Scaling to Non-Scaling Transition of Quantum Hall Systems

J. T. Chalker, Matthew P. A. Fisher, S. M. Girvin +1 more

Romain Vasseur

The critical behavior of quantum Hall transitions in two-dimensional disordered electronic systems can be described by a class of non-unitary conformal field theories with logarithmic correlations. Using the replica trick and the underlying symmetries of these quantum critical points, the author shows how to construct non-perturbatively disorder-averaged observables in terms of Green's functions that scale logarithmically at criticality.

C.A. Lütken, G.G. Ross

M. A. Baranov, I. S. Burmistrov, Adrianus M. M. Pruisken

We consider the Finkelstein action describing a system of spin-polarized or spinless electrons in $2+2\ensuremath{\epsilon}$ dimensions, in the presence of disorder as well as the Coulomb interactions. We extend the renormalization-group analysis of our previous work and evaluate the metal-insulator transition of the electron gas to second order in an $\ensuremath{\epsilon}$ expansion. We obtain the complete scaling behavior of physical observables like the conductivity and the specific heat with varying frequency, temperature, and/or electron density. We extend the results for the interacting electron gas in $2+2\ensuremath{\epsilon}$ dimensions to include the quantum critical behavior of the plateau transitions in the quantum Hall regime. Although these transitions have a very different microscopic origin and are controlled by a topological term in the action $(\ensuremath{\theta}$ term), the quantum critical behavior is in many ways the same in both cases. We show that the two independent critical exponents of the quantum Hall plateau transitions, previously denoted as $\ensuremath{\nu}$ and p, control not only the scaling behavior of the conductances ${\ensuremath{\sigma}}_{\mathrm{xx}}$ and ${\ensuremath{\sigma}}_{\mathrm{xy}}$ at finite temperatures T, but also the non-Fermi-liquid behavior of the specific heat ${(c}_{v}\ensuremath{\propto}{T}^{p}).$ To extract the numerical values of $\ensuremath{\nu}$ and p it is necessary to extend the experiments on transport to include the specific heat of the electron gas.

H. P. Wei, Daniel C. Tsui, Adrianus M. M. Pruisken

Wellington da Cruz

We consider the quantum phase transitions of fractons in correspondence with the quantum phase transitions of the fractional quantum Hall effect-FQHE. We have that the Hall states can be modelled by fractons, known as charge-flux systems which satisfy a fractal distribution function associated with a fractal von Neumann entropy. In our formulation, the universality classes of the fractional quantum Hall transitions, are considered as fractal sets of dual topological quantum numbers filling factors labelled by the Hausdorff dimension $h$ ($1 < h < 2$) of the quantum paths of fractons. In this way we have defined, associated to these universality classes, a scaling exponent as $κ=1/h$, such that when $h$ runs into its interval of definition, we obtain $ 1 \gtrsim κ\gtrsim 0.5 $. The behavior of this scaling exponent, topological in character, is in agreement with some experimental values claimed in the literature. Thus, according to our approach we have a fractal scaling theory for the FQHE which distinguishes diverse universality classes for the fractional quantum Hall transitions.