We study the influence of short-range electron-electron interactions on scaling behavior near the integer quantum Hall plateau transitions. Short-range interactions are known to be irrelevant at the renormalization group fixed point which represents the transition in the noninteracting system. We find, nevertheless, that transport properties change discontinuously when interactions are introduced. Most importantly, in the thermodynamic limit the conductivity at finite temperature is zero without interactions, but nonzero in the presence of arbitrarily weak interactions. In addition, scaling as a function of frequency $\ensuremath{\omega}$ and temperature T is determined by the scaling variable $\ensuremath{\omega}{/T}^{p}$ (where p is the exponent for the temperature dependence of the inelastic scattering rate) and not by $\ensuremath{\omega}/T,$ as it would be at a conventional quantum phase transition described by an interacting fixed point. We express the inelastic exponent p and the thermal exponent
zT in terms of the scaling dimension $\ensuremath{-}\ensuremath{\alpha}<0$ of the interaction strength and the dynamical exponent z (which has the value
z=2), obtaining $p=1+2\ensuremath{\alpha}/z$ and
zT=2/p.We study the consequences of the random mass, random scalar potential, and random vector potential on the line of fixed points between integer and/or fractional quantum Hall states and an insulator. This line of fixed points was first identified in a clean Dirac fermion system with both Chern-Simon coupling and Coulomb interaction [Phys. Rev. Lett. 80, 5409 (1998)]. By performing a renormalization-group analysis in
1/N (N is the number of species of Dirac fermions) and the variances of three disorders ${\ensuremath{\Delta}}_{M},{\ensuremath{\Delta}}_{V},{\ensuremath{\Delta}}_{A},$ we find that ${\ensuremath{\Delta}}_{M}$ is irrelevant along this line, and both ${\ensuremath{\Delta}}_{A}$ and ${\ensuremath{\Delta}}_{V}$ are marginal. With the presence of all three disorders, the pure fixed line is unstable. Setting Chern-Simon interaction to zero, we find one nontrivial line of fixed points in the $({\ensuremath{\Delta}}_{A},w)$ plane with dynamic exponent
z=1 and continuously changing \ensuremath{\nu}; it is stable against small $({\ensuremath{\Delta}}_{M},{\ensuremath{\Delta}}_{V})$ in a small range of the line
1<w<1.31, therefore it may be relevant to integer quantum Hall transition. Setting ${\ensuremath{\Delta}}_{M}=0,$ we find a fixed plane with
z=1, the part of this plane with $\ensuremath{\nu}>1$ is stable against small ${\ensuremath{\Delta}}_{M},$ therefore it may be relevant to fractional quantum Hall transition. Although we do not find a generic fixed point with all the couplings nonvanishing, we prove that the theory is renormalizable to the order ${(1/N)}^{2},(1/N)\ensuremath{\Delta},{\ensuremath{\Delta}}^{2},$ and we explore the interesting processes which describe the interferences between the Chern-Simon interaction, the Coulomb interaction, and the three kinds of disorders.
lattice model exhibiting the integer quantum Hall effect. We consider an interacting version of the model, and prove that for short-range interactions, smaller than the bandwidth, the Hall conductivity is quantized, for all the values of the parameters outside two critical curves, across which the model undergoes a 'topological' phase transition: the Hall coefficient remains integer and constant as long as we continuously deform the parameters without crossing the curves; when this happens, the Hall coefficient jumps abruptly to a different integer. Previous works were limited to the perturbative regime, in which the interaction is much smaller than the bare gap, so they were restricted to regions far from the critical lines. The non-renormalization of the Hall conductivity arises as a consequence of lattice conservation laws and of the regularity properties of the current-current correlations. Our method provides a full construction of the critical curves, which are modified ('dressed') by the electron-electron interaction. The shift of the transition curves manifests itself via apparent infrared divergences in the naive perturbative series, which we resolve via renormalization group methods.
We study the role of electron-electron interactions near integer and abelian fractional quantum Hall (QH) transitions using composite fermion (CF) representations. Interaction effects are encapsulated in CF theories as gauge fluctuations. Without gauge fluctuations, the CF system realizes a `dual' representation of the noninteracting QH transition. With gauge fluctuations, the system is governed by a gauged nonlinear sigma model (NLSM) with a $\ensuremath{\theta}\phantom{\rule{0.28em}{0ex}}\mathrm{term}$. While the transition is described by a strong-coupling fixed point of the NLSM, we are nevertheless able to deduce two of its properties. With
1/r interactions, (1) the transition has a dynamical exponent
z=1, and (2) all transitions are ``superuniversal'': fractional and integer QH transitions are in the same universality class. With short-range interactions,
z=2 and the fate of superuniversality remains unclear.
We study the consequences of long-range Coulomb interactions at the critical points between integer/fractional quantum Hall states and an insulator. We use low energy theories for such transitions in anyon gases in the presence of an external periodic potential. We find that Coulomb interactions are marginally irrelevant for the integer quantum Hall case. For the fractional case, depending upon the anyon statistics parameter, we find behavior similar to the integer case, or flow to a novel line of fixed points with exponents
z=1, $\ensuremath{\nu}>1$ stable against weak disorder in the position of the critical point, or runaway flow to strong coupling.
We study the effects of electron-electron interactions in disordered metals in and close to two dimensions (2D). We consider physical situations in which localization effects are suppressed. The field-theoretical renormalization-group (RG) calculation performed recently by Finkelstein is interpreted and rederived in terms of perturbative results. Surprisingly, except for the density of states, the scaling behavior is independent of the interaction range. We further extend the model to several new universality classes. In the presence of a strong magnetic field the metal is unstable in 2D and undergoes a metal-insulator transition in $d=2+\ensuremath{\epsilon}$. The conductivity exponent, defined by $\ensuremath{\sigma}\ensuremath{\sim}{(n\ensuremath{-}{n}_{c})}^{\ensuremath{\mu}}$, is universal with $\ensuremath{\mu}=1+O(\ensuremath{\epsilon})$ but
N(EF) depends not only on the range of the interaction but also on its strength for short-ranged interactions. In 2D the conductivity has a universal temperature dependence [$\ensuremath{\delta}\ensuremath{\sigma}(T)={\ensuremath{\sigma}}_{N}(2\ensuremath{-}2\mathrm{ln}2)\mathrm{ln}(T\ensuremath{\tau})$, ${\ensuremath{\sigma}}_{N}=\frac{{e}^{2}}{2{\ensuremath{\pi}}^{2}\ensuremath{\hbar}}$] if the interaction is Coulombic. If magnetic impurities (or strong spin-orbit scattering with a weak magnetic field) are present instead, the noninteracting fixed point is stable for short-ranged interactions ($\ensuremath{\mu}=\frac{1}{2}$). For the Coulomb interaction the interaction is relevant and drives a metal-insulator transition in $d=2+\ensuremath{\epsilon}$ with universal critical properties ($\ensuremath{\mu}=1$). In 2D the conductivity also has a universal temperature dependence [$\ensuremath{\delta}\ensuremath{\sigma}(T)={\ensuremath{\sigma}}_{N}\mathrm{ln}(T\ensuremath{\tau})$]. We also discuss the behavior of the dielectric constant on the insulator side and the frequency (temperature) dependence of the conductivity at criticality. Remarks are made on the relationship of the above to experiments.