Integer quantum Hall transition: An alternative approach and exact results

Emergence of nodal-knot transitions by disorder

Under certain symmetries, degenerate points in three-dimensional metals form one-dimensional nodal lines. These nodal lines sometimes exhibit intricate knotted structures and have been studied in various contexts. As one of the most common physical perturbations, disorder effects often trigger novel quantum phase transitions. For nodal-knot phases, whether disorder can drive knot transitions remains an open and intriguing question. Employing renormalization-group calculations, we demonstrate that nodal-knot transitions emerge in the presence of weak disorder. Specifically, both chemical-potential-type and magnetic-type disorders can induce knot transitions, resulting in the emergence of distinct knot topologies. The transition can be quantitatively characterized by changes in topological invariants such as the knot Wilson loop integrals. Our findings open up a new avenue for manipulating the topology of nodal-knot phases through disorder effects.

Nonzero Berry curvature dipole, magnetic gapped edge states and persistent spin texture in a rotational symmetry preserved van der Waals magnetic topological insulator

Berry curvature dipole and persistent spin textures are unconventional quantum marvels with paramount relevance towards nonlinear transport phenomena and futuristic spintronic applications. Here, we demonstrate that a magnetic van der Waals (vdW) heterobilayer, comprised of semiconducting monolayers 1H-MoSe2 (nonmagnetic) and ferromagnet 1H-VSe2, exhibits a finite Berry curvature dipole (DBcd) in the absence of strain or twist perturbation, where each constituent layer displays no DBcd owing to three-fold (C3v) rotational symmetry. A change in sign of DBcd is observed over an energy range, suggesting its oscillatory nature. Further, persistent spin texture (PST) with small spin canting and a prominent magnetic gap of 190 meV have emerged in the edge state spectrum of the vdW system. We find the emerged topological properties are crystal facet dependent. Moreover, an exotic quantum state of intrinsic spin-valley locking at two inequivalent K valleys with distinct spin identity is observed in the dispersion relation. We explore the subtle role of spin-valley locking in the generation of a finite DBcd, which has not been previously discussed in this context. Such quantum states promote right and left handed circular polarization and are perceived as a binary catalogue for information encoding and energy. Further, the role of symmetry breaking in the observed phenomena of the AB-stacked vdW heterobilayer is discussed. We propose that both time reversal symmetry breaking and spin-valley locking are essential for inducing nonzero DBcd in an AB stacked vdW heterostructure, where the three-fold (C3) rotational symmetry is preserved along the z-axis only.

Conformal Invariance and Multifractality at Anderson Transitions in Arbitrary Dimensions

Multifractals arise in various systems across nature whose scaling behavior is characterized by a continuous spectrum of multifractal exponents Δ_{q}. In the context of Anderson transitions, the multifractality of critical wave functions is described by operators O_{q} with scaling dimensions Δ_{q} in a field-theory description of the transitions. The operators O_{q} satisfy the so-called Abelian fusion expressed as a simple operator product expansion. Assuming conformal invariance and Abelian fusion, we use the conformal bootstrap framework to derive a constraint that implies that the multifractal spectrum Δ_{q} (and its generalized form) must be quadratic in its arguments in any dimension d≥2.

Mechanical, electronic, optical, piezoelectric and ferroic properties of strained graphene and other strained monolayers and multilayers: an update

This is an update of a previous review (Naumiset al2017Rep. Prog. Phys.80096501). Experimental and theoretical advances for straining graphene and other metallic, insulating, ferroelectric, ferroelastic, ferromagnetic and multiferroic 2D materials were considered. We surveyed (i) methods to induce valley and sublattice polarisation (P) in graphene, (ii) time-dependent strain and its impact on graphene's electronic properties, (iii) the role of local and global strain on superconductivity and other highly correlated and/or topological phases of graphene, (iv) inducing polarisationPon hexagonal boron nitride monolayers via strain, (v) modifying the optoelectronic properties of transition metal dichalcogenide monolayers through strain, (vi) ferroic 2D materials with intrinsic elastic (σ), electric (P) and magnetic (M) polarisation under strain, as well as incipient 2D multiferroics and (vii) moiré bilayers exhibiting flat electronic bands and exotic quantum phase diagrams, and other bilayer or few-layer systems exhibiting ferroic orders tunable by rotations and shear strain. The update features the experimental realisations of a tunable two-dimensional Quantum Spin Hall effect in germanene, of elemental 2D ferroelectric bismuth, and 2D multiferroic NiI2. The document was structured for a discussion of effects taking place in monolayers first, followed by discussions concerning bilayers and few-layers, and it represents an up-to-date overview of exciting and newest developments on the fast-paced field of 2D materials.

Quantum Diffusion in the Lowest Landau Level of Disordered Graphene

Electronic transport in the lowest Landau level of disordered graphene sheets placed in a homogeneous perpendicular magnetic field is a long-standing and cumbersome problem which defies a conclusive solution for several years. Because the modeled system lacks an intrinsic small parameter, the theoretical picture is infested with singularities and anomalies. We propose an analytical approach to the conductivity based on the analysis of the diffusive processes, and we calculate the density of states, the diffusion coefficient and the static conductivity. The obtained results are not only interesting from the purely theoretical point of view but have a practical significance as well, especially for the development of the novel high-precision calibration devices.

Coexistence of Quantum Hall and Quantum Anomalous Hall Phases in Disordered MnBi_{2}Te_{4}

In most cases, to observe quantized Hall plateaus, an external magnetic field is applied in intrinsic magnetic topological insulators MnBi_{2}Te_{4}. Nevertheless, whether the nonzero Chern number (C≠0) phase is a quantum anomalous Hall (QAH) state, or a quantum Hall (QH) state, or a mixing state of both is still a puzzle, especially for the recently observed C=2 phase [Deng et al., Science 367, 895 (2020)SCIEAS0036-807510.1126/science.aax8156]. In this Letter, we propose a physical picture based on the Anderson localization to understand the observed Hall plateaus in disordered MnBi_{2}Te_{4}. Rather good consistency between the experimental and numerical results confirms that the bulk states are localized in the absence of a magnetic field and a QAH edge state emerges with C=1. However, under a strong magnetic field, the lowest Landau band formed with the localized bulk states, survives disorder, together with the QAH edge state, leading to a C=2 phase. Eventually, we present a phase diagram of a disordered MnBi_{2}Te_{4} which indicates more coexistence states of QAH and QH to be verified by future experiments.

Coulomb Instabilities of a Three-Dimensional Higher-Order Topological Insulator

Topological insulators (TIs) are an exciting discovery because of their robustness against disorder and interactions. Recently, second-order TIs have been attracting increasing attention, because they host topologically protected 1D hinge states in 3D or 0D corner states in 2D. A significantly critical issue is whether the second-order TIs also survive interactions, but it is still unexplored. We study the effects of weak Coulomb interactions on a 3D second-order TI, with the help of renormalization-group calculations. We find that the 3D second-order TIs are always unstable, suffering from two types of topological phase transitions. One is from second-order TI to TI, the other is to normal insulator. The first type is accompanied by emergent time-reversal and inversion symmetries and has a dynamical critical exponent κ=1. The second type does not have the emergent symmetries but has nonuniversal dynamical critical exponents κ<1. Our results may inspire more inspections on the stability of higher-order topological states of matter and related novel quantum criticalities.

Delocalization Transition of a Disordered Axion Insulator

The axion insulator is a higher-order topological insulator protected by inversion symmetry. We show that, under quenched disorder respecting inversion symmetry on average, the topology of the axion insulator stays robust, and an intermediate metallic phase in which states are delocalized is unavoidable at the transition from an axion insulator to a trivial insulator. We derive this conclusion from general arguments, from classical percolation theory, and from the numerical study of a 3D quantum network model simulating a disordered axion insulator through a layer construction. We find the localization length critical exponent near the delocalization transition to be ν=1.42±0.12. We further show that this delocalization transition is stable even to weak breaking of the average inversion symmetry, up to a critical strength. We also quantitatively map our quantum network model to an effective Hamiltonian and we find its low-energy k·p expansion.

Critical Behavior and Universal Signature of an Axion Insulator State

Recently, the search for an axion insulator state in the ferromagnetic-3D topological insulator (TI) heterostructure and MnBi_{2}Te_{4} has attracted intense interest. However, its detection remains difficult in experiments. We systematically investigate the disorder-induced phase transition of the axion insulator state in a 3D TI with antiparallel magnetization alignment surfaces. It is found that there exists a 2D disorder-induced phase transition on the surfaces of the 3D TI which shares the same universality class with the quantum Hall plateau to plateau transition. Then, we provide a phenomenological theory which maps the random mass Dirac Hamiltonian of the axion insulator state into the Chalker-Coddington network model. Therefore, we propose probing the axion insulator state by investigating the universal signature of such a phase transition in the ferromagnetic-3D TI heterostructure and MnBi_{2}Te_{4}. Our findings not only show a global phase diagram of the axion insulator state, but also stimulate further experiments to probe it.

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.

A Supersymmetric Hierarchical Model for Weakly Disordered 3d Semimetals

In this paper, we study a hierarchical supersymmetric model for a class of gapless, three-dimensional, weakly disordered quantum systems, displaying pointlike Fermi surface and conical intersections of the energy bands in the absence of disorder. We use rigorous renormalization group methods and supersymmetry to compute the correlation functions of the system. We prove algebraic decay of the two-point correlation function, compatible with delocalization. A main technical ingredient is the multiscale analysis of massless bosonic Gaussian integrations with purely imaginary covariances, performed via iterative stationary phase expansions.

Scaling behavior of the quantum phase transition from a quantum-anomalous-Hall insulator to an axion insulator

The phase transitions from one plateau to the next plateau or to an insulator in quantum Hall and quantum anomalous Hall (QAH) systems have revealed universal scaling behaviors. A magnetic-field-driven quantum phase transition from a QAH insulator to an axion insulator was recently demonstrated in magnetic topological insulator sandwich samples. Here, we show that the temperature dependence of the derivative of the longitudinal resistance on magnetic field at the transition point follows a characteristic power-law that indicates a universal scaling behavior for the QAH to axion insulator phase transition. Similar to the quantum Hall plateau to plateau transition, the QAH to axion insulator transition can also be understood by the Chalker-Coddington network model. We extract a critical exponent κ ~ 0.38 ± 0.02 in agreement with recent high-precision numerical results on the correlation length exponent of the Chalker-Coddington model at ν ~ 2.6, rather than the generally-accepted value of 2.33.

Sign Change in the Anomalous Hall Effect and Strong Transport Effects in a 2D Massive Dirac Metal Due to Spin-Charge Correlated Disorder

The anomalous Hall effect (AHE) is highly sensitive to disorder in the metallic phase. Here we show that statistical correlations between the charge-spin disorder sectors strongly affect the conductivity and the sign or magnitude of AHE. As the correlation between the charge and gauge-mass components increases, so does the AHE, achieving its universal value, and even exceeding it, although the system is an impure metal. The AHE can change sign when the anticorrelations reverse the sign of the effective Dirac mass, a possible mechanism behind the sign change seen in recent experiments.

Electron Traversal Times in Disordered Graphene Nanoribbons

Using the partition-free time-dependent Landauer-Büttiker formalism for transient current correlations, we study the traversal times taken for electrons to cross graphene nanoribbon (GNR) molecular junctions. We demonstrate electron traversal signatures that vary with disorder and orientation of the GNR. These findings can be related to operational frequencies of GNR-based devices and their consequent rational design.

Photonic Anomalous Quantum Hall Effect

We experimentally realize a photonic analogue of the anomalous quantum Hall insulator using a two-dimensional (2D) array of coupled ring resonators. Similar to the Haldane model, our 2D array is translation invariant, has a zero net gauge flux threading the lattice, and exploits next-nearest neighbor couplings to achieve a topologically nontrivial band gap. Using direct imaging and on-chip transmission measurements, we show that the band gap hosts topologically robust edge states. We demonstrate a topological phase transition to a conventional insulator by frequency detuning the ring resonators and thereby breaking the inversion symmetry of the lattice. Furthermore, the clockwise or the counterclockwise circulation of photons in the ring resonators constitutes a pseudospin degree of freedom. The two pseudospins acquire opposite hopping phases, and their respective edge states propagate in opposite directions. These results are promising for the development of robust reconfigurable integrated nanophotonic devices for applications in classical and quantum information processing.

Bond-Disordered Spin Liquid and the Honeycomb Iridate H_{3}LiIr_{2}O_{6}: Abundant Low-Energy Density of States from Random Majorana Hopping

The 5d-electron honeycomb compound H_{3}LiIr_{2}O_{6} [K. Kitagawa et al., Nature (London) 554, 341 (2018)NATUAS0028-083610.1038/nature25482] exhibits an apparent quantum spin liquid state. In this intercalated spin-orbital compound, a remarkable pileup of low-energy states was experimentally observed in specific heat and spin relaxation. We show that a bond-disordered Kitaev model can naturally account for this phenomenon, suggesting that disorder plays an essential role in its theoretical description. In the exactly soluble Kitaev model, we obtain, via spin fractionalization, a random bipartite hopping problem of Majorana fermions in a random flux background. This has a divergent low-energy density of states of the required power-law form N(E)∝E^{-ν} with a drifting exponent which takes on the value ν≈1/2 for relatively strong bond disorder. Breaking time-reversal symmetry removes the divergence of the density of states, as does applying a magnetic field in experiment. We discuss the implication of our scenario, both for future experiments and from a broader perspective.

A methodical study of quantum phase engineering in topological crystalline insulator SnTe and related alloys

A detailed analysis of the perturbation effects on the quantum phase of SnTe(001) surface states.

Disorder-Driven Quantum Transition in Relativistic Semimetals: Functional Renormalization via the Porous Medium Equation

In the presence of randomness, a relativistic semimetal undergoes a quantum transition towards a diffusive phase. A standard approach relates this transition to the U(N) Gross-Neveu model in the limit of N→0. We show that the corresponding fixed point is infinitely unstable, demonstrating the necessity to include fluctuations beyond the usual Gaussian approximation. We develop a functional renormalization group method amenable to include these effects and show that the disorder distribution renormalizes following the so-called porous medium equation. We find that the transition is controlled by a nonanalytic fixed point drastically different from that of the U(N) Gross-Neveu model. Our approach provides a unique mechanism of spontaneous generation of a finite density of states and also characterizes the scaling behavior of the broad distribution of fluctuations close to the transition. It can be applied to other problems where nonanalytic effects may play a role, such as the Anderson localization transition.

Fractional Excitonic Insulator

We argue that a correlated fluid of electrons and holes can exhibit a fractional quantum Hall effect at zero magnetic field analogous to the Laughlin state at filling 1/m. We introduce a variant of the Laughlin wave function for electrons and holes and show that for m=1 it is the exact ground state of a free fermion model that describes p_{x}+ip_{y} excitonic pairing. For m>1 we develop a simple composite fermion mean field theory, and we present evidence that our wave function correctly describes this phase. We derive an interacting Hamiltonian for which our wave function is the exact ground state, and we present physical arguments that the m=3 state can be realized in a system in which energy bands with angular momentum that differ by 3 cross at the Fermi energy. This leads to a gapless state with (p_{x}+ip_{y})^{3} excitonic pairing, which we argue is conducive to forming the fractional excitonic insulator in the presence of interactions. Prospects for numerics on model systems and band structure engineering to realize this phase in real materials are discussed.

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).

Role of four-fermion interaction and impurity in the states of two-dimensional semi-Dirac materials

We study the effects of four-fermion interaction and impurity on the low-energy states of 2D semi-Dirac materials by virtue of the unbiased renormalization group approach. The coupled flow equations that govern the energy-dependent evolutions of all correlated interaction parameters are derived after taking into account one-loop corrections from the interplay between four-fermion interaction and impurity. Whether and how four-fermion interaction and impurity influence the low-energy properties of 2D semi-Dirac materials are discreetly explored and addressed attentively. After carrying out the standard renormalization group analysis, we find that both trivial insulating and nontrivial semimetal states are qualitatively stable against all four kinds of four-fermion interactions. However, while switching on both four-fermion interaction and impurity, certain insulator-semimetal phase transitions and the distance of Dirac nodal points can be respectively induced and modified due to their strong interplay and intimate competition. Moreover, several non-Fermi liquid behaviors that deviate from the conventional Fermi liquids are exhibited at the lowest-energy limit.

Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-Abelian bosonization to truncated spectrum methods

We review two important non-perturbative approaches for extracting the physics of low-dimensional strongly correlated quantum systems. Firstly, we start by providing a comprehensive review of non-Abelian bosonization. This includes an introduction to the basic elements of conformal field theory as applied to systems with a current algebra, and we orient the reader by presenting a number of applications of non-Abelian bosonization to models with large symmetries. We then tie this technique into recent advances in the ability of cold atomic systems to realize complex symmetries. Secondly, we discuss truncated spectrum methods for the numerical study of systems in one and two dimensions. For one-dimensional systems we provide the reader with considerable insight into the methodology by reviewing canonical applications of the technique to the Ising model (and its variants) and the sine-Gordon model. Following this we review recent work on the development of renormalization groups, both numerical and analytical, that alleviate the effects of truncating the spectrum. Using these technologies, we consider a number of applications to one-dimensional systems: properties of carbon nanotubes, quenches in the Lieb-Liniger model, 1  +  1D quantum chromodynamics, as well as Landau-Ginzburg theories. In the final part we move our attention to consider truncated spectrum methods applied to two-dimensional systems. This involves combining truncated spectrum methods with matrix product state algorithms. We describe applications of this method to two-dimensional systems of free fermions and the quantum Ising model, including their non-equilibrium dynamics.

Thermal Transport in the Kitaev Model

In conventional insulating magnets, heat is carried by magnons and phonons. In contrast, when the magnets harbor a quantum spin liquid state, emergent quasiparticles from the fractionalization of quantum spins can carry heat. Here, we investigate unconventional thermal transport yielded by such exotic carriers, in both longitudinal and transverse components, for the Kitaev model, whose ground state is exactly shown to be a quantum spin liquid with fractional excitations described as itinerant Majorana fermions and localized Z_{2} fluxes. We find that the longitudinal thermal conductivity exhibits a single peak at a high temperature, while the nonzero frequency component has a peak at a low temperature, reflecting the spin fractionalization. On the other hand, we show that the transverse thermal conductivity is induced by the magnetic field in a wide temperature range up to the energy scale of the bare exchange coupling; while increasing temperature, the transverse response divided by temperature decreases from the quantized value expected for the topologically nontrivial ground state and shows nonmonotonic temperature dependence. These characteristic behaviors provide experimentally accessible evidence of fractional excitations in the proximity to the Kitaev quantum spin liquid.

Diffusive and arrested transport of atoms under tailored disorder

Ultracold atoms in optical lattices offer a unique platform for investigating disorder-driven phenomena. While static disordered site potentials have been explored in a number of experiments, a more general, dynamical control over site-energy and off-diagonal tunnelling disorder has been lacking. The use of atomic quantum states as synthetic dimensions has introduced the spectroscopic, site-resolved control necessary to engineer more tailored realisations of disorder. Here, we present explorations of dynamical and tunneling disorder in an atomic system by controlling laser-driven dynamics of atomic population in a momentum-space lattice. By applying static tunnelling phase disorder to a one-dimensional lattice, we observe ballistic quantum spreading. When the applied disorder fluctuates on time scales comparable to intersite tunnelling, we instead observe diffusive atomic transport, signalling a crossover from quantum to classical expansion dynamics. We compare these observations to the case of static site-energy disorder, where we directly observe quantum localisation.

Cold atom quantum simulation has had challenges in realising the tailored, dynamic types of disorder relevant to real materials. Here, the authors use synthetic momentum-space lattices to engineer spatially and dynamically controlled disorder to observe ballistic, diffusive, and arrested atomic transport.

Quantum Loop Topography for Machine Learning

Despite rapidly growing interest in harnessing machine learning in the study of quantum many-body systems, training neural networks to identify quantum phases is a nontrivial challenge. The key challenge is in efficiently extracting essential information from the many-body Hamiltonian or wave function and turning the information into an image that can be fed into a neural network. When targeting topological phases, this task becomes particularly challenging as topological phases are defined in terms of nonlocal properties. Here, we introduce quantum loop topography (QLT): a procedure of constructing a multidimensional image from the "sample" Hamiltonian or wave function by evaluating two-point operators that form loops at independent Monte Carlo steps. The loop configuration is guided by the characteristic response for defining the phase, which is Hall conductivity for the cases at hand. Feeding QLT to a fully connected neural network with a single hidden layer, we demonstrate that the architecture can be effectively trained to distinguish the Chern insulator and the fractional Chern insulator from trivial insulators with high fidelity. In addition to establishing the first case of obtaining a phase diagram with a topological quantum phase transition with machine learning, the perspective of bridging traditional condensed matter theory with machine learning will be broadly valuable.

Accurate Determination of the Quasiparticle and Scaling Properties Surrounding the Quantum Critical Point of Disordered Three-Dimensional Dirac Semimetals

Exploiting the enabling power of the Lanczos method in momentum space, we determine accurately the quasiparticle and scaling properties of disordered three-dimensional Dirac semimetals surrounding the quantum critical point separating the semimetal and diffusive metal regimes. We unveil that the imaginary part of the quasiparticle self-energy obeys a common power law before, at, and after the quantum phase transition, but the power law is nonuniversal, whose exponent is dependent on the disorder strength. More intriguingly, whereas a common power law is also found for the real part of the self-energy before and after the phase transition, a distinctly different behavior is identified at the critical point, characterized by the existence of a nonanalytic logarithmic singularity. This nonanalytical correction serves as the very basis for the unusual power-law behaviors of the quasiparticles and many other physical properties surrounding the quantum critical point. Our approach also allows the ready and reliable determination of the scaling properties of the correlation length and dynamical exponents. We further show that the central findings are valid for both uncorrelated and correlated disorder distributions and should be directly comparable with future experimental observations.

Finite-size scaling in a 2D disordered electron gas with spectral nodes

We study the DC conductivity of a weakly disordered 2D electron gas with two bands and spectral nodes, employing the field theoretical version of the Kubo-Greenwood conductivity formula. Disorder scattering is treated within the standard perturbation theory by summing up ladder and maximally crossed diagrams. The emergent gapless (diffusion) modes determine the behavior of the conductivity on large scales. We find a finite conductivity with an intermediate logarithmic finite-size scaling towards smaller conductivities but do not obtain the logarithmic divergence of the weak-localization approach. Our results agree with the experimentally observed logarithmic scaling of the conductivity in graphene with the formation of a plateau near [Formula: see text].

Large discrete jumps observed in the transition between Chern states in a ferromagnetic topological insulator

A striking prediction in topological insulators is the appearance of the quantized Hall resistance when the surface states are magnetized. The surface Dirac states become gapped everywhere on the surface, but chiral edge states remain on the edges. In an applied current, the edge states produce a quantized Hall resistance that equals the Chern number C = ±1 (in natural units), even in zero magnetic field. This quantum anomalous Hall effect was observed by Chang et al. With reversal of the magnetic field, the system is trapped in a metastable state because of magnetic anisotropy. We investigate how the system escapes the metastable state at low temperatures (10 to 200 mK). When the dissipation (measured by the longitudinal resistance) is ultralow, we find that the system escapes by making a few very rapid transitions, as detected by large jumps in the Hall and longitudinal resistances. Using the field at which the initial jump occurs to estimate the escape rate, we find that raising the temperature strongly suppresses the rate. From a detailed map of the resistance versus gate voltage and temperature, we show that dissipation strongly affects the escape rate. We compare the observations with dissipative quantum tunneling predictions. In the ultralow dissipation regime, two temperature scales (T 1 ~ 70 mK and T 2 ~ 145 mK) exist, between which jumps can be observed. The jumps display a spatial correlation that extends over a large fraction of the sample.

Unconventional band structure for a periodically gated surface of a three-dimensional topological insulator

The surface states of the three-dimensional (3D) topological insulators are described by a two-dimensional (2D) massless dirac equation. A gate-voltage-induced one-dimensional potential barrier on such surfaces creates a discrete bound state in the forbidden region outside the dirac cone. Even for a single barrier it is shown that such a bound state can create an electrostatic analogue of Shubnikov de Haas oscillation which can be experimentally observed for relatively smaller size samples. However, when these surface states are exposed to a periodic arrangement of such gate-voltage-induced potential barriers, the band structure of the same was significantly modified. This is expected to significantly alter the properties of the macroscopic system. We also suggest that, within suitable limits, the system may offer ways to control electron spin electrostatically, which may be practically useful.

Critical transport in weakly disordered semiconductors and semimetals

Motivated by Weyl semimetals and weakly doped semiconductors, we study transport in a weakly disordered semiconductor with a power-law quasiparticle dispersion ξ_{k}∝k^{α}. We show, that in 2α dimensions short-correlated disorder experiences logarithmic renormalization from all energies in the band. We study the case of a general dimension d using a renormalization group, controlled by an ϵ=2α-d expansion. Above the critical dimensions, conduction exhibits a localization-delocalization phase transition or a sharp crossover (depending on the symmetries of the Hamiltonian) as a function of disorder strength. We utilize this analysis to compute the low-temperature conductivity in Weyl semimetals and weakly doped semiconductors near and below the critical disorder point.

Effect of chiral symmetry on chaotic scattering from Majorana zero modes

In many of the experimental systems that may host Majorana zero modes, a so-called chiral symmetry exists that protects overlapping zero modes from splitting up. This symmetry is operative in a superconducting nanowire that is narrower than the spin-orbit scattering length, and at the Dirac point of a superconductor-topological insulator heterostructure. Here we show that chiral symmetry strongly modifies the dynamical and spectral properties of a chaotic scatterer, even if it binds only a single zero mode. These properties are quantified by the Wigner-Smith time-delay matrix Q=-iℏS^{†}dS/dE, the Hermitian energy derivative of the scattering matrix, related to the density of states by ρ=(2πℏ)^{-1}TrQ. We compute the probability distribution of Q and ρ, dependent on the number ν of Majorana zero modes, in the chiral ensembles of random-matrix theory. Chiral symmetry is essential for a significant ν dependence.

Chiral anomaly and transport in Weyl metals

We present an overview of our recent work on transport phenomena in Weyl metals, which may be connected to their nontrivial topological properties, particularly to chiral anomaly. We argue that there are two basic phenomena, which are related to chiral anomaly in Weyl metals: anomalous Hall effect (AHE) and chiral magnetic effect (CME). While AHE is in principle present in any ferromagnetic metal, we demonstrate that a magnetic Weyl metal is distinguished from an ordinary ferromagnetic metal by the absence of the extrinsic and the Fermi surface part of the intrinsic contributions to the AHE, as long as the Fermi energy is sufficiently close to the Weyl nodes. The AHE in a Weyl metal is thus shown to be a purely intrinsic, universal property, fully determined by the location of the Weyl nodes in the first Brillouin zone. In other words, a ferromagnetic Weyl metal may be thought of as the only example of a ferromagnetic metal with a purely intrinsic AHE. We further develop a fully microscopic theory of diffusive magnetotransport in Weyl metals. We derive coupled diffusion equations for the total and axial (i.e. node-antisymmetric) charge densities and show that chiral anomaly manifests as a magnetic-field-induced coupling between them. We demonstrate that an experimentally-observable consequence of CME in magnetotransport in Weyl metals is a quadratic negative magnetoresistance, which will dominate all other contributions to magnetoresistance under certain conditions and may be regarded as a smoking-gun transport characteristic, unique to Weyl metals.

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.

Plasmonics in Dirac systems: from graphene to topological insulators

Recent developments in the emerging field of plasmonics in graphene and other Dirac systems are reviewed and a comprehensive introduction to the standard models and techniques is given. In particular, we discuss intrinsic plasmon excitation of single and bilayer graphene via hydrodynamic equations and the random phase approximation, but also comment on double and multilayer structures. Additionally, we address Dirac systems in the retardation limit and also with large spin–orbit coupling including topological insulators. Finally, we summarize basic properties of the charge, current and photon linear response functions in an appendix.

Conductance fluctuations in semiconductor nanostructures

Conductance fluctuations have been seen in semiconductors and graphene for quite some time. It has generally been believed that a universality existed in which the conductance variance was the same for variations in energy and magnetic field, although some experiments have questioned this view. Here, we use numerical simulations to show that fluctuations in magneto-conductance are typically smaller than those in energy by as much as a factor of 3. Moreover, the amplitude of the fluctuations in each case varies with the strength of the random potential.

Weyl semimetal in a topological insulator multilayer

We propose a simple realization of the three-dimensional (3D) Weyl semimetal phase, utilizing a multilayer structure, composed of identical thin films of a magnetically doped 3D topological insulator, separated by ordinary-insulator spacer layers. We show that the phase diagram of this system contains a Weyl semimetal phase of the simplest possible kind, with only two Dirac nodes of opposite chirality, separated in momentum space, in its band structure. This Weyl semimetal has a finite anomalous Hall conductivity and chiral edge states and occurs as an intermediate phase between an ordinary insulator and a 3D quantum anomalous Hall insulator. We find that the Weyl semimetal has a nonzero dc conductivity at zero temperature, but Drude weight vanishing as T(2), and is thus an unusual metallic phase, characterized by a finite anomalous Hall conductivity and topologically protected edge states.

Symmetries of multifractal spectra and field theories of Anderson localization

We uncover the field-theoretical origin of symmetry relations for multifractal spectra at Anderson transitions and at critical points of other disordered systems. We show that such relations follow from the conformal invariance of the critical theory, which implies their general character. We also demonstrate that for the Anderson localization problem the entire probability distribution for the local density of states possesses a symmetry arising from the invariance of correlation functions of the underlying nonlinear σ model with respect to the Weyl group of the target space of the model.

Renormalization group aspects of graphene

Graphene is a two-dimensional crystal of carbon atoms with fascinating electronic and morphological properties. The low-energy excitations of the neutral, clean system are described by a massless Dirac Hamiltonian in (2+1) dimensions, which also captures the main electronic and transport properties. A renormalization group analysis sheds light on the success of the free model: owing to the special form of the Fermi surface that reduces to two single points in momentum space, short-range interactions are irrelevant and only gauge interactions such as long-range Coulomb or effective disorder can play a role in the low-energy physics. We review these features and briefly discuss other aspects related to disorder and to the bilayer material along the same lines.

Localization properties of random-mass Dirac fermions from real-space renormalization group

Localization properties of random-mass Dirac fermions for a realization of mass disorder, commonly referred to as the Cho-Fisher model, are studied on the D-class chiral network. We show that a simple renormalization group (RG) description captures accurately a rich phase diagram: thermal metal and two insulators with quantized σ(xy), as well as transitions (including critical exponents) between them. Our main finding is that, even with small transmission of nodes, the RG block exhibits a sizable portion of perfect resonances. Delocalization occurs by proliferation of these resonances to larger scales. Evolution of the thermal conductance distribution towards a metallic fixed point is synchronized with evolution of signs of transmission coefficients, so that delocalization is accompanied with sign percolation.

Backscattering of Dirac fermions in HgTe quantum wells with a finite gap

The density-dependent mobility of n-type HgTe quantum wells with inverted band ordering has been studied both experimentally and theoretically. While semiconductor heterostructures with a parabolic dispersion exhibit an increase in mobility with carrier density, high-quality HgTe quantum wells exhibit a distinct mobility maximum. We show that this mobility anomaly is due to backscattering of Dirac fermions from random fluctuations of the band gap (Dirac mass). Our findings open new avenues for the study of Dirac fermion transport with finite and random mass, which so far has been hard to access.

Aspects of the theory of graphene

Following a brief review of the device-friendly features of graphene, recent work on its Green's functions with and without a normal magnetic field are discussed, for an infinite graphene sheet and also for a quantum dot, with analyses of the Landau-quantized energy spectra of the sheet and dot. The random phase approximation dielectric response of graphene is reviewed and discussed in connection with the van der Waals interactions of a graphene sheet with atoms/molecules and with a second graphene sheet in a double layer. Energy-loss spectroscopy for a graphene sheet subject to both parallel and perpendicular particle probes of its dynamic, non-local response properties are also treated. Furthermore, we discuss recent work on the coupling of a graphene plasmon and a surface plasmon, yielding a collective plasma mode that is linear in wavenumber. Finally, we discuss the unusual aspects of graphene conduction and recent work on diffusive charge transport in graphene, in both the DC and AC regimes.

Thinnest two-dimensional nanomaterial-graphene for solar energy

Graphene is a rapidly rising star in materials science. This two-dimensional material exhibits unique properties, such as low resistance, excellent optical transmittance, and high mechanical and chemical stabilities. These exceptional advantages possess great promise for its potential applications in photovoltaic devices. In this Review, we present the status of graphene research for solar energy with emphasis on solar cells. Firstly, the preparation and properties of graphene are described. Secondly, applications of graphene as transparent conductive electrodes and counter electrodes are presented. Thirdly, graphene-based electron- (or hole) accepting materials for solar energy conversion are evaluated. Fourthly, the promoting effect of graphene on photovoltaic devices and the photocatalytic property of graphene-semiconductor composites are discussed. Finally, the challenges to increase the power conversion efficiency of graphene-based solar cells are explored.

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.

Disorder in a quantum spin liquid: flux binding and local moment formation

We study the consequences of disorder in the Kitaev honeycomb model, considering both site dilution and exchange randomness. We show that a single vacancy binds a flux and induces a local moment. This moment is polarized by an applied field h: in the gapless phase, for small h the local susceptibility diverges as χ(h)∼ln(1/h); for a pair of nearby vacancies on the same sublattice, this even increases to χ(h)∼1/(h[ln(1/h)](3/2)). By contrast, weak exchange randomness does not qualitatively alter the susceptibility but has its signature in the heat capacity, which in the gapless phase is power law in temperature with an exponent dependent on disorder strength.

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.

Dynamic conductivity in graphene beyond linear response

The independence of the dynamic conductivity of intrinsic graphene of frequency takes its origin in the compensation of the vanishing density of states by the diverging matrix element of the corresponding interband transition. The applicability of the linear response approach, however, breaks down when this matrix element becomes comparable with the inverse electron lifetime. We show that the physics of the ac conductivity in this regime is determined by Rabi oscillations and obtain it beyond the first-order perturbation theory. Under strong applied electric fields, the induced current eventually saturates at a value determined by the frequency and the lifetime. We also calculate the electromagnetic response of a graphene sheet and find that the optical transparency is increased by the nonlinear effects and make experimental predictions.

Optical conductivity and electron-hole pair creation in graphene

The optical conductivity of graphene is studied in detail. It is shown how crossover from nonlinear to linear behaviour of the current density occurs when the frequency of the external electric field is increased. We also study electron-hole pair creation in bilayer graphene and show that its rate in a static electric field is determined by a power law as for single-layer graphene but with a different exponent. Subsequently, transport in carbon nanotubes is studied.