Miraging a Majorana

The image of a Majorana mode located on the focus of an elliptical corral of free electrons is studied. The Majorana mode may be taken at the edge of a topological wire superimposed on the two-dimensional electron gas. At low energies the states of the wire are ignored except for the Majorana mode. Usual tunneling to a fermionic mode is compared. In the favorable cases, tunneling to a Majorana mode leads to an enhanced mirage effect and a spectral weight mainly confined around the foci, in comparison to the tunneling to a fermionic mode. The Majorana character of the image is shown computing the self-conjugacy.Comment: Submission to SciPost, 1 reference adde

Edge mode dynamics of quenched topological wires

The fermionic and Majorana edge mode dynamics of various topological systems is compared, after a sudden global quench of the Hamiltonian parameters takes place. Attention is focused on the regimes where the survival probability of an edge state has oscillations either due to critical or off-critical quenches. The nature of the wave functions and the overlaps between the eigenstates of different points in parameter space determine the various types of behaviors, and the distinction due to the Majorana nature of the excitations plays a lesser role. Performing a sequence of quenches it is shown that the edge states, including Majorana modes, may be switched off and on. Also, the generation of Majoranas due to quenching from a trivial phase is discussed.Comment: 17 pages, 21 figure

Zero energy modes in a superconductor with ferromagnetic adatom chains and quantum phase transitions

We study Majorana zero energy modes (MZEM) that occur in a s-wave superconducting surface, at the ends of a ferromagnetic (FM) chain of adatoms, in the presence of Rashba spin-orbit interaction (SOI) considering both non self-consistent and self-consistent superconducting order. We find that in the self-consistent solution the average superconducting gap function over the adatom sites has a discontinuous drop with increasing exchange interaction at the same critical value where the topological phase transition occurs. We also study the MZEM for both treatments of superconducting order and find that the decay length is a linear function of the exchange coupling strength, chemical potential and superconducting order. For wider FM chains the MZEM occur at smaller exchange couplings and the slope of the decay length as a function of exchange coupling grows with chain width. Thus we suggest experimental detection of different delocalization of MZEM in chains of varying widths. We discuss similarities and differences between the MZEM for the two treatments of the superconducting order.Comment: 10 pages, 10 figure

Entanglement modes and topological phase transitions in superconductors

Topological insulators and topological superconductors display various topological phases that are characterized by different Chern numbers or by gapless edge states. In this work we show that various quantum information methods such as the von Neumann entropy, entanglement spectrum, fidelity, and fidelity spectrum may be used to detect and distinguish topological phases and their transitions. As an example we consider a two-dimensional $p$-wave superconductor, with Rashba spin-orbit coupling and a Zeeman term. The nature of the phases and their changes are clarified by the eigenvectors of the $k$-space reduced density matrix. We show that in the topologically nontrivial phases the highest weight eigenvector is fully aligned with the triplet pairing state. A signature of the various phase transitions between two points on the parameter space is encoded in the $k$-space fidelity operator.Comment: 17 pages, 19 figure

Fate of Majorana fermions and Chern numbers after a quantum quench

The stability of Majorana fermions at the edges of a two-dimensional topological supercondutor is studied, after quenches to either non-topological phases or other topological phases. Both instantaneous and slow quenches are considered. In general, the Majorana modes decay and, in the case of instantaneous quenches, their revival times scale to infinity as the system size grows. Considering fast quantum quenches within the same topological phase, leads, in some cases, to robust edge modes. Quenches to a topological $Z_2$ phase reveal some robustness of the Majorana fermions. Comparing strong spin-orbit coupling with weak spin-orbit coupling, it is found that the Majorana fermions are fairly robust, if the pairing is not aligned with the spin-orbit Rashba coupling. It is also shown that the Chern number remains invariant after the quench, until the propagation of the mode along the transverse direction reaches the middle point, beyond which the Chern number oscillates between increasing values. In some cases, the time average Chern number seems to converge to the appropriate value, but often the decay is very slow. The effect of varying the rate of change in slow quenches is also analysed. It is found that the defect production is non-universal and does not follow the Kibble-Zurek scaling with the quench rate, as obtained before for other systems with topological edge states.Comment: 14 pages, 19 figure

Spinor Description of a General Spin-J System

We consider a spin coherent states description of a general quantum spin system. It is shown that it is possible to use the spin-1/2 representation to study the general spin-J case. We identify the 1/2 spinor components as the homogeneous coordinates of the projective space associated to the complex variable that labels the coherent states and establish a relation between the two-component spinor and the bosonic Schwinger representation of a spin operator. We rewrite the equations of motion, obtained from the path integral for the evolution operator or partition function, in terms of the 1/2 spinor and define the effective Hamiltonian of its evolution

Charge and spin edge currents in two-dimensional Floquet topological superconductors

A time periodic driving on a topologically trivial system induces edge modes and topological properties. In this work we consider triplet and singlet superconductors subject to periodic variations of the chemical potential, spin-orbit coupling and magnetization, in both topologically trivial and nontrivial phases, and study their influence on the charge and spin currents that propagate along the edges of the two-dimensional system, for moderate to large driving frequencies. Currents associated with the edge modes are induced in the trivial phases and enhanced in the topological phases. In some cases there is a sign reversal of the currents as a consequence of the periodic driving. The edge states associated with the finite quasi-energy states at the edge of the Floquet zone are in general robust, while the stability of the zero quasi-energy states depends on the parameters. Also, the spin polarization of the Floquet spectrum quasi-energies is strong as for the unperturbed topological phases. It is found that in some cases the unperturbed edge states are immersed in a continuum of states due to the perturbation, particularly if the driving frequency is not large enough. However, their contribution to the edge currents and spin polarization is still significant.Comment: 15 pages, 12 figure

Haldane gap in a S=1 exchange model with long-range interactions

The ground-state properties of the S=1 Haldane-Shastry model are studied using a modified Lanczos algorithm and diagonalizing exactly small chains. We find evidence that, as for the antiferromagnetic Heisenberg model, the spectrum shows a gap, in contrast to the S=1/2 case. The correlation functions decay exponentially for large m. We find that the correlation functions for the Haldane-Shastry model decay faster than for the Heisenberg model. We estimate the infinite system limit for the ground-state energy, value of the gap and correlation functions.Comment: Latex, 18 pages plus 7 figures (postscript

Vanishing k-space fidelity and phase diagram's bulk-edge-bulk correspondence

The fidelity between two infinitesimally close states or the fidelity susceptibility of a system are known to detect quantum phase transitions. Here we show that the k-space fidelity between two states far from each other and taken deep inside (bulk) of two phase s, generically vanishes at the k-points where there are gapless points in the energy spectrum that give origin to the lines (edges) separating the phases in the phase diagram. We consider a general case of two-band models and present a sufficient condition for the existence of gapless points, given there are pairs of parameter points for which the fidelity between the corresponding states is zero. By presenting an explicit counter-example, we showed that the sufficient condition is not necessary. Further, we showed that, unless the set of parameter points is suitably constrained, the existence of gapless points generically imply the accompanied pairs of parameter points with vanishing fidelity. Also, we showed the connection between the vanishing fidelity and gapless points on a number of concrete examples (topological triplet superconductor, topological insulator, 1d Kitaev model of spinless fermions, BCS superconductor, Ising model in a transverse field, graphene and Haldane Chern insulator), as well as for the more general case of Dirac-like Hamiltonians. We also briefly discuss the relation between the vanishing fidelity and gapless points at finite temperatures

Edge and bulk localization of Floquet topological superconductors

We study the bulk and edge properties of a driven Kitaev chain, where the driving is performed as instantaneous quenches of the on-site energies. We identify three periodic driving regimes: low period, which is equivalent to a static model, with renormalized parameters obtained from the Baker-Campbell-Hausdorff (BCH) expansion; intermediate period, where the first order BCH expansion breaks down; and high period when the quasienergy gap at $\omega/2$ closes. We investigate the dynamical localization properties for the case of quasiperiodic potential driving as a function of its amplitude and the pairing strength, obtaining regimes with extended, critical and localized bulk states, if the driving is performed at high frequencies. In these, we characterize wave-packet propagation, obtaining ballistic, subdiffusive and absence of spreading, respectively. In the intermediate period regime, we find an additional region in the phase diagram with a mobility edge between critical and localized states. Further, we investigate the stability of these phases under time-aperiodicity on the drivings, observing that the system eventually thermalizes: It results in featureless random states which can be described by the symmetry of the Hamiltonian. In a system with open edges, we find that both Majorana and fermionic localized edge modes can be engineered with a spatially quasiperiodic potential. Besides, we demonstrate the possibility of creating multiple Majorana $0$ and $\pi$ modes in a driven setting, even if the underlying static Hamiltonian is in its trivial phase. Lastly, we study the robustness of the Majorana modes against the aperiodicity in the driving period, showing that the ones created via quasiperiodic potential are more robust to the decoherence. Moreover, we find an example where Majorana mode is robust, provided that it is chosen from a special point in the topological region.Comment: 11+1 pages, 11+1 figure