It is shown that a sufficiently pure mesoscopic metallic ring will exhibit a magnetic phase transition to a low-temperature state with a persistent orbital current in zero external field (to a flux-phase state). The transition temperature and the spontaneous persistent current depend strongly on the geometry of the ring and of the Fermi surface.PACS numbers: 72.10.BgThe conductance across mesoscopic cylinders and rings made of normal metals such as Mg, Cu, or Au was found to be a periodic function of the magnetic flux enclosed by the ring, with periods h/2e and/or h/e. {~4 This implies the existence of persistent currents around rings made of nonsuperconducting metals. 2,5 "" 9 Such currents were detected recently in the magnetization of a collection of isolated mesoscopic rings. ,0The flux O, which drives the persistent current /, is the sum of the externally applied flux <£> e and of the flux 0/ from the persistent current itself, 0=0^4-0/. Most theoretical discussions neglect the second term, 2,6 which is justified for the experimental structures realized so far. However, at a sufficiently low level of elastic defect scattering, at sufficiently low temperature, and at small 0 e , 0/ can become of order <& e or even larger. This raises the possibility of orbital magnetic phase transitions. In this paper we show that in the so-called ballistic regime, a mesoscopic ring will make a transition from a hightemperature state with zero current to a low-temperature state with a persistent current already in the absence of an external field. ] l The magnitude of this spontaneous current and of the transition temperature T c depends strongly on the external geometry and on that of the Fermi surface. By model calculations for a thick torus with R « 400 A, with the Fermi velocity of Cu and with a flat Fermi surface, we find 0 < T c < 70 K. For a spherical Fermi surface with otherwise the same parameters we estimate 7Vs of a few tens of mK. We discuss the reductions of T c by inelastic and by elastic scattering.We consider a metallic torus with radius R and with circular cross section nidi I) 1 , where R and d are on a mesoscopic length scale (10 2 -10 4 lattice constants). Some elastic scattering by surface roughness and by defects within the material is admitted; 2,12 however, we assume l e > 2KR for the elastic mean free path ("ballistic" regime). A magnetic field lies parallel to the axis of symmetry, small enough to leave the radius of the Landau orbit at the Fermi energy very large compared to d [£<200 T (Ref. 8)]. It will then mainly affect the phase of the azimuthal orbitals, by shifting it according to the semiclassical boundary condition dr< p p n (r)p tpn =(r) = §dr^A^(r) depends on r. However, the integral §dr^p n ir)p^n taken inside the torus on circular paths with different r must have the same value nh, if p n ir) is continuous, at 5=0 a...

We extend the multifractal analysis of the statistics of critical wave functions

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We present two novel approaches to establish the local density of states as an order parameter field for the Anderson transition problem. We first demonstrate for 2D quantum Hall systems the validity of conformal scaling relations which are characteristic of order parameter fields. Second we show the equivalence between the critical statistics of eigenvectors of the Hamiltonian and of the transfer matrix, respectively. Based on this equivalence we obtain the order parameter exponent α0 ≈ 3.4 for 3D quantum Hall systems.The absence of diffusion in coherent disordered electron systems is known as Anderson localization [1]. In dimensions d > 2 a disorder induced localizationdelocalization (LD) transition occurs quite generally at some value of the Fermi energy [2]. In d = 2 all states are localized unless a certain amount of spin-orbit scattering [3] or a strong magnetic field is present [4,5]. The LD transitions of independent (spinless) 2D electrons subject to a strong perpendicular magnetic field are located at the Landau energies. These transitions are generally believed to be responsible for the integer quantum Hall effect [6,5].In general, LD transitions are characterized by the critical exponent ν of the localization length [7] and by the multifractal f (α) spectrum of the local amplitudes of critical eigenstates [8,9]. The f (α) spectrum describes the statistics and scaling behavior of the local density of states (LDOS). Although the average density of states does not reflect the LD transition, the typical value (i.e. geometrically averaged) of the LDOS does: it vanishes with an exponent β typ = (α 0 − d)ν on approaching the LD transition point, where α 0 > d is the maximum position of f (α). It is thus tempting to interpret the LDOS as an order parameter field of the LD problem [8] (see also [10,11]). Our aim is to support this interpretation by establishing two characteristic features of order parameter fields for the LDOS: First, scaling exponents of the order parameter field are related to the critical exponents of the corresponding spatial correlation functions. These correlation functions show conformal invariance. Second, the scaling exponents are universal in the sense of (one-parameter) scaling theory, i.e. any local quantity containing contributions from the relevant scaling field shows asymptotically the same spectrum of scaling exponents.In this article we demonstrate that the f (α) spectrum of critical eigenstates is related to correlation functions in different geometries by conformal invariance. We derive the conformal mapping relations appropriate for a multifractal situation and check them numerically in 2D quantum Hall systems (QHS). Furthermore, we have calculated numerically f (α) for the local components of transfer matrix eigenvectors in 2D QHS and show that it coincides with f (α) of the Hamiltonian eigenstates. Thus, these two local quantities share the same spectrum of scaling exponents although their microscopic construction is quite different. Our findings support the identificati...

Scattering theoretical network models for general coherent wave mechanical systems with quenched disorder are investigated. We focus on universality classes for two dimensional systems with no preferred orientation: Systems of spinless waves undergoing scattering events with broken or unbroken time reversal symmetry and systems of spin 1/2 waves with time reversal symmetric scattering. The phase diagram in the parameter space of scattering strengths is determined. The model breaking time reversal symmetry contains the critical point of quantum Hall systems but, like the model with unbroken time reversal symmetry, only one attractive fixed point, namely that of strong localization. Multifractal exponents and quasi-one-dimensional localization lengths are calculated numerically and found to be related by conformal invariance. Furthermore, they agree quantitatively with theoretical predictions. For non-vanishing spin scattering strength the spin 1/2 systems show localization-delocalization transitions. Large intensity fluctuations and spatial localization are fascinating universal features in any coherent wave mechanical system subject to quenched disorder. A modeling that covers essential symmetries and characteristic length scales, but does not rely on particular dispersion relations and specific details is highly desirable in various fields of theoretical physics, e.g. in optics, mesoscopic electronics, and quantum chaos (see e.g. In contrast to the CC-model here we deal with systems in the absence of fields that would introduce a handedness (chirality). However we do allow for the breaking of time reversal symmetry. In particular, we address three classes of non-chiral (NC) 2D NWMs with respect to their quantum localization properties. The first model describes time reversal symmetric scattering and is denoted as O2NC-model, where 'O' stands for 'orthogonal' (a corresponding Hamiltonian can be diagonalized by orthogonal matrices). Second, a similar model with broken time reversal symmetry ('U' for 'unitary') is introduced denoted as U2NC-model. It describes e.g. the motion of mesoscopic (spinless) electrons in the presence of random magnetic fields with zero mean and disorder potentials. The third model deals with time reversal symmetric scattering also in spin degrees of freedom. It is denoted as S2NC-model ('S' for 'symplectic') and a detailed analysis for this model is given in [10]. Here we focus on the U/O2NC-model, discuss their construction and the phase diagram in detail, and present a quantitative check of analytical results for quasi-1D localization lengths, multifractal exponents and conformal invariance in the weak localization regime [9].Quite generally, a NWM can be constructed as follows. Take a regular network of N sites and N bonds. Each bond α carries propagating wave modes (n + α incoming modes and n − α outgoing modes) represented by complex amplitudes, ψ n ± α . On the sites unitary S-matrices map incoming to outgoing amplitudes. The elements of each S-matrix are (in general) random quantit...