Mutual Information and Correlations across Topological Phase Transitions in Topologically Ordered Graphene Zigzag Nanoribbons.

Graphene zigzag nanoribbons, initially in a topologically ordered state, undergo a topological phase transition into crossover phases distinguished by quasi-topological order. We computed mutual information for both the topologically ordered phase and its crossover phases, revealing the following results: (i) In the topologically ordered phase, A-chirality carbon lines strongly entangle with B-chirality carbon lines on the opposite side of the zigzag ribbon. This entanglement persists but weakens in crossover phases. (ii) The upper zigzag edge entangles with non-edge lines of different chirality on the opposite side of the ribbon. (iii) Entanglement increases as more carbon lines are grouped together, regardless of the lines' chirality. No long-range entanglement was found in the symmetry-protected phase in the absence of disorder.

New disordered anyon phase of doped graphene zigzag nanoribbon.

We investigate interacting disordered zigzag nanoribbons at low doping, using the Hubbard model to treat electron interactions within the density matrix renormalization group and Hartree-Fock method. Extra electrons that are inserted into an interacting disordered zigzag nanoribbon divide into anyons. Furthermore, the fractional charges form a new disordered anyon phase with a highly distorted edge spin density wave, containing numerous localized magnetic moments residing on the zigzag edges, thereby displaying spin-charge separation and a strong non-local correlation between the opposite zigzag edges. We make the following new predictions, which can be experimentally tested: (1) In the low doping case and weak disorder regime, the soft gap in the tunneling density of states is replaced by a sharp peak at the midgap energy with two accompanying peaks. The [Formula: see text] fractional charges that reside on the boundary of the zigzag edges are responsible for these peaks. (2) We find that the midgap peak disappears as the doping concentration increases. The presence of [Formula: see text] fractional charges will be strongly supported by the detection of these peaks. Doped zigzag ribbons may also exhibit unusual transport, magnetic, and inter-edge tunneling properties.

Soliton Fractional Charges in Graphene Nanoribbon and Polyacetylene: Similarities and Differences.

An introductory overview of current research developments regarding solitons and fractional boundary charges in graphene nanoribbons is presented. Graphene nanoribbons and polyacetylene have chiral symmetry and share numerous similar properties, e.g., the bulk-edge correspondence between the Zak phase and the existence of edge states, along with the presence of chiral boundary states, which are important for charge fractionalization. In polyacetylene, a fermion mass potential in the Dirac equation produces an excitation gap, and a twist in this scalar potential produces a zero-energy chiral soliton. Similarly, in a gapful armchair graphene nanoribbon, a distortion in the chiral gauge field can produce soliton states. In polyacetylene, a soliton is bound to a domain wall connecting two different dimerized phases. In graphene nanoribbons, a domain-wall soliton connects two topological zigzag edges with different chiralities. However, such a soliton does not display spin-charge separation. The existence of a soliton in finite-length polyacetylene can induce formation of fractional charges on the opposite ends. In contrast, for gapful graphene nanoribbons, the antiferromagnetic coupling between the opposite zigzag edges induces integer boundary charges. The presence of disorder in graphene nanoribbons partly mitigates antiferromagnetic coupling effect. Hence, the average edge charge of gap states with energies within a small interval is e / 2 , with significant charge fluctuations. However, midgap states exhibit a well-defined charge fractionalization between the opposite zigzag edges in the weak-disorder regime. Numerous occupied soliton states in a disorder-free and doped zigzag graphene nanoribbon form a solitonic phase.

Topological Gap States of Rectangular Armchair Ribbon.

We consider a rectangular graphene armchair ribbon with an excitation gap. The boundary of this system consists of two short zigzag edges and two long armchair edges. Within such a ribbon, topological gap states exist that are localized along the zigzag edges. The end charge of these states is an integer, which can be related to the Zak phase of the periodic armchair ribbon constructed from the rectangular armchair ribbon by connecting its zigzag edges together. In this paper, we provide an explicit analytical computation of the Zak phase of a periodic armchair ribbon, and show that its value is consistent with the integer end charges that are computed numerically. In the presence of a staggered potential, non-integer end charges are possible. We discuss its relation to the Zak phase.

Coulomb Impurity Problem of Graphene in Strong Coupling Regime in Magnetic Fields.

We investigate the Coulomb impurity problem of graphene in strong coupling limit in the presence of magnetic fields. When the strength of the Coulomb potential is sufficiently strong the electron of the lowest energy boundstate of the n = 0 Landau level may fall to the center of the potential. To prevent this spurious effect the Coulomb potential must be regularized. The scaling function for the inverse probability density of this state at the center of the impurity potential is computed in the strong coupling regime. The dependence of the computed scaling function on the regularization parameter changes significantly as the strong coupling regime is approached.

Stability of Anomalous States of a Local Potential in Graphene.

Graphene Landau levels have discrete energies consisting zero energy chiral states and non-zero energy states with mixed chirality. Each Landau level splits into discrete energies when a localized potential is present. A simple scaling analysis suggests that a localized potential can act as a strong perturbation, and that it can be even more singular in graphene than in ordinary two-dimensional systems of massful electrons. Parabolic, Coulomb, and Gaussian potentials in graphene may have anomalous boundstates whose probability density has a sharp peak inside the potential and a broad peak of size magnetic length l outside the potential. The n = 0 Landau level with zero energy has only one anomalous state while the n = ±1 Landau levels with non-zero energy have two (integer quantum number n is related to the quantized Landau level energies). These anomalous states can provide a new magnetospectroscopic feature in impurity cyclotron resonances of graphene. In the present work we investigate quantitatively the conditions under which the anomalous states can exist. These results may provide a guide in searching for anomalous states experimentally.

Impurity cyclotron resonance of anomalous Dirac electrons in graphene.

We have investigated a new feature of impurity cyclotron resonances common to various localized potentials of graphene. A localized potential can interact with a magnetic field in an unexpected way in graphene. It can lead to formation of anomalous boundstates that have a sharp peak with a width R in the probability density inside the potential and a broad peak of size magnetic length ℓ outside the potential. We investigate optical matrix elements of anomalous states and find that they are unusually small and depend sensitively on the magnetic field. The effect of many-body interactions on their optical conductivity is investigated using a self-consistent time-dependent Hartree-Fock approach. For a completely filled Landau level we find that an excited electron-hole pair, originating from the optical transition between two anomalous impurity states, is nearly uncorrelated with other electron-hole pairs, although it displays substantial exchange self-energy effects. This absence of correlation is a consequence of a small vertex correction in comparison to the difference between renormalized transition energies computed within the one electron-hole pair approximation. However, an excited electron-hole pair originating from the optical transition between a normal and an anomalous impurity state can be substantially correlated with other electron-hole states with a significant optical strength.

Optical properties of Dirac electrons in a parabolic well.

A single electron transitor may be fabricated using qunatum dots. A good model for the confinement potential of a quantum dot is a parabolic well. Here we consider such a parabolic dot made of graphene. Recently, we found counter intuitively that resonant quasi-boundstates of both positive and negative energies exist in the energy spectrum. The presence of resonant quasi-boundstates of negative energies is a unique property of massless Dirac fermions. As magnetic field B gets smaller the energy width of these states become broader and for sufficiently weak value of B resonant quasi-bound states disappear into a quasi-continuum. In the limit of small B resonant and nonresonant states transform into discrete anomalous states with a narrow probability density peak inside the well and another broad peak under the potential barrier. In this paper we compute the optical strength between resonant quasi-bound states as a function of B, and investigate how the signature of resonant quasi-bound states of Dirac electrons may appear in optical measurements.

Can a repulsive potential in graphene have boundstates in a magnetic field?

We consider Klein tunneling through a repulsive and cylindrical potential with range R and strength V. Recently it was found that, in the strong coupling regime R/l < 1, the repulsive potential can have bound states peaked inside the potential with tails extending over l mean square root of 2(N+1), where N is Landau level (LL) index and f is the magnetic length. The presence of these bound states is a consequence of a subtle interplay between Klein tunneling and quantization effect of magnetic fields. Because of the presence of these bound states the effective coupling between the repulsive potential and an electron can be attractive. Here we show that this effect is a consequence of singular interaction between the repulsive potential and an electron that cannot be captured in perturbative approaches.

Confinement and deconfinement in the potential of antidot arrays of a massless Dirac electron in magnetic fields.

We have investigated the effect of inter-Landau level mixing on confinement/deconfinement in antidot potentials of states with energies less than the potential height of the antidot array. We find that, depending on the ratio between the size of the antidot R and the magnetic length [Formula: see text], probability densities display confinement or deconfinement in antidot potentials (B is the magnetic field). When R/ℓ < 1 inter-Landau level mixing is strong and probability densities with energy less than the potential height are non-chiral and localized inside antidot potentials. However, in the strong magnetic field limit R/ℓ â‰« 1, where inter-Landau level mixing is small, they are delocalized outside antidot potentials, and are chiral for N = 0 Landau level (LL) states while non-chiral for N = 1. In the non-trivial crossover regime R/ℓ âˆ¼ 1 localized and delocalized probability densities coexist. States that are delocalized outside antidots when R/ℓ > 1 form a nearly degenerate band and their probability densities are independent of k, in contrast to the case of R/ℓ < 1.

Dirac electrons of a split-gate Hall bar.

In this work we study several unusual properties of Klein tunneling through the abrupt and flat barriers of a split-gate Hall bar system of graphene. We show that Klein tunneling of Dirac electrons can be rather strong in such a system, and that a significant electron density can be present under the barrier. It can be shown that the probability wavefunctions for large angular momenta are identical to the probability wavefunctions of the same angular momenta in the absence of the potential barrier, i.e., it is as if the barrier does not exist and the Klein tunneling is complete. This is a unique effect in a magnetic field. We propose that STM measurements may be used to detect the presence of such a density. We have also investigated drift velocity of electrons as the center of probability wavefunction varies from outside to inside of the flat potential barrier, and find a significant deviation from the semiclassical result.

Coupling of surface and lowest Landau level states in a rectangular graphene dot.

A rectangular graphene dot with two zigzag edges and two armchair edges have electronic states in the presence of a magnetic field that are localized on the zigzag edges with non zero values of the wavefunction inside the dot. We have investigated the dependence of these wavefunctions on the size of the dot, and explain the physical origin of them in terms of surface and the lowest Landau level (LLL) states of infinitely long nanoribbons. We find that the armchair edges play a crucial role by coupling the surface and LLL states.

Room-temperature charge stability modulated by quantum effects in a nanoscale silicon island.

We report on transport measurement performed on a room-temperature-operating ultrasmall Coulomb blockade devices with a silicon island of sub5 nm. The charge stability at 300K exhibits a substantial change in slopes and diagonal size of each successive Coulomb diamond, but remarkably its main feature persists even at low temperature down to 5.3K except for additional Coulomb peak splitting. This key feature of charge stability with additional fine structures of Coulomb peaks are successfully modeled by including the interplay between Coulomb interaction, valley splitting, and strong quantum confinement, which leads to several low-energy many-body excited states for each dot occupancy. These excited states become enhanced in the sub5 nm ultrasmall scale and persist even at 300K in the form of cluster, leading to the substantial modulation of charge stability.

Electronic properties of a graphene antidot in magnetic fields.

We report on several unusual properties of a graphene antidot created by a piecewise constant potential in a magnetic field. We find that the total probability of finding the electron in the barrier can be nearly one while it is almost zero outside the barrier. In addition, for each electron state of a graphene antidot there is a dot state with exactly the same wavefunction but with a different energy. This symmetry is a consequence of Klein tunneling of Dirac electrons. Moreover, in zigzag nanoribbons we find strong coupling between some antidot states and zigzag edge states. Experimental tests of these effects are proposed.

Coulomb blockade and kondo effect in a quantum Hall antidot.

We propose a general capacitive model for an antidot, which has two localized edge states with different spins in the quantum Hall regime. The capacitive coupling of localized excess charges, which are generated around the antidot due to magnetic flux quantization, and their effective spin fluctuation can result in Coulomb blockade, h/(2e) Aharonov-Bohm oscillations, and the Kondo effect. The resultant conductance is in qualitative agreement with recent experimental data.