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arXiv:cond-mat/9906454 · Physical review. B, Condensed matter

Short-range interactions and scaling near integer quantum Hall transitions

Ziqiang Wang, Matthew P. A. Fisher, S. M. Girvin, J. T. Chalker

Published Mar 15, 2000• 51 citations
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Abstract

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 ${z}_{T}$ 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 ${z}_{T}=2/p.$

DOI
10.1103/physrevb.61.8326

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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 \ensuremathω\ensuremath{\omega}\ensuremathω and temperature T is determined by the scaling variable \ensuremathω/Tp\ensuremath{\omega}{/T}^{p}\ensuremathω/Tp (where p is the exponent for the temperature dependence of the inelastic scattering rate) and not by \ensuremathω/T,\ensuremath{\omega}/T,\ensuremathω/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{z}_{T}zT​ in terms of the scaling dimension \ensuremath−\ensuremathα<0\ensuremath{-}\ensuremath{\alpha}<0\ensuremath−\ensuremathα<0 of the interaction strength and the dynamical exponent z (which has the value z=2),z=2),z=2), obtaining p=1+2\ensuremathα/zp=1+2\ensuremath{\alpha}/zp=1+2\ensuremathα/z and zT=2/p.{z}_{T}=2/p.zT​=2/p.
We study the effects of the electron-electron interaction on the critical properties of the plateau transitions in the integer quantum Hall effect. We find the renormalization group dimension associated with short-range interactions to be $\ensuremath{-}0.66\ifmmode\pm\else\textpm\fi{}0.04$. Thus the noninteracting fixed point (characterized z=2z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2z=2 and $\ensuremath{\nu}\ensuremath{\approx}2.3$) is stable. For the Coulomb interaction, we find the correlation effect is a marginal perturbation at a Hartree-Fock fixed point ( z=1z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1z=1, $\ensuremath{\nu}\ensuremath{\approx}2.3$) by dimension counting. Further calculations are needed to determine its stability upon loop corrections.
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/r1/r1/r interactions, (1) the transition has a dynamical exponent z=1z=1z=1, and (2) all transitions are ``superuniversal'': fractional and integer QH transitions are in the same universality class. With short-range interactions, z=2z=2z=2 and the fate of superuniversality remains unclear.
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)N({E}_{F})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.
As a unified theory of integer and fractional quantum Hall plateau transitions, a nonperturbative theory of the two-parameter scaling renormalization group function is presented. By imposing global symmetries known as ``the law of corresponding states'', we seek a possible form of renormalization group flows. Asking for consistency with the result from weak-localization perturbation theory, such restriction is so intense that we can analytically determine its concrete form. Accordingly, the critical exponent ννν and the irrelevant scaling index yyy are obtained analytically and turn out to be irrational. Their values (ν≈2.1ν\approx 2.1ν≈2.1, y≈0.3y\approx 0.3y≈0.3) agree favorably well with experiments and numerics.
We introduce and analyze a class of model systems to study transitions in the integer quantum Hall effect (IQHE). Even without disorder our model exhibits an IQHE transition as a control parameter is varied. We find that the transition is in the two-dimensional Ising universality class and compute all associated exponents and critical transport properties. The fixed point has time-reversal, particle-hole, and parity invariance. We then consider the effect of quenched disorder on the IQHE transition and find the following. (i) Randomness in the control parameter (which breaks all the above symmetries) translates into bond randomness in the Ising model and is hence marginally irrelevant. The transition may equally well be viewed as a quantum percolation of edge states localized on equipotentials. The absence of random-phase factors for the edge states is responsible for the nongeneric (Ising) critical properties. (ii) For a random magnetic field (which preserves particle-hole symmetry in every realization) the model exhibits an exactly solvable fixed line, described in terms of a product of a Luttinger liquid and an SU(n) spin chain. While exponents vary continuously along the fixed line, the longitudinal conductivity is constant due to a general conformal sum rule for Kac-Moody algebras (derived here), and is computed exactly. We also obtain a closed expression for the extended zero-energy wave function for every realization of disorder and compute its exact multifractal spectrum f(\ensuremath{\alpha}) and the exponents of all participation ratios. One point on the fixed line corresponds to a recently proposed model by Gade and Wegner. (iii) The model in the presence of a random on-site potential scales to a strong disorder regime, which is argued to be described by a symplectic nonlinear-sigma-model fixed point. (iv) We find a plausible global phase diagram in which all forms of disorder are simultaneously considered. In this generic case, the presence of random-phase factors in the edge-state description indicates that the transition is described by a Chalker-Coddington model, with a so far analytically inaccessible fixed point.
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=1z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1z=1, $\ensuremath{\nu}>1$ stable against weak disorder in the position of the critical point, or runaway flow to strong coupling.