Quantum criticality and minimal conductivity in graphene with long-range disorder

Enhanced Quantum Metric due to Vacancies in Graphene

Random vacancies in a graphene monolayer induce defect states that are known to form a narrow impurity band centered around zero energy at half filling. We use a space-resolved formulation of the quantum metric and establish a strong enhancement of the electronic correlations in this impurity band. The enhancement is primarily due to strong correlations between pairs of vacancies situated on different sublattices at anomalously large spatial distances. We trace the strong enhancement to both the multifractal vacancy wave functions, which ties the system exactly at the Anderson insulator transition for all defect concentrations, and preserving the chiral symmetry.

Large Diamagnetism and Electromagnetic Duality in Two-Dimensional Dirac Electron System

A Dirac electron system in solids mimics relativistic quantum physics that is compatible with Maxwell's equations, with which we anticipate unified electromagnetic responses. We find a large orbital diamagnetism only along the interplane direction and a nearly temperature-independent electrical conductivity of the order of e^{2}/h per plane for the new 2D Dirac organic conductor, α-(BETS)_{2}I_{3}, where BETS is bis(ethylenedithio)tetraselenafulvalene. Unlike conventional electrons in solids whose nonrelativistic effects bifurcate electric and magnetic responses, the observed orbital diamagnetism scales with the electrical conductivity in a wide temperature range. This demonstrates that an electromagnetic duality that is valid only within the relativistic framework is revived in solids.

Interacting chiral electrons at the 2D Dirac points: a review

The pseudo-relativistic chiral electrons in 2D graphene and 3D topological semimetals, known as the massless Dirac or Weyl fermions, constitute various intriguing issues in modern condensed-matter physics. In particular, the issues linked to the Coulomb interaction between the chiral electrons attract great attentions due to their unusual features, namely, the interaction is not screened and has a long-ranged property near the charge-neutrality point, in clear contrast to its screened and short-ranged properties in the conventional correlated materials. In graphene, this long-range interaction induces an anomalous logarithmic renormalization of the Fermi velocity, which causes a nonlinear reshaping of its Dirac cone. In addition, for strong interactions, it even leads to the predictions of an excitonic condensation with a spontaneous mass generation. The interaction, however, would seem to be not that large in graphene, so that the latter phenomenon appears to have not yet been observed. Contrastingly, the interaction is probably large in the pressurized organic materialα-(BEDT-TTF)₂I₃, where a 2D massless-Dirac-fermion phase emerges next to a correlated insulating phase. Therefore, an excellent testing ground would appear in this material for the studies of both the velocity renormalization and the mass generation, as well as for those of the short-range electronic correlations. In this review, we give an overview of the recent progress on the understanding of such interacting chiral electrons in 2D, by placing particular emphasis on the studies in graphene andα-(BEDT-TTF)₂I₃. In the first half, we briefly summarize our current experimental and theoretical knowledge about the interaction effects in graphene, then turn attentions to the understanding inα-(BEDT-TTF)₂I₃, and highlight its relevance to and difference from graphene. The second half of this review focusses on the studies linked to the nuclear magnetic resonance experiments and the associated model calculations inα-(BEDT-TTF)₂I₃. These studies allow us to discuss the anisotropic reshaping of a tilted Dirac cone together with various electronic correlations, and the precursor excitonic dynamics growing prior to a condensation. We see these provide unique opportunities to resolve the momentum dependence of the spin excitations and fluctuations that are strongly influenced by the long-range interaction near the Dirac points.

Criticality of Two-Dimensional Disordered Dirac Fermions in the Unitary Class and Universality of the Integer Quantum Hall Transition

Two-dimensional (2D) Dirac fermions are a central paradigm of modern condensed matter physics, describing low-energy excitations in graphene, in certain classes of superconductors, and on surfaces of 3D topological insulators. At zero energy E=0, Dirac fermions with mass m are band insulators, with the Chern number jumping by unity at m=0. This observation lead Ludwig et al. [Phys. Rev. B 50, 7526 (1994)PRBMDO0163-182910.1103/PhysRevB.50.7526] to conjecture that the transition in 2D disordered Dirac fermions (DDF) and the integer quantum Hall transition (IQHT) are controlled by the same fixed point and possess the same universal critical properties. Given the far-reaching implications for the emerging field of the quantum anomalous Hall effect, modern condensed matter physics, and our general understanding of disordered critical points, it is surprising that this conjecture has never been tested numerically. Here, we report the results of extensive numerics on the phase diagram and criticality of 2D DDF in the unitary class. We find a critical line at m=0, with an energy-dependent localization length exponent. At large energies, our results for the DDF are consistent with state-of-the-art numerical results ν_{IQH}=2.56-2.62 from models of the IQHT. At E=0, however, we obtain ν_{0}=2.30-2.36 incompatible with ν_{IQH}. This result challenges conjectured relations between different models of the IQHT, and several interpretations are discussed.

Dynamics of impurity in the environment of Dirac fermions

We study the dynamics of a nonmagnetic impurity interacting with the surface states of a 3D and 2D topological insulator (TI). Employing the linked cluster technique we develop a formalism for obtaining the Green's function of the mobile impurity interacting with the low-energy Dirac fermions. We show that for the non-recoil case in 2D, the Green's function in the long-time limit has a power-law decay in time implying the breakdown of the quasiparticle description of the impurity. The spectral function in turn exhibits a weak power-law singularity. In the recoil case, however, the reduced phase-space for scattering processes implies a non-zero quasiparticle weight and the presence of a coherent part in the spectral function. Performing a weak coupling analysis we find that the mobility of the impurity reveals aT-3/2divergence at low temperatures. In addition, we show that the Green's function of an impurity interacting with the helical edge modes (surface states of 2D TI) exhibit power-law decay in the long-time limit for both the non-recoil and recoil case (with low impurity momentum), indicating the break down of the quasiparticle picture. However, for impurity with high momentum, the quasiparticle picture is restored. The mobility of the heavy impurity interacting with the helical edge modes exhibits unusual behaviour. It has an exponential divergence at low temperatures which can be tuned to a power-law divergence,T-4, by the application of the magnetic field.

Sticking of atomic hydrogen on graphene

Recent years have witnessed an ever growing interest in the interactions between hydrogen atoms and a graphene sheet. Largely motivated by the possibility of modulating the electric, optical and magnetic properties of graphene, a huge number of studies have appeared recently that added to and enlarged earlier investigations on graphite and other carbon materials. In this review we give a glimpse of the many facets of this adsorption process, as they emerged from these studies. The focus is on those issues that have been addressed in detail, under carefully controlled conditions, with an emphasis on the interplay between the adatom structures, their formation dynamics and the electric, magnetic and chemical properties of the carbon sheet.

Critical Percolation without Fine-Tuning on the Surface of a Topological Superconductor

We present numerical evidence that most two-dimensional surface states of a bulk topological superconductor (TSC) sit at an integer quantum Hall plateau transition. We study TSC surface states in class CI with quenched disorder. Low-energy (finite-energy) surface states were expected to be critically delocalized (Anderson localized). We confirm the low-energy picture, but find instead that finite-energy states are also delocalized, with universal statistics that are independent of the TSC winding number, and consistent with the spin quantum Hall plateau transition (percolation).

Disorder enabled band structure engineering of a topological insulator surface

Three-dimensional topological insulators are bulk insulators with Z₂ topological electronic order that gives rise to conducting light-like surface states. These surface electrons are exceptionally resistant to localization by non-magnetic disorder, and have been adopted as the basis for a wide range of proposals to achieve new quasiparticle species and device functionality. Recent studies have yielded a surprise by showing that in spite of resisting localization, topological insulator surface electrons can be reshaped by defects into distinctive resonance states. Here we use numerical simulations and scanning tunnelling microscopy data to show that these resonance states have significance well beyond the localized regime usually associated with impurity bands. At native densities in the model Bi₂X₃ (X=Bi, Te) compounds, defect resonance states are predicted to generate a new quantum basis for an emergent electron gas that supports diffusive electrical transport.

The surface electrons in a topological insulator are resistant to localization by nonmagnetic disorder, but are affected by lattice disorder. Here, the authors show that resonance states near lattice defects on the surface have significance beyond the localized regime usually associated with impurity bands.

Lévy Flights due to Anisotropic Disorder in Graphene

We study transport properties of graphene with anisotropically distributed on-site impurities (adatoms) that are randomly placed on every third line drawn along carbon bonds. We show that stripe states characterized by strongly suppressed backscattering are formed in this model in the direction of the lines. The system reveals Lévy-flight transport in the stripe direction such that the corresponding conductivity increases as the square root of the system length. Thus, adding this type of disorder to clean graphene near the Dirac point strongly enhances the conductivity, which is in stark contrast with a fully random distribution of on-site impurities, which leads to Anderson localization. The effect is demonstrated both by numerical simulations using the Kwant code and by an analytical theory based on the self-consistent T-matrix approximation.

Topological Protection from Random Rashba Spin-Orbit Backscattering: Ballistic Transport in a Helical Luttinger Liquid

The combination of Rashba spin-orbit coupling and potential disorder induces a random current operator for the edge states of a 2D topological insulator. We prove that charge transport through such an edge is ballistic at any temperature, with or without Luttinger liquid interactions. The solution exploits a mapping to a spin 1/2 in a time-dependent field that preserves the projection along one randomly undulating component (integrable dynamics). Our result is exact and rules out random Rashba backscattering as a source of temperature-dependent transport, absent integrability-breaking terms.

Observation of Anderson localization in ultrathin films of three-dimensional topological insulators

Anderson localization, the absence of diffusive transport in disordered systems, has been manifested as hopping transport in numerous electronic systems, whereas in recently discovered topological insulators it has not been directly observed. Here, we report experimental demonstration of a crossover from diffusive transport in the weak antilocalization regime to variable range hopping transport in the Anderson localization regime with ultrathin (Bi_{1-x}Sb_{x})_{2}Te_{3} films. As disorder becomes stronger, negative magnetoconductivity due to the weak antilocalization is gradually suppressed, and eventually, positive magnetoconductivity emerges when the electron system becomes strongly localized. This work reveals the critical role of disorder in the quantum transport properties of ultrathin topological insulator films, in which theories have predicted rich physics related to topological phase transitions.

Disordered Weyl Semimetals and Their Topological Family

We develop a topological theory for disordered Weyl semimetals in the framework of the gauge invariance of the replica formalism and boundary-bulk correspondence of Chern insulators. An anisotropic topological θ term is analytically derived for the effective nonlinear σ model. It is this nontrivial topological term that ensures that the bulk transverse transport of Weyl semimetals is robust against disorders. Moreover, we establish a general diagram that reveals the intrinsic relations among topological terms in the nonlinear σ models and gauge response theories, respectively, for (2n+2)-dimensional topological insulators, (2n+1)-dimensional chiral fermions, (2n+1)-dimensional chiral semimetals, and (2n)-dimensional topological insulators with n being a positive integer.

Multiple quantum phases in graphene with enhanced spin-orbit coupling: from the quantum spin Hall regime to the spin Hall effect and a robust metallic state

We report an intriguing transition from the quantum spin Hall phase to the spin Hall effect upon segregation of thallium adatoms adsorbed onto a graphene surface. Landauer-Büttiker and Kubo-Greenwood simulations are used to access both edge and bulk transport physics in disordered thallium-functionalized graphene systems of realistic sizes. Our findings not only quantify the detrimental effects of adatom clustering in the formation of the topological state, but also provide evidence for the emergence of spin accumulation at opposite sample edges driven by spin-dependent scattering induced by thallium islands, which eventually results in a minimum bulk conductivity ∼4e²/h, insensitive to localization effects.

Metal-insulator transition in graphene on boron nitride

Electrons in graphene aligned with hexagonal boron nitride are modeled by Dirac fermions in a correlated random-mass landscape subject to a scalar- and vector-potential disorder. We find that the system is insulating in the commensurate phase since the average mass deviates from zero. At the transition the mean mass is vanishing and electronic conduction in a finite sample can be described by a critical percolation along zero-mass lines. In this case graphene at the Dirac point is in a critical state with the conductivity sqrt[3]e(2)/h. In the incommensurate phase the system behaves as a symplectic metal.

Quantum Hall criticality and localization in graphene with short-range impurities at the Dirac point

We explore the longitudinal conductivity of graphene at the Dirac point in a strong magnetic field with two types of short-range scatterers: adatoms that mix the valleys and "scalar" impurities that do not mix them. A scattering theory for the Dirac equation is employed to express the conductance of a graphene sample as a function of impurity coordinates; an averaging over impurity positions is then performed numerically. The conductivity σ is equal to the ballistic value 4e2/πh for each disorder realization, provided the number of flux quanta considerably exceeds the number of impurities. For weaker fields, the conductivity in the presence of scalar impurities scales to the quantum-Hall critical point with σ≃4×0.4e2/h at half filling or to zero away from half filling due to the onset of Anderson localization. For adatoms, the localization behavior is also obtained at half filling due to splitting of the critical energy by intervalley scattering. Our results reveal a complex scaling flow governed by fixed points of different symmetry classes: remarkably, all key manifestations of Anderson localization and criticality in two dimensions are observed numerically in a single setup.

Quantum interference and Aharonov-Bohm oscillations in topological insulators

Topological insulators (TIs) have an insulating bulk but a metallic surface. In the simplest case, the surface electronic structure of a three-dimensional (3D) TI is described by a single two-dimensional (2D) Dirac cone. A single 2D Dirac fermion cannot be realized in an isolated 2D system with time-reversal symmetry, but rather owes its existence to the topological properties of the 3D bulk wavefunctions. The transport properties of such a surface state are of considerable current interest; they have some similarities with graphene, which also realizes Dirac fermions, but have several unique features in their response to magnetic fields. In this review we give an overview of some of the main quantum transport properties of TI surfaces. We focus on the efforts to use quantum interference phenomena, such as weak anti-localization and the Aharonov-Bohm effect, to verify in a transport experiment the Dirac nature of the surface state and its defining properties. In addition to explaining the basic ideas and predictions of the theory, we provide a survey of recent experimental work.

Coulomb drag in graphene near the Dirac point

We study Coulomb drag in graphene near the Dirac point, focusing on the regime of interaction-dominated transport. We establish a novel, graphene-specific mechanism of Coulomb drag based on fast interlayer thermalization, inaccessible by standard perturbative approaches. Using the quantum kinetic equation framework, we derive a hydrodynamic description of transport in double-layer graphene in terms of electric and energy currents. In the clean limit the drag becomes temperature independent. In the presence of disorder the drag coefficient at the Dirac point remains nonzero due to higher-order scattering processes and interlayer disorder correlations. At low temperatures (diffusive regime) these contributions manifest themselves in the peak in the drag coefficient centered at the neutrality point with a magnitude that grows with lowering temperature.

Topology, delocalization via average symmetry and the symplectic Anderson transition

A field theory of the Anderson transition in two-dimensional disordered systems with spin-orbit interactions and time-reversal symmetry is developed, in which the proliferation of vortexlike topological defects is essential for localization. The sign of vortex fugacity determines the Z(2) topological class of the localized phase. There are two distinct fixed points with the same critical exponents, corresponding to transitions from a metal to an insulator and a topological insulator, respectively. The critical conductivity and correlation length exponent of these transitions are computed in an N=1-[symbol: see text] expansion in the number of replicas, where for small [symbol: see text] the critical points are perturbatively connected to the Kosterlitz-Thouless critical point. Delocalized states, which arise at the surface of weak topological insulators and topological crystalline insulators, occur because vortex proliferation is forbidden due to the presence of symmetries that are violated by disorder, but are restored by disorder averaging.

Interaction-mediated surface-state instability in disordered three-dimensional topological superconductors with spin SU2 symmetry

We show that arbitrarily weak interparticle interactions destabilize the surface states of 3D topological superconductors with spin SU(2) invariance (symmetry class CI) in the presence of nonmagnetic disorder. The conduit for the instability is disorder-induced wave function multifractality. We argue that time-reversal symmetry breaks spontaneously at the surface, so that topologically protected states do not exist for this class. The interaction-stabilized surface phase is expected to exhibit ferromagnetic order, or to reside in an insulating plateau of the spin quantum Hall effect.

Quantum transport and two-parameter scaling at the surface of a weak topological insulator

Weak topological insulators have an even number of Dirac cones in their surface spectrum and are thought to be unstable to disorder, which leads to an insulating surface. Here we argue that the presence of disorder alone will not localize the surface states; rather, the presence of a time-reversal symmetric mass term is required for localization. Through numerical simulations, we show that in the absence of the mass term the surface always flow to a stable metallic phase and the conductivity obeys a one-parameter scaling relation, just as in the case of a strong topological insulator surface. With the inclusion of the mass, the transport properties of the surface of a weak topological insulator follow a two-parameter scaling form.

Surface-quantized anomalous Hall current and the magnetoelectric effect in magnetically disordered topological insulators

We study theoretically the role of quenched magnetic disorder at the surface of topological insulators by numerical simulation and scaling analysis based on the massive Dirac fermion model. This addresses the problem of Anderson localization on chiral anomaly. It is found that all the surface states are localized, while the transverse conductivity is quantized to be ±e2/2h as long as the Fermi energy is within the bulk gap. This greatly facilitates the realization of the topological magnetoelectric effect proposed by Qi et al. [Phys. Rev. B 78, 195424 (2008)] with the surface magnetization direction being controlled by the simultaneous application of magnetic and electric fields.

Characteristics of level-spacing statistics in chaotic graphene billiards

A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions.

Two-dimensional graphene with structural defects: elastic mean free path, minimum conductivity, and Anderson transition

Quantum transport properties of disordered graphene with structural defects (Stone-Wales and divacancies) are investigated using a realistic π-π* tight-binding model elaborated from ab initio calculations. Mean free paths and semiclassical conductivities are then computed as a function of the nature and density of defects (using an order-N real-space Kubo-Greenwood method). By increasing the defect density, the decay of the semiclassical conductivities is predicted to saturate to a minimum value of 4e2/πh over a large range (plateau) of carrier density (>0.5×10(14)  cm(-20). Additionally, strong contributions of quantum interferences suggest that the Anderson localization regime could be experimentally measurable for a defect density as low as 1%.

Diffusion and criticality in undoped graphene with resonant scatterers

A general theory is developed to describe graphene with an arbitrary number of isolated impurities. The theory provides a basis for an efficient numerical analysis of the charge transport and is applied to calculate the Dirac-point conductivity σ of graphene with resonant scatterers. In the case of smooth resonant impurities the symmetry class is identified as DIII and σ grows logarithmically with increasing impurity concentration. For vacancies (or strong on-site potential impurities, class BDI) σ saturates at a constant value that depends on the vacancy distribution among two sublattices.

Interaction-induced criticality in Z(2) topological insulators

We study interaction effects in topological insulators with strong spin-orbit coupling. We find that the interplay of nontrivial topology and Coulomb repulsion induces a novel critical state on the surface of a three-dimensional topological insulator. Remarkably, this interaction-induced criticality, characterized by a universal value of conductivity, emerges without any adjustable parameters. Further, we predict a direct quantum-spin-Hall transition in two dimensions that occurs via a similar critical state.

Disorder and electronic transport in graphene

In this review, we provide an account of the recent progress in understanding electronic transport in disordered graphene systems. Starting from a theoretical description that emphasizes the role played by band structure properties and lattice symmetries, we describe the nature of disorder in these systems and its relation to transport properties. While the focus is primarily on theoretical and conceptual aspects, connections to experiments are also included. Issues such as short- versus long-range disorder, localization (strong and weak), the carrier density dependence of the conductivity, and conductance fluctuations are considered and some open problems are pointed out.

Electronic properties of a biased graphene bilayer

We study, within the tight-binding approximation, the electronic properties of a graphene bilayer in the presence of an external electric field applied perpendicular to the system-a biased bilayer. The effect of the perpendicular electric field is included through a parallel plate capacitor model, with screening correction at the Hartree level. The full tight-binding description is compared with its four-band and two-band continuum approximations, and the four-band model is shown to always be a suitable approximation for the conditions realized in experiments. The model is applied to real biased bilayer devices, made out of either SiC or exfoliated graphene, and good agreement with experimental results is found, indicating that the model is capturing the key ingredients, and that a finite gap is effectively being controlled externally. Analysis of experimental results regarding the electrical noise and cyclotron resonance further suggests that the model can be seen as a good starting point for understanding the electronic properties of graphene bilayer. Also, we study the effect of electron-hole asymmetry terms, such as the second-nearest-neighbour hopping energies t' (in-plane) and γ(4) (inter-layer), and the on-site energy Δ.

Charge transport in graphene with resonant scatterers

The full counting statistics for the charge transport through an undoped graphene sheet in the presence of strong potential impurities is studied. Treating the scattering off the impurity in the s-wave approximation, we calculate the impurity correction to the cumulant generating function. This correction is universal provided the impurity strength is tuned to a resonant value. In particular, the conductance of the sample acquires a correction of 16e{2}/(pi{2}h) per resonant impurity.

Topological surface States in three-dimensional magnetic insulators

An electron moving in a magnetically ordered background feels an effective magnetic field that can be both stronger and more rapidly varying than typical externally applied fields. One consequence is that insulating magnetic materials in three dimensions can have topologically nontrivial properties of the effective band structure. For the simplest case of two bands, these "Hopf insulators" are characterized by a topological invariant as in quantum Hall states and Z2 topological insulators, but instead of a Chern number or parity, the underlying invariant is the Hopf invariant that classifies maps from the three-sphere to the two-sphere. This Letter gives an efficient algorithm to compute whether a given magnetic band structure has nontrivial Hopf invariant, a double-exchange-like tight-binding model that realizes the nontrivial case, and a numerical study of the surface states of this model.

Quantum Hall effect of massless dirac fermions in a vanishing magnetic field

The effect of strong long-range disorder on the quantization of the Hall conductivity sigma{xy} in graphene is studied numerically. It is shown that increasing Landau-level mixing progressively destroys all plateaus in sigma{xy} except the plateaus at sigma{xy}=-/+e{2}/2h (per valley and per spin). The critical state at the Dirac point is robust to strong disorder and belongs to the universality class of the conventional plateau transitions in the integer quantum Hall effect. We propose that the breaking of time-reversal symmetry by ripples in graphene can realize this quantum critical point in a vanishing magnetic field.

Topological delocalization of two-dimensional massless Dirac fermions

The beta function of a two-dimensional massless Dirac Hamiltonian subject to a random scalar potential, which, e.g., underlies theoretical descriptions of graphene, is computed numerically. Although it belongs to, from a symmetry standpoint, the two-dimensional symplectic class, the beta function monotonically increases with decreasing conductance. We also provide an argument based on the spectral flows under twisting boundary conditions, which shows that none of the states of the massless Dirac Hamiltonian can be localized.

Z2 topological term, the global anomaly, and the two-dimensional symplectic symmetry class of Anderson localization

We discuss, for a two-dimensional Dirac Hamiltonian with a random scalar potential, the presence of a Z2 topological term in the nonlinear sigma model encoding the physics of Anderson localization in the symplectic symmetry class. The Z2 topological term realizes the sign of the Pfaffian of a family of Dirac operators. We compute the corresponding global anomaly, i.e., the change in the sign of the Pfaffian by studying a spectral flow numerically. This Z2 topological effect can be relevant to graphene when the impurity potential is long ranged and, also, to the two-dimensional boundaries of a three-dimensional lattice model of Z2 topological insulators in the symplectic symmetry class.

One-parameter scaling at the dirac point in graphene

We numerically calculate the conductivity sigma of an undoped graphene sheet (size L) in the limit of a vanishingly small lattice constant. We demonstrate one-parameter scaling for random impurity scattering and determine the scaling function beta(sigma)=dlnsigma/dlnL. Contrary to a recent prediction, the scaling flow has no fixed point (beta>0) for conductivities up to and beyond the symplectic metal-insulator transition. Instead, the data support an alternative scaling flow for which the conductivity at the Dirac point increases logarithmically with sample size in the absence of intervalley scattering--without reaching a scale-invariant limit.