There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Annual review of condensed matter physics

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Electronic Transport In Mesoscopic Systems

A possible sighting of Majorana states Nearly 80 years ago, the Italian physicist Ettore Majorana proposed the existence of an unusual type of particle that is its own antiparticle, the so-called Majorana fermion. The search for a free Majorana fermion has so far been unsuccessful, but bound Majorana-like collective excitations may exist in certain exotic superconductors. Nadj-Perge et al. created such a topological superconductor by depositing iron atoms onto the surface of superconducting lead, forming atomic chains (see the Perspective by Lee). They then used a scanning tunneling microscope to observe enhanced conductance at the ends of these chains at zero energy, where theory predicts Majorana states should appear. Science, this issue p. 602; see also p. 547 Scanning tunneling microscopy is used to observe signatures of Majorana states at the ends of iron atom chains. [Also see Perspective by Lee] Majorana fermions are predicted to localize at the edge of a topological superconductor, a state of matter that can form when a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin-orbit interaction. With the goal of realizing a one-dimensional topological superconductor, we have fabricated ferromagnetic iron (Fe) atomic chains on the surface of superconducting lead (Pb). Using high-resolution spectroscopic imaging techniques, we show that the onset of superconductivity, which gaps the electronic density of states in the bulk of the Fe chains, is accompanied by the appearance of zero-energy end-states. This spatially resolved signature provides strong evidence, corroborated by other observations, for the formation of a topological phase and edge-bound Majorana fermions in our atomic chains.

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Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor

We show that the pseudorelativistic physics of graphene near the Fermi level can be extended to three dimensional (3D) materials. Unlike in phase transitions from inversion symmetric topological to normal insulators, we show that particular space groups also allow 3D Dirac points as symmetry protected degeneracies. We provide criteria necessary to identify these groups and, as an example, present ab initio calculations of β-cristobalite BiO(2) which exhibits three Dirac points at the Fermi level. We find that β-cristobalite BiO(2) is metastable, so it can be physically realized as a 3D analog to graphene.

Dirac Semimetal in Three Dimensions

Current understanding of higher-order topological insulators (HOTIs) is based primarily on crystalline materials. Here, we propose that HOTIs can be realized in quasicrystals. Specifically, we show that two distinct types of second-order topological insulators (SOTIs) can be constructed on the quasicrystalline lattices (QLs) with different tiling patterns. One is derived by using a Wilson mass term to gap out the edge states of the quantum spin Hall insulator on QLs. The other is the quasicrystalline quadrupole insulator (QI) with a quantized quadrupole moment. We reveal some unusual features of the corner states (CSs) in the quasicrystalline SOTIs. We also show that the quasicrystalline QI can be simulated by a designed electrical circuit, where the CSs can be identified by measuring the impedance resonance peak. Our findings not only extend the concept of HOTIs into quasicrystals but also provide a feasible way to detect the topological property of quasicrystals in experiments.

Higher-Order Topological Insulators in Quasicrystals

Type-II Weyl semimetals have recently attracted intensive research interest because they host Lorentz-violating Weyl fermions as quasiparticles. The discovery of type-II Weyl semimetals evokes the study of type-II line-node semimetals (LNSMs) whose linear dispersion is strongly tilted near the nodal ring. We present here a study on the circularly polarized light-induced Floquet states in type-II LNSMs, as well as those in hybrid LNSMs that have a partially overtilted linear dispersion in the vicinity of the nodal ring. We illustrate that two distinct types of Floquet Weyl semimetal (WSM) states can be induced in periodically driven type-II and hybrid LNSMs, and the type of Floquet WSMs can be tuned by the direction and intensity of the incident light. We construct phase diagrams of light-irradiated type-II and hybrid LNSMs which are quite distinct from those of light-irradiated type-I LNSMs. Moreover, we show that photoinduced Floquet type-I and type-II WSMs can be characterized by the emergence of different anomalous Hall conductivities.

Floquet Weyl semimetals in light-irradiated type-II and hybrid line-node semimetals

We study theoretically the circularly polarized light-induced Floquet state in line-node semimetals with time-reversal symmetry and inversion symmetry. It is found that the Floquet state can show the photovoltaic anomalous Hall effect when an applied circularly polarized light creates a gap in the line node in the bulk and leaves Weyl point nodes. The Hall conductivity is sensitive to the location of the Fermi level: When the Fermi level is located at the node, the Hall conductivity depends on the radius of the line node and is nearly independent of the intensity of the light. Far from the line node, the Hall conductivity is dependent on the intensity of the light. The sensitive Fermi-level dependence of the Hall conductivity in the presence of a laser of weak intensity can have applications in phototransistors based on thin films of line-node semimetals.

Photovoltaic anomalous Hall effect in line-node semimetals

We systematically investigate the finite-size effects in non-Hermitian one-dimensional Su-Schrieffer-Heeger (SSH) and two-dimensional (2D) Chern insulator models. In the Hermitian SSH system, the finite-size energy gap is always real and shows a monotonic-exponential decay as the chain length grows. In contrast, for the non-Hermitian SSH model, the non-Hermitian intracell hoppings can modify the localization lengths of bulk and end states, giving rise to a complex finite-size energy gap that exhibits an oscillating exponential decay as the chain length grows. However, the imaginary staggered onsite potentials in the SSH model only change the end-state energy, leaving the localization lengths of the system unchanged. In this case, the finite-size energy gap can undergo a transition from real values to imaginary values. We observed similar phenomena for the finite-size effect in 2D non-Hermitian Chern insulator systems.

Finite-size effects in non-Hermitian topological systems

Majorana zero modes (MZMs) have been predicted to exist in a topological insulator (TI)/superconductor (SC) heterostructure. A recent spin-polarized scanning tunneling microscope (STM) experiment [Sun et al., Phys. Rev. Lett. 116, 257003 (2016)] has observed a spin-polarization dependence of the zero bias differential tunneling conductance at the center of a vortex core. Here, we consider a helical electron system described by a Rashba spin-orbit coupling Hamiltonian on a spherical surface with an $s$-wave superconducting pairing due to proximity effect. We examine the in-gap excitations of a pair of vortices with one at the north pole and the other at the south pole. While the MZM is not a spin eigenstate, the spin wave function of the MZM at the center of the vortex core, $r=0$, is parallel to the magnetic field, and the local Andreev reflection of the MZM is spin selective, namely, occurs only when the STM tip has the spin polarization parallel to the magnetic field, similar to the case in a one-dimensional nanowire [He et al., Phys. Rev. Lett. 112, 037001 (2014)]. The total local differential tunneling conductance consists of the normal term proportional to the local density of states and an additional term arising from the Andreev reflection. We also discuss the finite size effect, for which the MZM at the north pole is hybridized with the MZM at the south pole. We apply our theory to examine the recently reported spin-polarized STM experiments and show good agreement with the experiments.

Dong-Hui Xu

Papers

Higher-Order Topological Insulators in Quasicrystals

Floquet Weyl semimetals in light-irradiated type-II and hybrid line-node semimetals

Photovoltaic anomalous Hall effect in line-node semimetals

Finite-size effects in non-Hermitian topological systems

Theory of Spin-Selective Andreev Reflection in the Vortex Core of a Topological Superconductor