We formulate a continuum model to study the low-energy electronic structure of heterostructures formed by graphene on a strong three-dimensional topological insulator (TI) for the case of both commensurate and incommensurate stacking. The incommensurability can be due to a twist angle between graphene and the TI surface or a lattice mismatch between the two systems. We find that the proximity of the TI induces in graphene a strong enhancement of the spin-orbit coupling that can be tuned via the twist angle.The surfaces of strong three-dimensional (3D) topological insulators (TIs) [1] and graphene [2,3] have very similar low-energy electronic structures: the conduction and valence bands touch at isolated points, the Dirac points (DPs), and around these points the fermionic excitations are well described as massless two-dimensional (2D) chiral Dirac fermions for which the phase of a twostate quantum degree of freedom is locked with the momentum direction. However, there are also qualitative differences: (i) in graphene the chirality is associated with the sublattice degree of freedom whereas in a TI surface (TIS) it is associated with the electron spin; (ii) in graphene the number of DPs is even whereas in a TIS it is odd ; (iii) in TIs the electron-phonon scattering is much stronger than that in graphene. Therefore, the transport properties of graphene [4] and TIs are different in significant aspects: in graphene, because the intrinsic spin-orbit (SO) coupling is negligible [5][6][7][8], no quantum spin Hall effect is expected, contrary to the case in a TI; graphene has the highest room-temperature mobility, whereas TIs have very low mobilities. These facts, together with the recent experimental progress [9], motivated us to study graphene-TI heterostructures, in which the proximity to a TI is expected to enhance the SO coupling of graphene and create a novel 2D system with nontrivial spin textures and high, room-temperature, electron mobility. This approach to enhance the SO coupling in graphene appears to be more practical than previously proposed approaches [10] that rely on doping graphene with heavy adatoms.In this work, we study the low-energy electronic structure of heterostructures formed by one sheet of graphene placed on the conducting surface of a 3D TI. Our results show that not only can the proximity of a TIS enhance the SO coupling in graphene and bilayer graphene (BLG), but also that this enhancement can be controlled via the relative rotation, the twist angle, between the graphene lattice and the TI's lattice. The presence of a relative rotation, in general, induces an incommensurate stacking of the graphene and the substrate [11] [12-36]. As a consequence we develop and present a theory that is able to take into account the incommensurability between graphene and the TIS. This cannot be achieved via standard approaches, such as density functional theory [37], and tight-binding models, due to the computational cost of these approaches for incommensurate structures. A continuous model, on the other...

Recent pump-probe experiments demonstrate the possibility that Dirac materials may be driven into transient excited states describable by two chemical potentials, one for the electrons and one for the holes. Given the Dirac nature of the spectrum, such an inverted population allows the optical tunability of the density of states of the electrons and holes, effectively offering control of the strength of the Coulomb interaction. Here we discuss the feasibility of realizing transient excitonic instabilities in optically-pumped Dirac materials. We demonstrate, theoretically, the reduction of the critical coupling leading to the formation of a transient condensate of electron-hole pairs and identify signatures of this state. Furthermore, we provide guidelines for experiments by both identifying the regimes in which such exotic many-body states are more likely to be observed and estimating the magnitude of the excitonic gap for a few important examples of existing Dirac materials. We find a set of material parameters for which our theory predicts large gaps and high critical temperatures and which could be realized in future Dirac materials. We also comment on transient excitonic instabilities in three-dimensional Dirac and Weyl semimetals. This study provides the first example of a transient collective instability in driven Dirac materials.

We study the emergence of odd-frequency superconducting pairing in UPt3. Starting from a tight-binding model accounting for the nonsymmorphic crystal symmetry of UPt3 and assuming an order parameter in the E2u representation, we demonstrate that odd-frequency pairing arises very generally, as soon as inter-sublattice hopping or spin-orbit coupling is present. We also show that in the low temperature superconducting B phase, the presence of a chiral order parameter together with spin-orbit coupling, leads to additional odd-frequency pair amplitudes not present in the A or C phases. Furthermore, we show that a finite Kerr rotation in the B phase is only present if odd-ω pairing also exists.

We obtain the general conditions for the emergence of odd-frequency superconducting pairing in a two-dimensional (2D) electronic system proximity coupled to a superconductor, making minimal assumptions about both the 2D system and the superconductor. Using our general results we show that a simple heterostructure formed by a monolayer of a group VI transition metal dichalcogenide, such as molybdenum disulfide, and an s-wave superconductor with Rashba spin-orbit coupling exhibits odd-frequency superconducting pairing. Our results allow the identification of a new class of systems among van der Waals heterostructures in which odd-frequency superconductivity should be present.

Recent progress in the understanding of multiband superconductivity and its relationship to odd‐frequency pairing are reviewed herein. The discussion begins by reviewing the emergence of odd‐frequency pairing in a simple two‐band model, providing a brief pedagogical overview of the formalism. Several examples of multiband superconducting systems are examined, in each case describing both the origin of the band degree of freedom and the nature of the odd‐frequency pairing. Throughout, it is attempted to convey a unified picture of how odd‐frequency pairing emerges in these materials and propose that similar mechanisms are responsible for odd‐frequency pairing in several analogous systems: layered 2D heterostructures, double quantum dots, double nanowires, Josephson junctions, and systems described by isolated valleys in momentum space. In addition, experimental probes of odd‐frequency pairing in multiband systems are reviewed, focusing on hybridization gaps in the electronic density of states, paramagnetic Meissner effect, and Kerr effect.

Odd-frequency superconductivity represents a truly unconventional ordered state which, in contrast to conventional superconductivity, exhibits pair correlations which are odd in relative time and, hence, inherently dynamical. In this review article we provide an overview of recent advances in the study of odd-frequency superconducting correlations in one-dimensional systems. In particular, we focus on recent developments in the study of nanowires with Rashba spin-orbit coupling and metallic edges of two-dimensional topological insulators in proximity to conventional superconductors. These systems have recently elicited a great deal of interest due to their potential for realizing one-dimensional topological superconductivity whose edges can host Majorana zero modes. We also provide a detailed discussion of the intimate relationship between Majorana zero modes and odd-frequency pairing. Throughout this review, we highlight the ways in which odd-frequency pairing provides a deeper understanding of the unconventional superconducting correlations present in each of these intriguing systems and how the study and control of these states holds the potential for future applications. a

We show that Berezinskii's classification of the symmetries of Cooper pair amplitudes holds for driven systems even in the absence of translation invariance. We then consider a model Hamiltonian for a superconductor coupled to an external driving potential and, treating the drive as a perturbation, we investigate the corrections to the anomalous Green's function, density of states, and spectral function. We find that in the presence of an external drive the anomalous Green's function develops terms that are odd in frequency and that the same mechanism responsible for these odd-frequency terms generates additional features in the density of states and spectral function.

It was recently shown that odd-frequency superconducting pair amplitudes can be induced in conventional superconductors subjected to a spatially nonuniform time-dependent drive. It has also been shown that, in the presence of interband scattering, multiband superconductors will possess bulk odd-frequency superconducting pair amplitudes. In this work we build on these previous results to demonstrate that by subjecting a multiband superconductor with interband scattering to a timedependent drive even-frequency pair amplitudes can be converted to odd-frequency pair amplitudes and vice versa. We will discuss the physical conditions under which these pair symmetry conversions can be achieved and possible experimental signatures of their presence.