Abstract. We present a detailed study of the fidelity, the entanglement entropy, and the entanglement spectrum, for a dimerized chain of spinless fermionsa simplified Su-Schrieffer-Heeger (SSH) model-with open boundary conditions which is a well-known example for a model supporting a symmetry protected topological (SPT) phase. In the non-interacting case the Hamiltonian matrix is tridiagonal and the eigenvalues and -vectors can be given explicitly as a function of a single parameter which is known analytically for odd chain lengths and can be determined numerically in the even length case. From a scaling analysis of these data for essentially semi-infinite chains we obtain the fidelity susceptibility and show that it contains a boundary contribution which is different in the topologically ordered than in the topologically trivial phase. For the entanglement spectrum and entropy we confirm predictions from massive field theory for a block in the middle of an infinite chain but also consider blocks containing the edge of the chain. For the latter case we show that in the SPT phase additional entanglement-as compared to the trivial phase-is present which is localized at the boundary. Finally, we extend our study to the dimerized chain with a nearest-neighbour interaction using exact diagonalization, Arnoldi, and densitymatrix renormalization group methods and show that a phase transition into a topologically trivial charge-density wave phase occurs.

International audienceWe study the formation of Majorana states in quasi-one-dimensional (quasi-1D) and two-dimensional square lattices with open boundary conditions, with general anisotropic Rashba coupling, in the presence of an applied Zeeman field and in the proximity of a superconductor. For systems in which the length of the system is very large (quasi-1D) we calculate analytically the exact topological invariant, and we find a rich corresponding phase diagram which is strongly dependent on the width of the system. We compare our results with previous results based on a few-band approximation. We also investigate numerically open two-dimensional systems of finite length in both directions. We use the recently introduced generalized Majorana polarization, which can locally evaluate the Majorana character of a given state. We find that the formation of Majoranas depends strongly on the geometry of the system: for a very elongated wire the finite-size numerical phase diagram reproduces the analytical phase diagram for infinite systems, while if the length and the width are comparable, no Majorana states can form; however, one can show the formation of “quasi-Majorana” states that have a local Majorana character but no global Majorana symmetry

We study the solutions of generic Hamiltonians exhibiting particle-hole mixing. We show that there exists a universal quantity that can describe locally the Majorana nature of a given state. This pseudo-spin like two-component quantity is in fact a generalization of the Majorana polarization (MP) measure introduced in Ref. 1, which was applicable only for some models with specific spin and symmetry properties. We apply this to an open two-dimensional Kitaev system, as well as to a one-dimensional topological wire. We show that the MP characterization is a necessary and sufficient criterion to test whether a state is a Majorana or not, and use it to numerically determine the topological phase diagram.