Majorana fusion in interacting one-dimensional Kitaev chains

We employ a time-dependent real-space local density-of-states method to study the movement and fusion of Majorana zero modes in the 1D interacting Kitaev model, based on the time evolution of many-body states. We study the dynamics and both fusion channels of Majoranas using time-dependent potentials, either {\it Wall} or {\it Well}, focusing on the local density-of-states and charge-density of fermions varying with time. For a {\it Wall}, i.e. repulsive strong potential, after fusion of Majoranas the electron (or hole) forms at $\omega=0$, whereas for a {\it Well}, i.e. attractive deep potential, electron (or hole) forms at $\omega \sim -V$, where $V$ is the Coulomb repulsion. We also describe specific upper and lower limits on the Majorana movement needed to reduce non-adiabatic effects as well as to avoid poisoning due to decoherence, focusing on forming a full electron (or hole) after the fusion.Comment: 5 figure

Unexpected results for the non-trivial fusion of Majorana zero modes in interacting quantum-dot arrays

Motivated by recent experimental reports of Majorana zero modes (MZMs) in quantum-dot systems at the ``sweet spot'', where the electronic hopping $t_h$ is equal to the superconducting coupling $\Delta$, we study the time-dependent spectroscopy corresponding to the {\it non-trivial} fusion of MZMs. The expression non-trivial refers to the fusion of Majoranas from different original pairs of MZMs, each with well-defined parities. For the first time, we employ an experimentally accessible time-dependent real-space local density-of-states (LDOS) method to investigate the non-trivial MZMs fusion outcomes in canonical chains and in a Y-shape array of interacting electrons. In the case of quantum-dot chains where two pairs of MZMs are initially disconnected, after fusion we find equal-height peaks in the electron and hole components of the LDOS, signaling non-trivial fusion into both the vacuum $I$ and fermion $\Psi$ channels with equal weight. For $\pi$-junction quantum-dot chains, where the superconducting phase has opposite signs on the left and right portions of the chain, after the non-trivial fusion, surprisingly we observed the formation of an exotic two-site MZM near the center of the chain, coexisting with another single-site MZM. Furthermore, we also studied the fusion of three MZMs in the Y-shape geometry. In this case, after the fusion we observed the novel formation of another exotic multi-site MZM, with properties depending on the connection and geometry of the central region of the Y-shape quantum-dot array.Comment: 10 pages, 4 Figure

Transparent ZnO Thin Film Transistors on Glass and Plastic Substrates Using Post Sputtering Oxygen Passivation

Monolayer graphene under a strong perpendicular field exhibit quantum Hall ferromagnetism with spontaneously broken spin and valley symmetry. The approximate SU(4) spin/valley symmetry is broken by small lattice scale effects in the central Landau level corresponding to filling factors $\nu=0,\pm 1$. Notably the ground state at $\nu=0$ is believed to be a canted antiferromagnetic (AF) or a ferromagnetic (F) state depending on the component of the magnetic field parallel to the layer and the strength of small anisotropies. We study the skyrmions for the filling factors $\nu=\pm 1,0$ by using exact diagonalizations on the spherical geometry. If we neglect anisotropies we confirm the validity of the standard skyrmion picture generalized to four degrees of freedom. For filling factor $\nu=- 1$ the hole skyrmion is an infinite-size valley skyrmion with full spin polarization because it does not feel the anisotropies. The electron skyrmion is also always of infinite size. In the F phase it is always fully polarized while in the AF phase it undergoes continuous magnetization under increasing Zeeman energy. In the case of $\nu=0$ the skyrmion is always maximally localized in space both in F and AF phase. In the F phase it is fully polarized while in the AF it has also progressive magnetization with Zeeman energy. The magnetization process is unrelated to the spatial profile of the skyrmions contrary to the SU(2) case. In all cases the skyrmion physics is dominated by the competition between anisotropies and Zeeman effect but not directly by the Coulomb interactions, breaking universal scaling with the ratio Zeeman to Coulomb energy.Comment: 14 pages, 9 figures, v2 : comments on experiments added, clarifications. Published versio