Quantum Hall Transition Research Articles

Wavefunction statistics and multifractality at the spin quantum Hall transition

The statistical properties of wave functions at the critical point of the spin quantum Hall transition are studied. The main emphasis is put onto determination of the spectrum of multifractal exponents $\Delta_q$ governing the scaling of moments $<|\psi|^{2q}>\sim L^{-qd-\Delta_q}$ with the system size $L$ and the spatial decay of wave function correlations. Two- and three-point correlation functions are calculated analytically by means of mapping onto the classical percolation, yielding the values $\Delta_2=-1/4$ and $\Delta_3=-3/4$. The multifractality spectrum obtained from numerical simulations is given with a good accuracy by the parabolic approximation $\Delta_q\simeq q(1-q)/8$ but shows detectable deviations. We also study statistics of the two-point conductance $g$, in particular, the spectrum of exponents $X_q$ characterizing the scaling of the moments $<g^q >$. Relations between the spectra of critical exponents of wave functions ($\Delta_q$), conductances ($X_q$), and Green functions at the localization transition with a critical density of states are discussed.

Renormalization group approach to the energy level statistics at the integer quantum Hall transition

We apply the real-space renormalization group (RG) approach to investigate the energy level statistics at the integer quantum Hall (QH) transition. Within the RG approach the macroscopic array of saddle points of the Chalker–Coddington network is replaced by a fragment consisting of only five saddle points. Previously, we have demonstrated that the RG approach reproduces the distribution of the conductance at the transition, P( G), with very high accuracy. To assess the level statistics we analyze the phases of the transmission coefficients of the saddle points. We find that, at the transition, the nearest-neighbor energy level spacing distribution (LSD) exhibits well-pronounced level repulsion. We emphasize that a metal-like LSD emerges when the fixed point distribution P c of G is used. Studying the change of the LSD around the QH transition we observe scaling behavior. Using a one-parameter finite-size scaling analysis we are able to extract a critical exponent ν=2.38±0.04 of the localization length.

Renormalization group approach to energy level statistics at the integer quantum Hall transition

We extend the real-space renormalization group (RG) approach to the study of the energy level statistics at the integer quantum Hall (QH) transition. Previously it was demonstrated that the RG approach reproduces the critical distribution of the {\em power} transmission coefficients, i.e., two-terminal conductances, $P_{\text c}(G)$, with very high accuracy. The RG flow of $P(G)$ at energies away from the transition yielded the value of the critical exponent, $\nu$, that agreed with most accurate large-size lattice simulations. To obtain the information about the level statistics from the RG approach, we analyze the evolution of the distribution of {\em phases} of the {\em amplitude} transmission coefficient upon a step of the RG transformation. From the fixed point of this transformation we extract the critical level spacing distribution (LSD). This distribution is close, but distinctively different from the earlier large-scale simulations. We find that away from the transition the LSD crosses over towards the Poisson distribution. Studying the change of the LSD around the QH transition, we check that it indeed obeys scaling behavior. This enables us to use the alternative approach to extracting the critical exponent, based on the LSD, and to find $\nu=2.37\pm0.02$ very close to the value established in the literature. This provides additional evidence for the surprising fact that a small RG unit, containing only five nodes, accurately captures most of the correlations responsible for the localization-delocalization transition.

The Hausdorff dimension of fractal sets and fractional quantum Hall effect

We consider Farey series of rational numbers in terms of fractal sets labeled by the Hausdorff dimension with values defined in the interval 1< h<2 and associated with fractal curves. Our results come from the observation that the fractional quantum Hall effect-FQHE occurs in pairs of dual topological quantum numbers, the filling factors. These quantum numbers obey some properties of the Farey series and so we obtain that the universality classes of the quantum Hall transitions are classified in terms of h. The connection between Number Theory and Physics appears naturally in this context.

Multifractality at the spin quantum Hall transition

Statistical properties of critical wave functions at the spin quantum Hall transition are studied both numerically and analytically (via mapping onto the classical percolation). It is shown that the index $\ensuremath{\eta}$ characterizing the decay of wave function correlations is equal to 1/4, at variance with the ${r}^{\ensuremath{-}1/2}$ decay of the diffusion propagator. The multifractality spectra of eigenfunctions and of two-point conductances are found to be close to parabolic, ${\ensuremath{\Delta}}_{q}\ensuremath{\simeq}q(1\ensuremath{-}q)/8$ and ${X}_{q}\ensuremath{\simeq}q(3\ensuremath{-}q)/4.$

Real-Space Renormalization Group Approach to the Quantum Hall Transition

The integer quantum Hall (QH) transition is described well in terms of a delocalization-localization transition of the electron wavefunction. In contrast to a usual metalinsulator transition (MIT), the QH transition is characterized by a single extended state located exactly at the center = 0 of each Landau band. When approaching = 0, the localization length ξ of the electron wavefunction diverges according to a power law −ν , where defines the distance to the MIT for a suitable control parameter, e.g., the electron energy. On the theoretical side, the value of ν has been extracted from various numerical simulations, e.g., ν = 2.5 ± 0.5, 2.4 ± 0.2, 2.35± 0.03, and 2.39± 0.01. In experiments ν ≈ 2.3 has been obtained, e.g., from the frequency or the sample size dependence of the critical behavior of the resistance in the transition region at strong magnetic field. We study the critical properties of the integer QH transition by employing the real-space renormalization-group (RG) approach to the Chalker-Coddington (CC) network model. We calculate the critical distribution Pc(G) of the conductance and the critical exponent ν of the QH transition for two different RG units. This allows to demonstrate that the quality of the results crucially depends on the choice of the RG unit. The CC model describes a single QH transition using a chiral network consisting of electron trajectories along equipotential lines (links) and saddle points (SP’s) of the potential (nodes). Each SP acts as a scatterer and relates the wavefunction amplitudes in two incoming and two outgoing channels. It can be characterized by a 2×2 S matrix, which depends only on the transmission and reflection coefficients ti and ri. The links correspond to random phases Φj and reflect the randomness of the potential disorder in a sample. We consider two previously studied, different RG units on a regular 2D square lattice as shown in Fig. 1. One is constructed from 4 SP’s, the other consists of 5 SP’s. The RG unit should be chosen in a way such that the essential properties of the network are taken into account. In the course of our RG approach an RG unit is then mapped onto a new single super-SP using the analytical dependence t′ = f ({ti, ri}, {Φj}) (1)

Multifractality and Scaling at the Quantum Hall Transitions

Multifractality and Scaling at the Quantum Hall Transitions

ExactS-matrices for supersymmetric sigma models and the Potts model

We study the algebraic formulation of exact factorizable S-matrices for integrable two-dimensional field theories. We show that different formulations of the S-matrices for the Potts field theory are essentially equivalent, in the sense that they can be expressed in the same way as elements of the Temperley-Lieb algebra, in various representations. This enables us to construct the S-matrices for certain nonlinear sigma models that are invariant under the Lie ``supersymmetry'' algebras sl(m+n|n) (m=1,2; n>0), both for the bulk and for the boundary, simply by using another representation of the same algebra. These S-matrices represent the perturbation of the conformal theory at theta=pi by a small change in the topological angle theta. The m=1, n=1 theory has applications to the spin quantum Hall transition in disordered fermion systems. We also find S-matrices describing the flow from weak to strong coupling, both for theta=0 and theta=pi, in certain other supersymmetric sigma models.

New spin-orbit-induced universality class in the integer quantum Hall regime.

Using heuristic arguments and numerical simulations it is argued that the critical exponent nu describing the localization length divergence at the integer quantum-Hall transition is modified in the presence of spin-orbit scattering with short-range correlations. The exponent is very close to nu=4/3, the percolation correlation length exponent, consistent with the prediction of a semiclassical argument. In addition, a band of weakly localized states is conjectured.

Spin splitting of the excited-state subband inGaAs−Al0.3Ga0.7Asasymmetric two-layer systems

We track the variation of the Landau-level filling factors for the quantization of the Hall resistance in unbalanced double-quantum-well structures when the electron density is a controlled parameter. We observe that the filling factors associated with adjacent quantum Hall states differ by three for two or more consecutive quantum Hall transitions. The anomalous filling-factor increment results from simultaneous depopulations of a spin-degenerate Landau level of the ground-state subband and a spin-polarized Landau level of the excited-state subband. The locking of the topmost Landau levels to the Fermi level gives rise to the phenomenon in which the Zeeman splitting of the excited-state subband can be mistaken to be enormous enough to match the Landau-level separation of the ground-state subband. The temperature dependence of the quantum Hall states evidences, however, that the exchange enhancement of the g factor for the excited-state subband is rather similar to that for the ground-state subband.

Mobility-edge scaling at semiclassical and dissipative Hall transitions

The universal anomalous diffusion scaling is obtained for the semiclassical quantum Hall transition, which has been argued to describe samples with dissipation or correlated impurities. The results explain a discrepancy between existing numerics and the expected scaling $\eta = 2 x_1 = {1/2}$, which is violated because of a cancellation in the scaling function. The crossover with increasing observation time from semiclassical to quantum scaling is shown to explain a recent experiment which finds different scaling laws depending on how the localization length is extracted.

Electron-electron interactions, quantum Coulomb gap, and dynamical scaling near integer quantum Hall transitions

The effects of electron-electron interactions on tunneling into the bulk of a two-dimensional electron system are studied near the integer quantum Hall transitions. Taking into account the dynamical screening of the interactions in the critical conducting state, we show that the behavior of the tunneling density of states (TDOS) is significantly altered at low energies from its noninteracting counterpart. For the long-range Coulomb interaction, we demonstrate that the TDOS vanishes linearly at the Fermi level according to a quantum Coulomb gap form $\ensuremath{\nu}(\ensuremath{\omega}{)=C}_{Q}|\ensuremath{\omega}|{/e}^{4},$ with ${C}_{Q}$ a nonuniversal coefficient of a quantum-mechanical origin. In the case of short-range or screened Coulomb interactions, the TDOS is found to follow a power law $|\ensuremath{\omega}{|}^{\ensuremath{\alpha}},$ with \ensuremath{\alpha} proportional to the bare interaction strength. Since short-range interactions are known to be irrelevant perturbations at the noninteracting critical point, we predict that, upon scaling, the power law is smeared, leading to a finite zero-bias TDOS $\ensuremath{\nu}(\ensuremath{\omega})/\ensuremath{\nu}(0)=1+(|\ensuremath{\omega}|/{\ensuremath{\omega}}_{0}{)}^{\ensuremath{\gamma}},$ where \ensuremath{\gamma} is a universal exponent determined by the scaling dimension of short-ranged interactions. We also consider the case of quasi-one-dimensional (1D) samples with edges, i.e., the long Hall bar geometry, and find that the TDOS becomes dependent on the Hall conductance due to an altered boundary condition for diffusion. For short-range interactions, the TDOS of a quasi-1D strip with edges is linear near the Fermi level, with a slope inversely proportional to ${\ensuremath{\rho}}_{\mathrm{xx}}$ in the perturbative limit. These results are in qualitative agreement with the findings of bulk tunneling experiments. We discuss recent developments in understanding the role played by electron-electron interactions at the integer quantum Hall transitions and the implications of these results on the dynamical scaling of the transition width. We argue that for long-range Coulomb interactions, the existence of the quantum Coulomb gap in the quantum critical regime of the transition gives rise to the observed dynamical exponent $z=1.$

Renormalization group for network models of quantum Hall transitions

We analyze in detail the renormalization group flows which follow from the recently proposed all orders β functions for the Chalker–Coddington network model. The flows in the physical regime reach a true singularity after a finite scale transformation. Other flows are regular and we identify the asymptotic directions. One direction is in the same universality class as the disordered XY model. The all orders β function is computed for the network model of the spin quantum Hall transition and the flows are shown to have similar properties. It is argued that fixed points of general current–current interactions in 2d should correspond to solutions of the Virasoro master equation. Based on this we identify two coset conformal field theories osp(2 N|2 N) 1/ u(1) 0 and osp(4 N|4 N) 1/ su(2) 0 as possible fixed points and study the resulting multifractal properties. We also obtain a scaling relation between the typical amplitude exponent α 0 and the typical point contact conductance exponent X t which is expected to hold when the density of states is constant.

Mesoscopic effects in the quantum Hall regime

We report results of a study of (integer) quantum Hall transitions in a single or multiple Landau levels for non-interacting electrons in disordered two-dimensional systems, obtained by projecting a tight-binding Hamiltonian to the corresponding magnetic subbands. In finite-size systems, we find that mesoscopic effects often dominate, leading to apparent non-universal scaling behavior in higher Landau levels. This is because localization length, which grows exponentially with Landau level index, exceeds the system sizes amenable to the numerical study at present. When band mixing between multiple Landau levels is present, mesoscopic effects cause a crossover from a sequence of quantum Hall transitions for weak disorder to classical behavior for strong disorder. This behavior may be of relevance to experimentally observed transitions between quantum Hall states and the insulating phase at low magnetic fields.

Thermodynamic and tunneling density of states of the integer quantum Hall critical state

We examine the long wave length limit of the self-consistent Hartree-Fock approximation irreducible static density-density response function by evaluating the charge induced by an external charge. Our results are consistent with the compressibility sum rule and inconsistent with earlier work that did not account for consistency between the exchange-local-field and the disorder potential. We conclude that the thermodynamic density of states is finite, in spite of the vanishing tunneling density of states at the critical energy of the integer quantum Hall transition.

Insulator–quantum Hall liquid transitions in a two-dimensional electron gas using self-assembled InAs dots

We investigate the transport properties of two-dimensional electron gases (2DEG) formed in a GaAs/AlGaAs quantum well, where self-assembled InAs quantum dots were grown at the center of the GaAs well. Due to the resulting strain fields repulsive short-range scattering is experienced by the conduction electrons in the 2DEG. In a perpendicular magnetic field, there are transitions between quantum Hall liquids at filling factors ν=1 and 2 and the insulating phase. We show that the boundary of insulator–Quantum Hall transitions can be identified either by analysing the temperature-independent points in ρ xx or from the peaks in σ xx at low temperatures and both methods give similar results.

Integer quantum Hall transition in the presence of a long-range-correlated quenched disorder

We theoretically study the effect of long-ranged inhomogeneities on the critical properties of the integer quantum Hall transition. For this purpose we employ the real-space renormalization-group (RG) approach to the network model of the transition. We start by testing the accuracy of the RG approach in the absence of inhomogeneities, and infer the correlation length exponent $\ensuremath{\nu}=2.39$ from a broad conductance distribution. We then incorporate macroscopic inhomogeneities into the RG procedure. Inhomogeneities are modeled by a smooth random potential with a correlator which falls off with distance as a power law, ${r}^{\ensuremath{-}\ensuremath{\alpha}}.$ Similar to the classical percolation, we observe an enhancement of $\ensuremath{\nu}$ with decreasing \ensuremath{\alpha}. Although the attainable system sizes are large, they do not allow one to unambiguously identify a cusp in the $\ensuremath{\nu}(\ensuremath{\alpha})$ dependence at ${\ensuremath{\alpha}}_{c}=2/\ensuremath{\nu},$ as might be expected from the extended Harris criterion. We argue that the fundamental obstacle for the numerical detection of a cusp in the quantum percolation is the implicit randomness in the Aharonov-Bohm phases of the wave functions. This randomness emulates the presence of a short-range disorder alongside the smooth potential.

Multifractality of wave functions at the quantum Hall transition revisited

We investigate numerically the statistics of wave function amplitudes $\ensuremath{\psi}(\mathbf{r})$ at the integer quantum Hall transition. It is demonstrated that in the limit of a large system size the distribution function of $|\ensuremath{\psi}{|}^{2}$ is log-normal, so that the multifractal spectrum $f(\ensuremath{\alpha})$ is exactly parabolic. Our findings lend strong support to a recent conjecture for a critical theory of the quantum Hall transition.

Search for multiple-step integer quantum Hall transitions

Recent experiments in the integer quantum Hall regime seem to find direct transitions from a quantum Hall state with Hall conductance $\sigma_{xy} = n e^2/h $ with integer $n > 1$, to an insulating state in weak magnetic fields. We study this issue using a variation of the tight-binding lattice model for non-interacting electrons. Although quantum Hall transitions with change in Hall conductance $n e^2/ h$ with $n > 1$ do exist in our model for special tuning of parameters, they generically split into quantum Hall transitions with the Hall conductance changing by $e^2 / h$ at each transition. This suggests that a generic multiple step quantum Hall transition is incompatible with a non-interacting electron picture.

Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

We study two-dimensional nonlinear sigma models in which the target spaces are the coset supermanifolds U(n+m|n)/[ U(1)× U(n+m−1|n)]≅ CP n+m−1|n (projective superspaces) and OSp(2 n+ m|2 n)/OSp(2 n+ m−1|2 n)≅ S 2 n+ m−1|2 n (superspheres), n, m integers, −2⩽ m⩽2; these quantum field theories live in Hilbert spaces with indefinite inner products. These theories possess non-trivial conformally-invariant renormalization-group fixed points, or in some cases, lines of fixed points. Some of the conformal fixed-point theories can also be obtained within Landau–Ginzburg theories. We obtain the complete spectra (with multiplicities) of exact conformal weights of states (or corresponding local operators) in the isolated fixed-point conformal field theories, and at one special point on each of the lines of fixed points. Although the conformal weights are rational, the conformal field theories are not, and (with one exception) do not contain the affine versions of their superalgebras in their chiral algebras. The method involves lattice models that represent the strong-coupling region, which can be mapped to loop models, and then to a Coulomb gas with modified boundary conditions. The results apply to percolation, dilute and dense polymers, and other statistical mechanics models, and also to the spin quantum Hall transition in non-interacting fermions with quenched disorder.