Two exact relations between mutlifractal exponents are shown to hold at the critical point of the Anderson localization transition. The first relation implies a symmetry of the multifractal spectrum linking the exponents with indices q<1/2 to those with q>1/2. The second relation connects the wave-function multifractality to that of Wigner delay times in a system with a lead attached.

We investigate numerically the statistics of wave function amplitudes (r) at the integer quantum Hall transition. It is demonstrated that in the limit of a large system size the distribution function of ͉͉ 2 is log-normal, so that the multifractal spectrum f (␣) is exactly parabolic. Our findings lend strong support to a recent conjecture for a critical theory of the quantum Hall transition.In 1980 von Klitzing, Dorda, and Pepper discovered 1 that the Hall conductance xy of a two-dimensional electron gas develops plateaus at values quantized in units of e 2 /h. Twenty years later, the integer quantum Hall effect ͑IQHE͒ still constitutes one of the great challenges of condensedmatter physics. Initially, the effort focused on the physics of the Hall plateaus which is by now fairly well understood. 2,3 Then interest has shifted towards the transition region, where xy crosses over from one plateau to the next. However, here the situation is not as well resolved. It has been understood from early on that this is a second-order phase transition, and the scaling scenario has been confirmed in numerous experiments and computer simulations 4 yielding, e.g., the value Ӎ2.35 for the localization length exponent. In contrast, analytical approaches did not lead to quantitative predictions. The ultimate goal here is to identify the effective low-energy theory of the critical point ͑expected to be a conformal field theory͒ and to calculate critical exponents and other characteristics of the transition region. In particular, independence of the critical theory on microscopic parameters would establish universality of the critical exponents which is a matter of intensive and controversial discussions in the experimental literature.The earliest field-theoretical formulation of the problem was given by Pruisken 5 ͑see Ref. 6 for a more precise, supersymmetric version͒ and has the form of the nonlinear -model with a topological term. Since the latter is invisible in perturbation theory, one has to resort to nonperturbative means in order to address the critical behavior. Pruisken and co-workers were thus led to the dilute instanton gas approximation. 7 However, this approximation can only be justified in the weak-coupling limit, xx /(e 2 /h)ӷ1, and becomes uncontrolled in the critical region xx ϳe 2 /h. For this reason, no quantitative predictions for critical properties have been made within this approach. Another line of efforts was based on a mapping of the low-energy sector of Pruisken's model onto an antiferromagnetic superspin chain. 8 The superspin chain was also obtained by starting from the Chalker-Coddington network model of the IQHE. 9 However, attempts to find an analytical solution of the superspin chain problem remained unsuccessful.Recently, two papers appeared which may signify a breakthrough in the quest for the conformal critical theory of the IQHE. Zirnbauer 10 and, a few months later, Bhaseen et al.,11 proposed that this is a -model with the Wess-Zumino-Novikov-Witten ͑WZNW͒ term ⌫,where g belongs to a certain sy...

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 ∆q governing the scaling of moments |ψ| 2q ∼ L −qd−∆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 ∆2 = −1/4 and ∆3 = −3/4. The multifractality spectrum obtained from numerical simulations is given with a good accuracy by the parabolic approximation ∆q ≃ q(1 − q)/8 but shows detectable deviations. We also study statistics of the two-point conductance g, in particular, the spectrum of exponents Xq characterizing the scaling of the moments g q . Relations between the spectra of critical exponents of wave functions (∆q), conductances (Xq), and Green functions at the localization transition with a critical density of states are discussed.

We present an ultrahigh-precision numerical study of the spectrum of multifractal exponents Deltaq characterizing anomalous scaling of wave function moments |psi|2q at the quantum Hall transition. The result reads Deltaq=2q(1-q)[b0+b1(q-1/2)2+cdots, three dots, centered], with b0=0.1291+/-0.0002 and b1=0.0029+/-0.0003. The central finding is that the spectrum is not exactly parabolic: b1 not equal0. This rules out a class of theories of the Wess-Zumino-Witten type proposed recently as possible conformal field theories of the quantum Hall critical point.

We investigate numerically the quasiparticle density of states ̺(E) for a two-dimensional, disordered superconductor in which both time-reversal and spin-rotation symmetry are broken. As a generic single-particle description of this class of systems (symmetry class D), we use the Cho-Fisher version of the network model. This has three phases: a thermal insulator, a thermal metal, and a quantized thermal Hall conductor. In the thermal metal, we find a logarithmic divergence in ̺(E) as E → 0, as predicted from sigma model calculations. Finite-size effects lead to superimposed oscillations, as expected from random matrix theory. In the thermal insulator and quantized thermal Hall conductor, we find that ̺(E) is finite at E = 0. At the plateau transition between these phases, ̺(E) decreases toward zero as |E| is reduced, in line with the result ̺(E) ∼ |E| ln(1/|E|) derived from calculations for Dirac fermions with random mass.