We investigate the topological properties of Floquet-engineered twisted bilayer graphene above the so-called magic angle driven by circularly polarized laser pulses. Employing a full Moiré-unit-cell tight-binding Hamiltonian based on first-principles electronic structure, we show that the band topology in the bilayer, at twisting angles above $1.{05}^{\circ }$, essentially corresponds to the one of single-layer graphene. However, the ability to open topologically trivial gaps in this system by a bias voltage between the layers enables the full topological phase diagram to be explored, which is not possible in single-layer graphene. Circularly polarized light induces a transition to a topologically nontrivial Floquet band structure with the Berry curvature analogous to a Chern insulator. Importantly, the twisting allows for tuning electronic energy scales, which implies that the electronic bandwidth can be tailored to match realistic driving frequencies in the ultraviolet or midinfrared photon-energy regimes. This implies that Moiré superlattices are an ideal playground for combining twistronics, Floquet engineering, and strongly interacting regimes out of thermal equilibrium.

From a semiclassical perspective, the Bohr-Sommerfeld quantization of the closed cyclotron orbits for charged particles such as electrons in an external magnetic field gives rise to discrete Landau levels and a series of fascinating quantum Hall phenomena. Here, we consider topologically nontrivial physics from a distinct origin, where the cyclotron orbits take nontrivial knotting structures such as a trefoil knot. We present a scenario of a Weyl semimetal with a slab geometry, where the Fermi arcs on the opposing surfaces can cross without interfering with each other and form a knot together with the bulk Weyl nodes, and in an external magnetic field, the resulting chiral Landau levels. We provide a microscopic lattice model to realize a cyclotron orbit with an unconventional geometry of a trefoil knot and study the corresponding quantum oscillations. Interestingly, unlike the conventional ring-shaped cyclotron orbit, a trefoil knot is self-threading, allowing the magnetic field line along the cyclotron orbit to contribute to the overall Berry phase and therefore altering the external magnetic field for each quantization level. The cyclotron orbit knot offers an arena of the nontrivial knot theory in three spatial dimensions and its subsequent physical consequences.

In non-Hermitian crystals showing the non-Hermitian skin effect, ordinary Bloch band theory and Bloch topological invariants fail to correctly predict energy spectra, topological boundary states, and symmetry-breaking phase transitions in systems with open boundaries. Recently, it has been shown that a correct description requires one to extend Bloch band theory into the complex plane. A still open question is whether the non-Hermitian skin effect and non-Bloch symmetry-breaking phase transitions can be probed by real-space wave dynamics far from edges, which is entirely governed by ordinary Bloch bands. Here it is shown that the Lyapunov exponent in the long-time behavior of bulk wave dynamics can generally reveal non-Bloch symmetry-breaking phase transitions and the existence of the non-Hermitian skin effect.

We present a topological characterization of time-periodically driven two-band models in $2+1$ dimensions as Hopf insulators. The intrinsic periodicity of the Floquet system with respect to both time and the underlying two-dimensional momentum space constitutes a map from a three-dimensional torus to the Bloch sphere. As a result, we find that the driven system can be understood by appealing to a Hopf map that is directly constructed from the micromotion of the drive. Previously found winding numbers are shown to correspond to Hopf invariants, which are associated with linking numbers describing the topology of knots in three dimensions. Moreover, after being cast as a Hopf insulator, not only the Chern numbers but also the winding numbers of the Floquet topological insulator become accessible in experiments as linking numbers. We exploit this description to propose a feasible scheme for measuring the complete set of their Floquet topological invariants in optical lattices.

Topological spintronics aims to exploit the spin-momentum locking in the helical surface states of topological insulators for spin-orbit torque devices. We address a fundamental question that still remains unresolved in this context: Does the topological surface state alone produce the largest values of spin-charge conversion efficiency or can the strongly spin-orbit coupled bulk states also contribute significantly? By studying the Fermi level dependence of spin pumping in topological insulator/ferrimagnetic insulator bilayers, we show that the spin Hall conductivity is constant when the Fermi level is tuned across the bulk band gap, consistent with a full bulk band calculation. The results suggest a different perspective, wherein “bulk-surface correspondence” allows spin-charge conversion to be simultaneously viewed either as coming from the full bulk band, or from spin-momentum locking of the surface state.

The systematic diagnosis of band topology enabled by the method of “symmetry indicators” underlies the recent advances in the search for new materials realizing topological crystalline insulators. Such an efficient method has been missing for superconductors because the quasiparticle spectrum in the superconducting phase is not usually available. In this work, we establish symmetry indicators for weak-coupling superconductors that detect nontrivial topology based on the representations of the metallic band structure in the normal phase, assuming a symmetry property of the gap function. We demonstrate the applications of our formulas using examples of tight-binding models and density-functional-theory band structures of realistic materials.

Fracton topological phases host fractionalized topological quasiparticles with restricted mobility, with promising applications to fault-tolerant quantum computation. While a variety of exactly solvable fracton models have been proposed, there is a need for platforms to realize them experimentally. We show that a rich set of fracton phases emerges in interacting Majorana band models whose building blocks are within experimental reach. Specifically, our Majorana constructions overcome a principal obstacle, namely, the implementation of the complicated spin cluster interactions underlying fracton stabilizer codes. The basic building blocks of the proposed constructions include Coulomb blockaded Majorana islands and weak interisland Majorana hybridizations. This setting produces a wide variety of fracton states and promises numerous opportunities for probing and controlling fracton phases experimentally. Our approach also reveals the relation between fracton phases and Majorana fermion codes and further generates a hierarchy of fracton spin liquids.

We introduce and investigate material junctions as a generic and tunable electronic platform for the realization of exotic non-Hermitian (NH) topological states of matter, where the NH character is induced by the surface self-energy of a thermal reservoir. As a conceptually rich and immediately experimentally realizable example, we consider a three-dimensional topological insulator (TI) coupled to a ferromagnetic lead. Remarkably, the symmetry-protected TI is promoted in a dissipative fashion to a non-symmetry-protected NH Weyl phase with no direct Hermitian counterpart and which exhibits robustness against any perturbation. The transition between a gapped phase and the NH Weyl phase may be readily tuned experimentally with the magnetization direction of the ferromagnetic lead. Given the robustness of this exotic nodal phase, our general analysis also applies to, e.g., a two-dimensional electron gas close to criticality in proximity to a ferromagnetic lead. There, the predicted bulk Fermi arcs are directly amenable to surface spectroscopy methods such as angle-resolved photoemission spectroscopy.