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

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

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

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

Landau Levels and Quasiparticle Spectrum of Extreme Type-II Superconductors

At high magnetic fields and low temperatures, numerous extreme type-II superconductors exhibit Landau quantization of electronic motion. We present an analytic construction of the quasiparticle spectrum in this regime, based on the high-field expansion. The spectrum is gapless and is separated from the familiar low-field regime by a quantum level-crossing transition. Such low energy excitations should lead to observable effects in thermodynamics and transport.Comment: 5 pages REVTeX, 1 .ps figure included, inset available on request from [email protected]

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

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

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

Quasiparticle density of states of d-wave superconductors in a disordered vortex lattice

We calculate the density of states of a disordered inhomogeneous d-wave superconductor in a magnetic field. The field-induced vortices are assumed to be pinned at random positions and the effects of the scattering of the quasi-particles off the vortices are taken into account using the singular gauge transformation of Franz and Tesanovic. We find two regimes for the density of states: at very low energies the density of states follows a law \rho(\epsilon) \sim \rho_0 + |\epsilon|^{\alpha} where the exponent is close to 1. A good fit of the density of states is obtained at higher energies, excluding a narrow region around the origin, with a similar power law energy dependence but with \alpha close to 2. Both at low and at higher energies \rho_0 scales with the inverse of the magnetic length (\sqrt{B}).Comment: 5 REVTeX pages, 5 figures; improved scaling analysi