Aavishkar A. Patel

Cloning of Dirac fermions in graphene superlattices

Superlattices have attracted great interest because their use may make it possible to modify the spectra of two-dimensional electron systems and, ultimately, create materials with tailored electronic properties. In previous studies (see, for example, refs 1, 2, 3, 4, 5, 6, 7, 8), it proved difficult to realize superlattices with short periodicities and weak disorder, and most of their observed features could be explained in terms of cyclotron orbits commensurate with the superlattice. Evidence for the formation of superlattice minibands (forming a fractal spectrum known as Hofstadter’s butterfly) has been limited to the observation of new low-field oscillations and an internal structure within Landau levels. Here we report transport properties of graphene placed on a boron nitride substrate and accurately aligned along its crystallographic directions. The substrate’s moiré potential acts as a superlattice and leads to profound changes in the graphene’s electronic spectrum. Second-generation Dirac points appear as pronounced peaks in resistivity, accompanied by reversal of the Hall effect. The latter indicates that the effective sign of the charge carriers changes within graphene’s conduction and valence bands. Strong magnetic fields lead to Zak-type cloning of the third generation of Dirac points, which are observed as numerous neutrality points in fields where a unit fraction of the flux quantum pierces the superlattice unit cell. Graphene superlattices such as this one provide a way of studying the rich physics expected in incommensurable quantum systems and illustrate the possibility of controllably modifying the electronic spectra of two-dimensional atomic crystals by varying their crystallographic alignment within van der Waals heterostuctures.

Generic miniband structure of graphene on a hexagonal substrate

Using a general symmetry-based approach, we provide a classification of generic miniband structures for electrons in graphene placed on substrates with the hexagonal Bravais symmetry. In particular, we identify conditions at which the first moire miniband is separated from the rest of the spectrum by either one or a group of three isolated mini Dirac points and is not obscured by dispersion surfaces coming from other minibands. In such cases, the Hall coefficient exhibits two distinct alternations of its sign as a function of charge carrier density.

Quantum chaos on a critical Fermi surface

Significance All high-temperature superconductors exhibit a remarkable “strange metal” state above their critical temperatures. A theory of the strange metal is a prerequisite for a deeper understanding of high-temperature superconductivity, but the ubiquitous quasiparticle theory of normal metals cannot be extended to the strange metal. Instead, strange metals exhibit many-body chaos over the shortest possible time allowed by quantum theory. We characterize the quantum chaos in a model of fermions at nonzero density coupled to an emergent gauge field. We find a universal relationship between the chaos parameters and the experimentally measurable thermal diffusivity. Our results establish a connection between quantum dephasing and energy transport in states of quantum matter without quasiparticles. We compute parameters characterizing many-body quantum chaos for a critical Fermi surface without quasiparticle excitations. We examine a theory of N species of fermions at nonzero density coupled to a U(1) gauge field in two spatial dimensions and determine the Lyapunov rate and the butterfly velocity in an extended random-phase approximation. The thermal diffusivity is found to be universally related to these chaos parameters; i.e., the relationship is independent of N, the gauge-coupling constant, the Fermi velocity, the Fermi surface curvature, and high-energy details.

Floquet generation of Majorana end modes and topological invariants

We show how Majorana end modes can be generated in a one-dimensional system by varying some of the parameters in the Hamiltonian periodically in time. The specific model we consider is a chain containing spinless electrons with a nearest-neighbor hopping amplitude, a p-wave superconducting term and a chemical potential; this is equivalent to a spin-1/2 chain with anisotropic XY couplings between nearest neighbors and a magnetic field applied in the z-direction. We show that varying the chemical potential (or magnetic field) periodically in time can produce Majorana modes at the ends of a long chain. We discuss two kinds of periodic driving, periodic delta-function kicks and a simple harmonic variation with time. We discuss some distinctive features of the end modes such as the inverse participation ratio of their wave functions and their Floquet eigenvalues which are always equal to +/- 1 for time-reversal symmetric systems. For the case of periodic delta-function kicks, we use the effective Hamiltonian of a system with periodic boundary conditions to define two topological invariants. The first invariant is a well-known winding number while the second invariant has not appeared in the literature before. The second invariant is more powerful in that it always correctly predicts the numbers of end modes with Floquet eigenvalues equal to +1 and -1, while the first invariant does not. We find that the number of end modes can become very large as the driving frequency decreases. We show that periodic delta-function kicks in the hopping and superconducting terms can also produce end modes. Finally, we study the effect of electron-phonon interactions (which are relevant at finite temperatures) and a random noise in the chemical potential on the Majorana modes.

Quantum Butterfly Effect in Weakly Interacting Diffusive Metals

We study scrambling, an avatar of chaos, in a weakly interacting metal in the presence of random potential disorder. It is well known that charge and heat spread via diffusion in such an interacting disordered metal. In contrast, we show within perturbation theory that chaos spreads in a ballistic fashion. The squared anticommutator of the electron field operators inherits a light-cone like growth, arising from an interplay of a growth (Lyapunov) exponent that scales as the inelastic electron scattering rate and a diffusive piece due to the presence of disorder. In two spatial dimensions, the Lyapunov exponent is universally related at weak coupling to the sheet resistivity. We are able to define an effective temperature-dependent butterfly velocity, a speed limit for the propagation of quantum information, that is much slower than microscopic velocities such as the Fermi velocity and that is qualitatively similar to that of a quantum critical system with a dynamical critical exponent

DC resistivity at the onset of spin density wave order in two-dimensional metals

z > 1

Dirac edges of fractal magnetic minibands in graphene with hexagonal moiré superlattices

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Skipping and snake orbits of electrons: Singularities and catastrophes

The theory for the onset of spin density wave order in a metal in two dimensions flows to strong coupling, with strong interactions not only at the “hot spots,” but on the entire Fermi surface. We advocate the computation of dc transport in a regime where there is rapid relaxation to local equilibrium around the Fermi surface by processes which conserve total momentum. The dc resistivity is then controlled by weaker perturbations which do not conserve momentum. We consider variations in the local position of the quantum-critical point, induced by long-wavelength disorder, and find a contribution to the resistivity which is linear in temperature (up to logarithmic corrections) at low temperature. Scattering of fermions between hot spots, by short-wavelength disorder, leads to a residual resistivity and a correction which is linear in temperature.

Quench dynamics of edge states in 2-D topological insulator ribbons

We find a systematic reappearance of massive Dirac features at the edges of consecutive minibands formed at magnetic fields B_{p/q}=\frac{p}{q}\phi_0/S providing rational magnetic flux through a unit cell of the moire superlattice created by a hexagonal substrate for electrons in graphene. The Dirac-type features in the minibands at B=B_{p/q} determine a hierarchy of gaps in the surrounding fractal spectrum, and show that these minibands have topological insulator properties. Using the additional q-fold degeneracy of magnetic minibands at B_{p/q}, we trace the hierarchy of the gaps to their manifestation in the form of incompressible states upon variation of the carrier density and magnetic field.

Classical and quantum magneto-oscillations of current flow near a p-n junction in graphene

Near the sample edge, or a sharp magnetic field step, the drift of two-dimensional (2D) electrons in a magnetic field has the form of skipping and snake orbits. We show that families of skipping and snake orbits of electrons injected at one point inside a 2D metal generically exhibit caustics folds, cusps, and cusp triplets, and, in one exceptional case, an extreme section of the butterfly bifurcation. Periodic appearance of singularities along the +/- B interface leads to the magneto-oscillations of nonlocal conductance in multiterminal electronic devices.