I. C. Fulga

Flux-controlled quantum computation with Majorana fermions

uxes. We show that readout operations can also be fully ux-controlled, without requiring microscopic control over tunnel couplings. We identify the minimal circuit that can perform the initialization{braiding{measurement steps required to demonstrate non-Abelian statistics. We introduce the Random Access Majorana Memory, a scalable circuit that can perform a joint parity measurement on Majoranas belonging to a selection of topological qubits. Such multi-qubit measurements allow for the ecient creation of highly entangled states and simplify quantum error correction protocols by avoiding the need for ancilla qubits.

Spatially resolved edge currents and guided-wave electronic states in graphene

Experiments show that electron waves can be confined to and guided along the edges of monolayer and bilayer graphene sheets, analogous to the guiding of light waves in optical fibres.

Scattering formula for the topological quantum number of a disordered multimode wire

The topological quantum number Q of a superconducting or chiral insulating wire counts the number of stable bound states at the end points. We determine Q from the matrix r of reflection amplitudes from one of the ends, generalizing the known result in the absence of time-reversal and chiral symmetry to all five topologically nontrivial symmetry classes. The formula takes the form of the determinant, Pfaffian, or matrix signature of r, depending on whether r is a real matrix, a real antisymmetric matrix, or a Hermitian matrix. We apply this formula to calculate the topological quantum number of N coupled dimerized polymer chains, including the effects of disorder in the hopping constants. The scattering theory relates a topological phase transition to a conductance peak, of quantized height and with a universal (symmetry class independent) line shape. Two peaks which merge are annihilated in the superconducting symmetry classes, while they reinforce each other in the chiral symmetry classes.

Adaptive tuning of Majorana fermions in a quantum dot chain

We suggest a way to overcome the obstacles that disorder and high density of states pose to the creation of unpaired Majorana fermions in one-dimensional systems. This is achieved by splitting the system into a chain of quantum dots, which are then tuned to the conditions under which the chain can be viewed as an effective Kitaev model, so that it is in a robust topological phase with well-localized Majorana states in the outermost dots. The tuning algorithm that we develop involves controlling the gate voltages and the superconducting phases. Resonant Andreev spectroscopy allows us to make the tuning adaptive, so that each pair of dots may be tuned independently of the other. The calculated quantized zero bias conductance serves then as a natural proof of the topological nature of the tuned phase.

Scattering theory of topological insulators and superconductors

The topological invariant of a topological insulator (or superconductor) is given by the number of symmetry-protected edge states present at the Fermi level. Despite this fact, established expressions for the topological invariant require knowledge of all states below the Fermi energy. Here, we propose a way to calculate the topological invariant employing solely its scattering matrix at the Fermi level without knowledge of the full spectrum. Since the approach based on scattering matrices requires much less information than the Hamiltonian-based approaches (surface versus bulk), it is numerically more efficient. In particular, is better-suited for studying disordered systems. Moreover, it directly connects the topological invariant to transport properties potentially providing a new way to probe topological phases.

Triple point fermions in a minimal symmorphic model

Gapless topological phases of matter may host emergent quasiparticle excitations which have no analog in quantum field theory. This is the case of so called triple point fermions (TPF), quasiparticle excitations protected by crystal symmetries, which show fermionic statistics but have an integer (pseudo)spin degree of freedom. TPFs have been predicted in certain three-dimensional non-symmorphic crystals, where they are pinned to high symmetry points of the Brillouin zone. In this work, we introduce a minimal, three-band model which hosts TPFs protected only by the combination of a C4 rotation and an anti-commuting mirror symmetry. Unlike current non-symmorphic realizations, our model allows for TPFs which are anisotropic and can be created or annihilated pairwise. It provides a simple, numerically affordable platform for their study.

Statistical Topological Insulators

We define a class of insulators with gapless surface states protected from localization due to the statistical properties of a disordered ensemble, namely due to the ensembles invariance under a certain symmetry. We show that these insulators are topological, and are protected by a

Observation of Electron Coherence and Fabry–Perot Standing Waves at a Graphene Edge

\mathbb{Z}_2

Coexisting edge states and gapless bulk in topological states of matter.

invariant. Finally, we prove that every topological insulator gives rise to an infinite number of classes of statistical topological insulators in higher dimensions. Our conclusions are confirmed by numerical simulations.

Bimodal conductance distribution of Kitaev edge modes in topological superconductors

Electron surface states in solids are typically confined to the outermost atomic layers and, due to surface disorder, have negligible impact on electronic transport. Here, we demonstrate a very different behavior for surface states in graphene. We probe the wavelike character of these states by Fabry-Perot (FP) interferometry and find that, in contrast to theoretical predictions, these states can propagate ballistically over micron-scale distances. This is achieved by embedding a graphene resonator formed by gate-defined p-n junctions within a graphene superconductor-normal-superconductor structure. By combining superconducting Aharanov-Bohm interferometry with Fourier methods, we visualize spatially resolved current flow and image FP resonances due to p-n-p cavity modes. The coherence of the standing-wave edge states is revealed by observing a new family of FP resonances, which coexist with the bulk resonances. The edge resonances have periodicity distinct from that of the bulk states manifest in a repeated spatial redistribution of current on and off the FP resonances. This behavior is accompanied by a modulation of the multiple Andreev reflection amplitude on-and-off resonance, indicating that electrons propagate ballistically in a fully coherent fashion. These results, which were not anticipated by theory, provide a practical route to developing electron analog of optical FP resonators at the graphene edge.