638
M. Zaimi et al.
r < r c
r > r c
0.5
1
1.5
2
r
-3
-2
-1
1
2
3
Energies (N = 20)
r c
0.8
0.9
1
r < 1 r > 1
0.5
1.0
1.5
2.0
r
-3
-2
-1
1
2
3
E/t 2
E/t 2
Energies (N = 21)
Fig. 4 Energy spectra for two values of N as a function of r. The shaded regions are the bands
for N = ∞. Dotted lines outside the bands are edge states. The inset on the left focuses on the
transition between an edge state (to the left of the vertical broken line) and a non-edge state (to the
right, in the shaded region)
zero-energy edge state (the remainder of the spectrum being symmetric). This state
is confined to the left (right) edge for r < 1 (r > 1) with decay length l = 1/| log r|.
Figure 4 displays the spectra for N = 20 and 21.
4 Tripartite System: TLS-SSH-Chain
We now study the tripartite system displayed in Fig. 1. Although it is an infinitedimensional system, the effects of the SSH chain and semi-infinite chain on the
TLS can be nicely incorporated into a 2×2 effective Hamiltonian for the TLS; these
effects are simply given by a term added to the (1, 1) component of the Hamiltonian
[7, 8]:
2 → 2 + Σ SSH,∞ ≡
2 .
(9)
Here Σ SSH,∞ is proportional to the surface Green’s function of the combined SSH
chain and semi-infinite chain. This can be calculated analytically, although it is fairly
nasty. The result is [7]