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We describe the mean-field model of a one-dimensional topological superconductor with two orbitals per unit cell. Time-reversal symmetry is absent, but a nonsymmorphic symmetry, involving a translation by a fraction of the unit cell, mimics the role of time-reversal symmetry. As a result, the topological superconductor has $\mathbb{Z}_4$ topological phases, two which support Majorana bound states and two which do not, in agreement with a prediction based on K-theory classification [K. Shiozaki et al., Phys. Rev. B 93, 195413 (2016)]. As with the Kitaev chain, the presence of Majorana bound states gives rise to the $4\pi$-periodic Josephson effect. A random matrix with nonsymmorphic time-reversal symmetry may be block diagonalized, and every individual block has time-reversal symmetry described by one of the Gaussian orthogonal, unitary or symplectic ensembles. We show how this is manifested in the energy level statistics of a random system in the $\mathbb{Z}_4$ class as the spatial period of the nonsymmorphic symmetry is varied from much less than to of the order of the system size., Comment: 8 pages, 4 figures, plus supplementary 11 pages

We model the electronic properties of thin films of binary compounds with stacked layers where each layer is a two-dimensional honeycomb lattice with two atoms per unit cell. The two atoms per cell are assigned different onsite energies in order to consider six different stacking orders: ABC, ABA, AA, ABC$^{\prime}$, ABA$^{\prime}$, and AA$^{\prime}$. Using a minimal tight-binding model with nearest-neighbor hopping, we consider whether a fault in the texture of onsite energies in the vertical, stacking direction supports localized states, and we find localized states within the bulk band gap for ABC, ABA, and AA$^{\prime}$ stacking. Depending on the stacking type, parameter values, and whether the soliton is atomically sharp or a smooth texture, there are a range of different band structures including soliton bands that are either isolated or that hybridize with other states, such as surface states, and soliton bands that are either dispersive or flat, the latter yielding narrow features in the density of states. We discuss the relevance of our results to specific materials including graphene, hexagonal boron nitride and other binary compounds., Comment: 29 pages, 18 figures

We model the influence of an in-plane magnetic field on the orbital motion of electrons in rhombohedral graphene multilayers. For zero field, the low-energy band structure includes a pair of flat bands near zero energy which are localized on the surface layers of a finite thin film. For finite field, we find that the zero-energy bands persist and that level bifurcations occur at energies determined by the component of the in-plane wave vector $q$ that is parallel to the external field. The occurrence of level bifurcations is explained by invoking semiclassical quantization of the zero field Fermi surface of rhombohedral graphite. We find parameter regions with a single isoenergetic contour of Berry phase zero corresponding to a conventional Landau level spectrum and regions with two isoenergetic contours, each of Berry phase $\pi$, corresponding to a Dirac-like spectrum of levels. We write down an analogous one-dimensional tight-binding model and relate the persistence of the zero-energy bands in large magnetic fields to a soliton texture supporting zero-energy states in the Su-Schreiffer-Heeger model. We show that different states contributing to the zero-energy flat bands in rhombohedral graphene multilayers in a large field, as determined by the wave vector $q$, are localized on different bulk layers of the system, not just the surfaces., Comment: 12 pages, 11 figures

We classify Hermitian tight-binding models describing noninteracting electrons on a one-dimensional periodic lattice with two energy bands. To do this, we write a generalized Rice-Mele model with two orbitals per unit cell, including all possible complex-valued long-range hoppings consistent with Hermicity. We then apply different forms of time-reversal, charge-conjugation and chiral symmetry in order to constrain the parameters, resulting in an array of possible models in different symmetry classes. For each symmetry class, we define a single, canonical form of the Hamiltonian and identify models that are related to the canonical form by an off-diagonal unitary transformation in the atomic basis. The models have either symmorphic or nonsymmorphic nonspatial symmetries (time $T$, chiral and charge-conjugation). The nonsymmorphic category separates into two types of state of matter: an insulator with a $\mathbb{Z}_2$ topological index in the absence of nonsymmorphic time-reversal symmetry or, in the presence of nonsymmorphic time-reversal symmetry, a metallic state. The latter is an instance of Kramer's degeneracy with one degeneracy point in the Brillouin zone as opposed to no degeneracy points in symmorphic systems with $T^2 = 1$ and two in symmorphic systems with $T^2 = - 1$., Comment: 27 pages, 12 figures

The Rice-Mele model has two topological and spatially-inversion symmetric phases, namely the Su-Schrieffer-Heeger (SSH) phase with alternating hopping only, and the charge-density-wave (CDW) phase with alternating energies only. The chiral symmetry of the SSH phase is robust in position space, so that it is preserved in the presence of the ends of a finite system and of textures in the alternating hopping. However, the chiral symmetry of the CDW wave phase is nonsymmorphic, resulting in a breaking of the bulk topology by an end or a texture in the alternating energies. We consider the presence of solitons (textures in position space separating two degenerate ground states) in finite systems with open boundary conditions. We identify the parameter range under which an atomically-sharp soliton in the CDW phase supports a localized state which lies within the band gap, and we calculate the expectation value $p_y$ of the nonsymmorphic chiral operator for this state, and the soliton electric charge. As the spatial extent of the soliton increases beyond the atomic limit, the energy level approaches zero exponentially quickly or inversely proportionally to the width, depending on microscopic details of the soliton texture. In both cases, the difference of $p_y$ from one is inversely proportional to the soliton width, while the charge is independent of the width. We investigate the robustness of the soliton level in the presence of disorder and sample-to-sample parameter variations, comparing with a single soliton level in the SSH phase with an odd number of sites., Comment: 15 pages, 12 figures

We consider the heat equation for monolayer two-dimensional materials in the presence of heat flow into a substrate and Joule heating due to electrical current. We compare devices including a nanowire of constant width and a bow tie (or wedge) constriction of varying width, and we derive approximate one-dimensional heat equations for them; a bow tie constriction is described by the modified Bessel equation of zero order. We compare steady state analytic solutions of the approximate equations with numerical results obtained by a finite element method solution of the two-dimensional equation. Using these solutions, we describe the role of thermal conductivity, thermal boundary resistance with the substrate and device geometry. The temperature in a device at fixed potential difference will remain finite as the width shrinks, but will diverge for fixed current, logarithmically with width for the bow tie as compared to an inverse square dependence in a nanowire., Comment: 20 pages, 6 figures

We apply the mean-field Hartree Fock theory of gapped electronic states at charge neutrality in bilayer graphene to thin films of rhombohedral graphite with up to thirty layers. For the ground state, the order parameter (the separation of bands at the valley center) saturates to a constant non-zero value as the layer number increases, whereas the band gap decreases with layer number. We consider chiral symmetry breaking disorder in the form of random layer potentials and chiral preserving disorder in the form of random values of the interlayer coupling. The former reduces the magnitude of the mean band gap whereas the latter has a negligible effect, which is due to self-averaging within a film with a large number of layers. We determine the ground state in the presence of an individual stacking fault which results in two pairs of low-energy bands and we identify two separate order parameters. One of them determines the band gap at zero temperature, the other determines the critical temperature leading, overall, to a temperature dependence of the band gap that is distinct to that of pristine rhombohedral graphite. In the presence of stacking faults, each individual rhombohedral section with $m$ layers contributes a pair of low-energy flat bands producing a peak in the Berry curvature located at a characteristic $m$-dependent wave vector. The Chern number per spin-valley flavor for the filled valence bands in the ground state is equal in magnitude to the total number of layers divided by two, the same value as for pristine rhombohedral graphite., Comment: 16 pages, 15 figures

Topologically non-trivial states characterized by Berry curvature appear in a number of materials ranging from spin-orbit-coupling driven topological insulators to graphene. In multivalley conductors, such as mono- and bilayer graphene, despite a zero total Chern number for the entire Brillouin zone, Berry curvature with different signs concentrated in different valleys can affect the observable material's transport characteristics. Here we consider thin films of rhombohedral graphite, which appear to retain truly two-dimensional properties up to tens of layers of thickness and host two-dimensional electron states with a large Berry curvature, accompanied by a giant intrinsic magnetic moment carried by electrons. The size of Berry curvature and magnetization in the vicinity of each valley can be controlled by electrostatic gating leading to a tuneable anomalous Hall effect and a peculiar structure of the two-dimensional Landau level spectrum., Comment: 24 pages, 7 figures

The influence of nanostructuring and quantum confinement on the thermoelectric properties of materials has been extensively studied. While this has made possible multiple breakthroughs in the achievable figure of merit, classical confinement, and its effect on the local Seebeck coefficient has mostly been neglected, as has the Peltier effect in general due to the complexity of measuring small temperature gradients locally. Here we report that reducing the width of a graphene channel to 100 nm changes the Seebeck coefficient by orders of magnitude. Using a scanning thermal microscope allows us to probe the local temperature of electrically contacted graphene two-terminal devices or to locally heat the sample. We show that constrictions in mono- and bilayer graphene facilitate a spatially correlated gradient in the Seebeck and Peltier coefficient, as evidenced by the pronounced thermovoltage $V_{th}$ and heating/cooling response $\Delta T_{Peltier}$, respectively. This geometry dependent effect, which has not been reported previously in 2D materials, has important implications for measurements of patterned nanostructures in graphene and points to novel solutions for effective thermal management in electronic graphene devices or concepts for single material thermocouples.

We model the magnetic ratchet effect in bilayer graphene in which a dc electric current is produced by an ac electric field of frequency $\omega$ in the presence of a steady in-plane magnetic field and inversion-symmetry breaking. In bilayer graphene, the ratchet effect is tunable by an external metallic gate which breaks inversion symmetry. For zero in-plane magnetic field, we show that trigonal warping and inversion-symmetry breaking are able to produce a large dc valley current, but not a non-zero total dc charge current. For the magnetic ratchet in a tilted magnetic field, the perpendicular field component induces cyclotron motion with frequency $\omega_c$ and we find that the dc current displays cyclotron resonance at $\omega_c = \omega$, although this peak in the current is actually smaller than its value at $\omega_c = 0$. Second harmonic generation, however, is greatly enhanced by resonances at $\omega_c = \omega$ and $\omega_c = 2\omega$ for which the current is generally much larger than at $\omega_c = 0$., Comment: 13 pages, 2 figures, added section about valley currents

We explore the electronic ground states of Bernal-stacked multilayer graphenes using the Hartree-Fock mean-field approximation and the full-parameter band model. We find that the electron-electron interaction tends to open a band gap in multilayer graphenes from bilayer to 8-layer, while the nature of the insulating ground state sensitively depends on the band parameter $\gamma_2$, which is responsible for the semimetallic nature of graphite. In 4-layer graphene, particularly, the ground state assumes an odd-spatial-parity staggered phase at $\gamma_2 = 0$, while an increasing, finite value of $\gamma_2$ stabilizes a different state with even parity, where the electrons are attracted to the top layer and the bottom layer. The two phases are topologically distinct insulating states with different Chern numbers, and they can be distinguished by spin or valley Hall conductivity measurements. Multilayers with more than five layers also exhibit similar ground states with potential minima at the outermost layers, although the opening of a gap in the spectrum as a whole is generally more difficult than in 4-layer because of a larger number of energy bands overlapping at the Fermi energy., Comment: 8 pages, 6 figures

Close to charge neutrality, the low-energy properties of high-quality suspended devices based on atomically thin graphene layers are determined by electron-electron interactions. Bernal-stacked layers, in particular, have shown a remarkable even-odd effect with mono- and tri-layers remaining gapless conductors, and bi- and tetra-layers becoming gapped insulators. These observations $-$at odds with the established notion that (Bernal) trilayers and thicker multilayers are semi-metals$-$ have resulted in the proposal of a physical scenario leading to a surprising prediction, namely that even-layered graphene multilayers remain insulating irrespective of their thickness. Here, we present data from two devices that conform ideally to this hypothesis, exhibiting the behavior expected for Bernal-stacked hexa and octalayer graphene. Despite their large thickness, these multilayers are insulating for carrier density |n|<2-3x10$^{10}$cm$^{-2}$, possess an energy gap of approximately 1.5 meV at charge neutrality (in virtually perfect agreement with what is observed in bi and tetra layer graphene) and exhibit the expected integer quantum Hall effect. These findings indicate the soundness of our basic insights on the effect of electron interactions in Bernal graphene multilayers, show that graphene multilayers exhibit unusual and interesting physics that remains to be understood, and pose ever more pressing questions as to the microscopic mechanisms behind the semimetallic behavior of bulk graphite., Comment: 22 pages, 6 figures, To appear in 2D Materials

We consider the orbital effect of an in-plane magnetic field on electrons in bilayer graphene, deriving linear-in-field contributions to the low-energy Hamiltonian arising from the presence of either skew interlayer coupling or interlayer potential asymmetry, the latter being tunable by an external metallic gate. To illustrate the relevance of such terms, we consider the ratchet effect in which a dc current results from the application of an alternating electric field in the presence of an in-plane magnetic field and inversion-symmetry breaking. By comparison with recent experimental observations in monolayer graphene [C. Drexler et al., Nature Nanotech. 8, 104 (2013)], we estimate that the effect in bilayer graphene can be two orders of magnitude greater than that in monolayer, illustrating that the bilayer is an ideal material for the realization of optoelectronic effects that rely on inversion-symmetry breaking., Comment: 7 pages, 3 figures

The absence of an energy gap separating valence and conduction bands makes the low-energy electronic properties of graphene and its multi-layers sensitive to electron-electron interactions. In bilayers, for instance, interactions are predicted to open a gap at charge neutrality, turning the system into an insulator, as observed experimentally. In mono and (Bernal-stacked) trilayers, interactions, although still important, do not have an equally drastic effect, and these systems remain conducting at low temperature. It may be expected that interaction effects become weaker for thicker multilayers, whose behavior should eventually converge to that of graphite. Here we show that this expectation does not correspond to reality by investigating the case of Bernal-stacked tetralayer graphene (4LG). We reveal the occurrence of a robust insulating state in a narrow range of carrier densities around charge neutrality, incompatible with the behavior expected from the single-particle band structure. The phenomenology resembles that observed in bilayers, but the stronger conductance suppression makes the insulating state in 4LG visible at higher temperature. To account for our findings, we suggest a natural generalization of the interaction-driven, symmetry-broken states proposed for bilayers. This generalization also explains the systematic even-odd effect of interactions in Bernal-stacked layers of different thickness that is emerging from experiments, and has implications for the multilayer-to-graphite crossover., Comment: 15 pages, 5 figures, This is the original submitted version of the manuscript whose final accepted version, following the review/editorial process, will appear in Nature Communications

We study the electronic structure of multilayer graphenes with a mixture of Bernal and rhombohedral stacking and propose a general scheme to understand the electronic band structure of an arbitrary configuration. The system can be viewed as a series of finite Bernal graphite sections connected by stacking faults. We find that the low-energy eigenstates are mostly localized in each Bernal section, and, thus, the whole spectrum is well approximated by a collection of the spectra of independent sections. The energy spectrum is categorized into linear, quadratic and cubic bands corresponding to specific eigenstates of Bernal sections. The ensemble-averaged spectrum exhibits a number of characteristic discrete structures originating from finite Bernal sections or their combinations likely to appear in a random configuration. In the low-energy region, in particular, the spectrum is dominated by frequently-appearing linear bands and quadratic bands with special band velocities or curvatures. In the higher energy region, band edges frequently appear at some particular energies, giving optical absorption edges at corresponding characteristic photon frequencies., Comment: 13 pages, 10 figures

We review the electronic properties of bilayer graphene, beginning with a description of the tight-binding model of bilayer graphene and the derivation of the effective Hamiltonian describing massive chiral quasiparticles in two parabolic bands at low energy. We take into account five tight-binding parameters of the Slonczewski-Weiss-McClure model of bulk graphite plus intra- and interlayer asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree model of screening and band-gap opening due to interlayer asymmetry in the presence of external gates is presented. The tight-binding model is used to describe optical and transport properties including the integer quantum Hall effect, and we also discuss orbital magnetism, phonons and the influence of strain on electronic properties. We conclude with an overview of electronic interaction effects., Comment: review, 31 pages, 15 figures

The tight-binding model of electrons in graphene is reviewed. We derive low-energy Hamiltonians supporting massless Dirac-like chiral fermions and massive chiral fermions in monolayer and bilayer graphene, respectively, and we describe how their chirality is manifest in the sequencing of plateaus observed in the integer quantum Hall effect. The opening of a tuneable band gap in bilayer graphene in response to a transverse electric field is described, and we explain how Hartree theory may be used to develop a simple analytical model of screening., Comment: tutorial-style introduction, 40 pages, 17 figures

We show that the influence of spin-orbit (SO) coupling on the weak localization effect for electrons in graphene depends on the lack or presence of z/-z symmetry in the system. While for z/-z asymmetric SO coupling, disordered graphene should display a weak anti-localization behavior at lowest temperature, z/-z symmetric coupling leads to an effective saturation of decoherence time which can be partially lifted by an in-plane magnetic field, thus, tending to restore the weak localization effect., Comment: 5 pages, 1 figure

The Landau level spectra and the quantum Hall effect of ABA-stacked multilayer graphenes are studied in the effective mass approximation. The low-energy effective mass Hamiltonian may be partially diagonalized into an approximate block-diagonal form, with each diagonal block contributing parabolic bands except, in a multilayer with an odd number of layers, for an additional block describing Dirac-like bands with a linear dispersion. We fully include the band parameters and, taking into account the symmetry of the lattice, we analyze their affect on the block-diagonal Hamiltonian. Next-nearest layer couplings are shown to be particularly important in determining the low-energy spectrum and the phase diagram of the quantum Hall conductivity, by causing energy shifts, level anti-crossings, and valley splitting of the low-lying Landau levels., Comment: 9 pages, 4 figures

Since the lattices of ABA-stacked graphene multilayers with an even number of layers, as well as that of monolayer graphene, satisfy spatial-inversion symmetry, their electronic bands must be spin degenerate in the presence of time-inversion symmetry. In intrinsic monolayer and bilayer graphene, when symmetry is not broken by external fields, the only spin-orbit coupling present at low energy near the corner of the Brillouin zone is the Kane-Mele term, that opens a bulk energy gap but does not break the spin degeneracy of the energy bands [C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005)]. However, spin splitting is allowed in multilayers with an odd number of layers (greater than or equal to 3) because their lattices do not satisfy spatial inversion symmetry. We show that, in trilayer graphene, in addition to the Kane-Mele term, there is a second type of intrinsic spin-orbit coupling present at low energy near the corner of the Brillouin zone. It introduces a Zeeman-like spin splitting of the energy bands at each valley, with an opposite sign of the effective magnetic field in the two valleys. We estimate the magnitude of the effective field to be ~2T., Comment: 4 pages, 1 figure

We study spatial symmetry in general ABA-stacked multilayer graphene to illustrate how electronic spectra at the two valleys are related in a magnetic field. We show that the lattice of multilayers with an even number of layers, as well as that of monolayer graphene, satisfy spatial inversion symmetry, which rigorously guarantees valley degeneracy in the absence of time-reversal symmetry. A multilayer with an odd number of layers (three or more) lacks inversion symmetry, but there is another transformation imposing an approximate valley degeneracy, which arises because the low-energy Hamiltonian consists of separate monolayerlike and bilayerlike parts. We show that an external electrostatic potential generally breaks valley degeneracy in a magnetic field, in a markedly different manner in odd and even multilayers., Comment: 6 pages, 3 figures

The electronic band structure of ABC-stacked multilayer graphene is studied within an effective mass approximation. The electron and hole bands touching at zero energy support chiral quasiparticles characterized by Berry's phase N pi for N-layers, generalizing the low-energy band structure of monolayer and bilayer graphene. We investigate the trigonal-warping deformation of the energy bands and show that the Lifshitz transition, in which the Fermi circle breaks up into separate parts at low energy, reflects Berry's phase N pi. It is particularly prominent in trilayers, N=3, with the Fermi circle breaking into three parts at a relatively large energy that is related to next-nearest-layer coupling. For N=3, we study the effects of electrostatic potentials which vary in the stacking direction, and find that a perpendicular electric field, as well as opening an energy gap, strongly enhances the trigonal-warping effect. In magnetic fields, the N=3 Lifshitz transition is manifested as a coalescence of Landau levels into triply-degenerate levels., Comment: 9 pages, 5 figures

We propose a non-magnetic, pseudospin-based version of a spin valve, in which the pseudospin polarization in neighboring regions of a graphene bilayer is controlled by external gates. Numerical calculations demonstrate a large on-off ratio of such a device. This finding holds promise for the realization of pseudospintronics: a form of electronics based upon the manipulation of pseudospin analogous to the control of physical spin in spintronics applications., Comment: Published version, 4 pages, 3 figures

We calculate the electronic band structure of ABA-stacked trilayer graphene in the presence of external gates, using a self-consistent Hartree approximation to take account of screening. In the absence of a gate potential, there are separate pairs of linear and parabolic bands at low energy. A gate field perpendicular to the layers breaks mirror reflection symmetry with respect to the central layer and hybridizes the linear and parabolic low-energy bands, leaving a chiral Hamiltonian essentially different from that of monolayer or bilayer graphene. Using the self-consistent Born approximation, we find that the density of states and the minimal conductivity in the presence of disorder generally increase as the gate field increases, in sharp contrast with bilayer graphene., Comment: 5 pages, 3 figures

A tight binding model is used to calculate the band structure of bilayer graphene in the presence of a potential difference between the layers that opens a gap $\Delta$ between the conduction and valence bands. In particular, a self consistent Hartree approximation is used to describe imperfect screening of an external gate, employed primarily to control the density $n$ of electrons on the bilayer, resulting in a potential difference between the layers and a density dependent gap $\Delta (n)$. We discuss the influence of a finite asymmetry gap $\Delta (0)$ at zero excess density, caused by the screening of an additional transverse electric field, on observations of the quantum Hall effect., Comment: 4 pages, 3 eps figures

We derive an effective two-dimensional Hamiltonian to describe the low energy electronic excitations of a graphite bilayer, which correspond to chiral quasiparticles with a parabolic dispersion exhibiting Berry phase $2\pi$. Its high-magnetic-field Landau level spectrum consists of almost equidistant groups of four-fold degenerate states at finite energy and eight zero-energy states. This can be translated into the Hall conductivity dependence on carrier density, $\sigma_{xy}(N)$, which exhibits plateaus at integer values of $4e^{2}/h$ and has a ``double'' $8e^{2}/h$ step between the hole and electron gases across zero density, in contrast to $(4n+2)e^{2}/h$ sequencing in a monolayer., Comment: 4 pages, 4 eps figures

We present a theoretical study of degeneracy breaking due to short-ranged impurities in finite, single-wall, metallic carbon nanotubes. The effective mass model is used to describe the slowly varying spatial envelope wavefunctions of spinless electrons near the Fermi level at two inequivalent valleys (K-points) in terms of the four component Dirac equation for massless fermions, with the role of spin assumed by pseudospin due to the relative amplitude of the wave function on the sublattice atoms (``A'' and ``B''). Using boundary conditions at the ends of the tube that neither break valley degeneracy nor mix pseudospin eigenvectors, we use degenerate perturbation theory to show that the presence of impurities has two effects. Firstly, the position of the impurity with respect to the spatial variation of the envelope standing waves results in a sinusoidal oscillation of energy level shift as a function of energy. Secondly, the position of the impurity within the hexagonal graphite unit cell produces a particular 4 by 4 matrix structure of the corresponding effective Hamiltonian. The symmetry of this Hamiltonian with respect to pseudospin flip is related to degeneracy breaking and, for an armchair tube, the symmetry with respect to mirror reflection in the nanotube axis is related to pseudospin mixing., Comment: 20 pages, 10 eps figures