative are essentially temperature independent, which we explain by a temperature independent intrinsic length that obscures the expected universal scaling behavior of the zeroth Landau level.
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