Topological Origin of Zero-Energy Edge States in Particle-Hole Symmetric Systems

A criterion to determine the existence of zero-energy edge states is discussed for a class of particle-hole symmetric Hamiltonians. A “loop” in a parameter space is assigned for each one-dimensional bulk Hamiltonian, and its topological properties, combined with the chiral symmetry, play an essential role. It provides a unified framework to discuss zero-energy edge modes for several systems such as fully gapped superconductors, two-dimensional $d$-wave superconductors, and graphite ribbons. A variance of the Peierls instability caused by the presence of edges is also discussed.

Figure 1

(a) Continuously deforming ${\ell }_c$ into a loop ${\ell }^{*}\sim {\ell }_c$. During the deformation, the loop is kept on the 2D plane without crossing $\mathcal{O}$. (b) A possible energy spectrum during the deformation. A thick/broken line represents an edge mode localized at $R/L$.

Figure 2

Loops in $\mathbf{R}$ space and the energy spectrum of $d_{x^2-y^2}$-wave SC with (a) $(110)$ and (b) $(100)$ surfaces. Dotted squares show a choice of unit cell in Fourier transforming along the edges. The calculation is for $N_x=50$ for $(110)$ surfaces and $N_x=30$ for $(100)$ surfaces.

Figure 3

Loops in $\mathbf{R}$ space and the energy spectrum of a graphite ribbon with (a) zigzag, (b) bearded, and (c) armchair edges. The ovals indicate how to form a spinor $\mathbf{c}$ for each edge, and dotted squares show a choice of unit cell in Fourier transforming along the edges. The loops corresponding to a one-parameter family of Hamiltonians are (a) ${\mathbf{R}}_{k_y}(k_{x^{\prime }})=[\cos (k_y-k_{x^{\prime }})+1+\cos k_y,-\sin (k_y-k_{x^{\prime }})+\sin k_y,0]$, (b) $[\cos k_{x^{\prime }}+\cos (k_y-k_{x^{\prime }})+1,\sin k_{x^{\prime }}-\sin (k_y-k_{x^{\prime }}),0]$, (c) $[\cos (k_{x^{\prime }}+k_y)+\cos k_{x^{\prime }}+1,-\sin (k_{x^{\prime }}+k_y)+\sin k_{x^{\prime }},0]$. Here we have taken all the hopping integrals equal to unity, and $k_y$ is a wave number along the edges. The calculation is for $N_{x^{\prime }}=30$ for zigzag and bearded edges and $N_{x^{\prime }}=29$ for armchair edges.