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arXiv:cond-mat/9902231 · Physical Review Letters

Universal Scaling, Beta Function, and Metal-Insulator Transitions

D. N. Sheng, Zheng-Yu Weng

Published Jul 5, 1999• 7 citations
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Abstract

We demonstrate a universal scaling form of longitudinal resistance in the quantum critical region of metal-insulator transitions. This is based on numerical results of three-dimensional Anderson transitions (with and without magnetic field), two-dimensional quantum Hall plateau to insulator transition, as well as experimental data of the recently discovered two-dimensional metal-insulator transition. The associated reflection symmetry and a peculiar logarithmic form of the beta function exist over a wide range in which the resistance can change by more than 1 order of magnitude. Interesting implications for the two-dimensional metal-insulator transition are discussed.

DOI
10.1103/physrevlett.83.144

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We demonstrate a universal scaling form of longitudinal resistance in the quantum critical region of metal-insulator transitions. This is based on numerical results of three-dimensional Anderson transitions (with and without magnetic field), two-dimensional quantum Hall plateau to insulator transition, as well as experimental data of the recently discovered two-dimensional metal-insulator transition. The associated reflection symmetry and a peculiar logarithmic form of the beta function exist over a wide range in which the resistance can change by more than 1 order of magnitude. Interesting implications for the two-dimensional metal-insulator transition are discussed.
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We show that the quantum critical point associated with the metal-insulator transition phenomenon in two dimensions controls an extended critical region encompassing not only the usual quantum critical sector but also a range of the low-temperature insulator region. The extended range of criticality permits a unified analysis of data from the insulating region and quantum critical sector, allowing us to determine both the dynamical critical exponent zzz and the correlation length critical exponent $\ensuremath{\nu}$ from published data from a single experiment in the insulator critical region. We show that the critical exponents determined from the insulator sector consistently describe the temperature dependence of the resistance data from the same experiment in the quantum critical sector. This provides evidence for the presence of a quantum critical point in these systems.
Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $\ensuremath{\nu}$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulator) or nontrivial (a Z2{\mathbb{Z}}_{2}Z2​ insulator supporting a quantum spin Hall effect). This is not true of the boundary multifractal scaling exponents, which we show (numerically) to depend on whether the insulating state is topologically trivial or not.
The resistivities of the dilute, strongly interacting two-dimensional electron systems in the insulating phase of a silicon MOSFET are the same for unpolarized electrons in the absence of magnetic field and for electrons that are fully spin polarized by the presence of an in-plane magnetic field. In both cases the resistivity obeys Efros-Shklovskii variable range hopping $\ensuremath{\rho}(T)={\ensuremath{\rho}}_{0}\text{exp}[{({T}_{ES}/T)}^{1/2}]$, with TES{T}_{ES}TES​ and $1/{\ensuremath{\rho}}_{0}$ mapping onto each other if one applies a shift of the critical density nc{n}_{\text{c}}nc​ reported earlier. With and without magnetic field, the parameters TES{T}_{ES}TES​ and $1/{\ensuremath{\rho}}_{0}={\ensuremath{\sigma}}_{0}$ exhibit scaling consistent with critical behavior approaching a metal-insulator transition.