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Register a new userSurface and Edge States in Topological Semi-metals
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We study the topologically non-trivial semi-metals by means of the 6-band Kane model. Existence of surface states is explicitly demonstrated by calculating the LDOS on the material surface. In the strain free condition, surface states are divided into two parts in the energy spectrum, one part is in the direct gap, the other part including the crossing point of surface state Dirac cone is submerged in the valence band. We also show how uni-axial strain induces an insulating band gap and raises the crossing point from the valence band into the band gap, making the system a true topological insulator. We predict existence of helical edge states and spin Hall effect in the thin film topological semi-metals, which could be tested with future experiment. Disorder is found to significantly enhance the spin Hall effect in the valence band of the thin films.
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In an ordinary three-dimensional metal the Fermi surface forms a two-dimensional closed sheet separating the filled from the empty states. Topological semimetals, on the other hand, can exhibit protected one-dimensional Fermi lines or zero-dimensional Fermi points, which arise due to an intricate interplay between symmetry and topology of the electronic wavefunctions. Here, we study how reflection symmetry, time-reversal symmetry, SU(2) spin-rotation symmetry, and inversion symmetry lead to the topological protection of line nodes in three-dimensional semi-metals. We obtain the crystalline invariants that guarantee the stability of the line nodes in the bulk and show that a quantized Berry phase leads to the appearance of protected surfaces states with a nearly flat dispersion. By deriving a relation between the crystalline invariants and the Berry phase, we establish a direct connection between the stability of the line nodes and the topological surface states. As a representative example of a topological semimetal with line nodes, we consider Ca$_3$P$_2$ and discuss the topological properties of its Fermi line in terms of a low-energy effective theory and a tight-binding model, derived from ab initio DFT calculations. Due to the bulk-boundary correspondence, Ca$_3$P$_2$ displays nearly dispersionless surface states, which take the shape of a drumhead. These surface states could potentially give rise to novel topological response phenomena and provide an avenue for exotic correlation physics at the surface.
The edge states of a two-dimensional quantum spin Hall (QSH) insulator form a one-dimensional helical metal which is responsible for the transport property of the QSH insulator. Conceptually, such a one-dimensional helical metal can be attached to any scattering region as the usual metallic leads. We study the analytical property of the scattering matrix for such a conceptual multiterminal scattering problem in the presence of time reversal invariance. As a result, several theorems on the connectivity property of helical edge states in two-dimensional QSH systems as well as surface states of three-dimensional topological insulators are obtained. Without addressing real model details, these theorems, which are phenomenologically obtained, emphasize the general connectivity property of topological edge/surface states from the mere time reversal symmetry restriction.
With a generic lattice model for electrons occupying a semi-infinite crystal with a hard surface, we study the eigenstates of the system with a bulk band gap (or the gap with nodal points). The exact solution to the wave functions of scattering states is obtained. From the scattering states, we derive the criterion for the existence of surface states. The wave functions and the energy of the surface states are then determined. We obtain a connection between the wave functions of the bulk states and the surface states. For electrons in a system with time-reversal symmetry, with this connection, we rigorously prove the correspondence between the change of Kramers degeneracy of the surface states and the bulk time-reversal $Z_2$ invariant. The theory is applicable to systems of (topological) insulators, superconductors, and semi-metals. Examples for solving the edge states of electrons with/without the spin-orbit interactions in graphene with a hard zigzag edge and that in a two-dimensional $d$-wave superconductor with a (1,1) edge are given in appendices.
The non-Hermitian skin effect (NHSE) in non-Hermitian lattice systems depicts the exponential localization of eigenstates at systems boundaries. It has led to a number of counter-intuitive phenomena and challenged our understanding of bulk-boundary correspondence in topological systems. This work aims to investigate how the NHSE localization and topological localization of in-gap edge states compete with each other, with several representative static and periodically driven 1D models, whose topological properties are protected by different symmetries. The emerging insight is that at critical system parameters, even topologically protected edge states can be perfectly delocalized. In particular, it is discovered that this intriguing delocalization occurs if the real spectrum of the systems edge states falls on the same systems complex spectral loop obtained under the periodic boundary condition. We have also performed sample numerical simulation to show that such delocalized topological edge states can be safely reconstructed from time-evolving states. Possible applications of delocalized topological edge states are also briefly discussed.
The Landau bands of mirror symmetric 2D Dirac semi-metals (for example odd-layers of ABA-graphene) can be identified by their parity with respect to mirror symmetry. This symmetry facilitates a new class of counter-propagating Hall states at opposite but equal electron and hole filling factors $| u_{pm}|=1/m$ ({it m} odd). Here, we propose a Laughlin-like correlated liquid wavefunction, at the charge neutrality point, that exhibits fractionally charged quasi-particle/hole pair excitation of opposite parity. Using a bosonized one-dimensional edge state theory, we show that the longitudinal conductance of this state, $sigma_{xx} = 2e^2/(m h)$, is robust to short-ranged inter-mode interactions.
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