Determining topological order from a local ground state correlation function

Topological insulators are physically distinguishable from normal insulators only near edges and defects, while in the bulk there is no clear signature to their topological order. In this work we show that the Z index of topological insulators and the Z index of the integer quantum Hall effect manifest themselves locally. We do so by providing an algorithm for determining these indices from a local equal time ground-state correlation function at any convenient boundary conditions. Our procedure is unaffected by the presence of disorder and can be naturally generalized to include weak interactions. The locality of these topological indices implies bulk-edge correspondence theorem.Comment: 7 pages, 3 figures. Major changes: the paper was divided into sections, the locality of the order in 3D topological insulators is also discusse

Topological Pumping over a Photonic Fibonacci Quasicrystal

Quasiperiodic lattices have recently been shown to be a non-trivial topological phase of matter. Charge pumping -- one of the hallmarks of topological states of matter -- was recently realized for photons in a one-dimensional (1D) off-diagonal Harper model implemented in a photonic waveguide array. The topologically nontrivial 1D Fibonacci quasicrystal (QC) is expected to facilitate a similar phenomenon, but its discrete nature and lack of pumping parameter hinder the experimental study of such topological effects. In this work we overcome these obstacles by utilizing a family of topologically equivalent QCs which ranges from the Fibonacci QC to the Harper model. Implemented in photonic waveguide arrays, we observe the topological properties of this family, and perform a topological pumping of photons across a Fibonacci QC.Comment: 5 pages, 4 figures, comments are welcom

Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

One-dimensional quasiperiodic systems, such as the Harper model and the Fibonacci quasicrystal, have long been the focus of extensive theoretical and experimental research. Recently, the Harper model was found to be topologically nontrivial. Here, we derive a general model that embodies a continuous deformation between these seemingly unrelated models. We show that this deformation does not close any bulk gaps, and thus prove that these models are in fact topologically equivalent. Remarkably, they are equivalent regardless of whether the quasiperiodicity appears as an on-site or hopping modulation. This proves that these different models share the same boundary phenomena and explains past measurements. We generalize this equivalence to any Fibonacci-like quasicrystal, i.e., a cut and project in any irrational angle.Comment: 7 pages, 2 figures, minor change

Majorana fermions of a two-dimensional Px+iPy superconductor

To investigate Majorana fermionic excitations of a $p_x+ip_y$ superconductor, the Bogoliubov-de-Gennes equation is solved on a sphere for two cases: (i) a vortex-antivortex pair at opposite poles and (ii) an edge near the south pole and an antivortex at the north pole. The vortex cores support a state of two Majorana fermions, the energy of which decreases exponentially with the radius of the sphere, independently of a moderate disorder potential. The tunneling conductance of an electron into the superconductor near the position of a vortex is computed for finite temperature, and is compared to the case of an {\it s}-wave superconductor. The zero bias conductance peak of the antivortex is half that of the vortex. This effect can be used as a probe of the order parameter symmetry, and as a direct measurement of the Majorana fermion.Comment: 15 pages, 17 figure

Testing for Majorana Zero Modes in a Px+iPy Superconductor at High Temperature by Tunneling Spectroscopy

Directly observing a zero energy Majorana state in the vortex core of a chiral superconductor by tunneling spectroscopy requires energy resolution better than the spacing between core states $\Delta^2/eF$. We show that nevertheless, its existence can be decisively tested by comparing the temperature broadened tunneling conductance of a vortex with that of an antivortex even at temperatures $T >> \Delta^2/eF$.Comment: 5 pages, 4 figure

Topological States and Adiabatic Pumping in Quasicrystals

The unrelated discoveries of quasicrystals and topological insulators have in turn challenged prevailing paradigms in condensed-matter physics. We find a surprising connection between quasicrystals and topological phases of matter: (i) quasicrystals exhibit nontrivial topological properties and (ii) these properties are attributed to dimensions higher than that of the quasicrystal. Specifically, we show, both theoretically and experimentally, that one-dimensional quasicrystals are assigned two-dimensional Chern numbers and, respectively, exhibit topologically protected boundary states equivalent to the edge states of a two-dimensional quantum Hall system.We harness the topological nature of these states to adiabatically pump light across the quasicrystal. We generalize our results to higher-dimensional systems and other topological indices. Hence, quasicrystals offer a new platform for the study of topological phases while their topology may better explain their surface properties.Comment: 10 pages, 5 figures, 4 appendice

The strong side of weak topological insulators

Three-dimensional topological insulators are classified into "strong" (STI) and "weak" (WTI) according to the nature of their surface states. While the surface states of the STI are topologically protected from localization, this does not hold for the WTI. In this work we show that the surface states of the WTI are actually protected from any random perturbation that does not break time-reversal symmetry, and does not close the bulk energy gap. Consequently, the conductivity of metallic surfaces in the clean system remains finite even in the presence of strong disorder of this type. In the weak disorder limit the surfaces are found to be perfect metals, and strong surface disorder only acts to push the metallic surfaces inwards. We find that the WTI differs from the STI primarily in its anisotropy, and that the anisotropy is not a sign of its weakness but rather of its richness.Comment: 12 pages, 7 figure