Bulk-edge correspondence in the trimer Su-Schrieffer-Heeger model

A remarkable feature of the trimer Su-Schrieffer-Heeger (SSH3) model is that it supports localized edge states. However, in contrast to the dimer version of the model, a change in the total number of edge states in SSH3 without mirror-symmetry is not necessarily associated with a phase transition, i.e., a closing of the band gap. As such, the topological invariant predicted by the 10-fold way classification does not always coincide with the total number of edge states present. Moreover, although Zak's phase remains quantized for the case of a mirror-symmetric chain, it is known that it fails to take integer values in the absence of this symmetry and thus it cannot play the role of a well-defined bulk invariant in the general case. Attempts to establish a bulk-edge correspondence have been made via Green's functions or through extensions to a synthetic dimension. Here we propose a simple alternative for SSH3, utilizing the previously introduced sublattice Zak's phase, which also remains valid in the absence of mirror symmetry and for noncommensurate chains. The defined bulk quantity takes integer values, is gauge invariant, and can be interpreted as the difference of the number of edge states between a reference and a target Hamiltonian. Our derivation further predicts the exact corrections for finite open chains, is straightforwardly generalizable, and invokes a chiral-like symmetry present in this model.

Figure 1

(a) The unit cell of an SSH3 tight-binding model with couplings $u, v$ (intracell), and $w$ (intercell). $A, B$, and $C$ denote the three sublattices while the length of the unit cell is $d$, which we set to $d=1$. (b) The spectrum of the bulk Hamiltonian of SSH3 in the first Brillouin zone. Here, all hoppings are different (no degeneracies present). The spectrum is symmetric with respect to the point $(k=\pi /2,E=0)$, as indicated.

Figure 2

Continuation of energy spectrum with respect to the intercell hopping coupling $w$ for a chain of 30 sites ($n=10$ unit cells). (a) A mirror-symmetric chain ($u=v=6$). Notice that the emergence of the edge states does not occur exactly at $w=6$, which would be the case in the thermodynamics limit; this is due to the finite size of the system. The exact way to calculate the finite-size corrections is presented later in Sec. 3c. (b) A chain with all couplings different ($u=3$ and $v=6$)—no mirror symmetry is present.

Figure 3

The spatial profile of the four edge states exhibited by an SSH3 lattice. One can observe that pairs of edge states localized on the same edge (corresponding to opposite energies) are related by a sign change of the amplitude over half of the sites. Moreover, all four edge states do not have support over one out of the three sublattices.

Figure 4

Plot of the momentum shift ${\varphi }_{\lambda }\left(k\right)$ (red-dashed line) from its analytical expression (see Appendix pp3) for the middle band in two different coupling regimes for a finite chain with $N=10$ unit cells. (a) In the trivial regime, one can observe that this bulk quantity intersects with the $\left\{F_n\left(k\right)\right\}$ lines (blue) 10 times in the reduced Brillouin zone $k\in (0,\pi )$. (b) When in the edge-states regime, momentum shift intersects eight times with the possible solutions of the finite problem. Two states have left the band. (c) Finite-size effect, where NS Zak's phase is nonzero but the same number of intersections occurs as in the trivial regime. This kind of finite-size effect vanishes in the thermodynamic limit ($N\to \infty$).

Figure 5

Plot of the limiting lines from relation (19) where the transition happens for the finite system, in the parametric space of $a$ and $b$, for the bottom and middle band. Chains have $N=2,10,1000$ (dashed line, dotted line, and solid line, respectively) unit cells. Notice how, at the thermodynamic limit, the curves approach the lines predicted by (17a). Comparing the current figure with Fig. 6 it becomes evident that the number of edge states matches with the values predicted analytically by NS Zak's phase.

Figure 6

Parameter regimes with a different number of edge states for an SSH3 chain with $3N$ sites, as predicted by the introduced bulk quantity. Dashed lines represent the mirror-symmetric paths while the path with the dotted lines is not constrained by mirror symmetry.

Figure 7

The boundary conditions of an open chain for the cases of $3N$ and $3N+1$ sites. Since they are not the same, a different NS Zak's phase should be utilized in each case.

Figure 8

Continuation diagram for a finite SSH3 chain with $3N+1$ sites.

Figure 9

Comparison of NS Zak's phase with Zak's phase in the case of an SSH4 without mirror symmetry. While Zak's phase is not quantized, NS Zak's is quantized and predicts correctly the number of edge states that each band contributes. The spectral diagram corresponds to a finite chain with $4N$ (where $N=20$) sites. Due to that, the NS Zak's corresponding to the first sublattice has been utilized. The bulk invariants for the first and second band are presented. The results for the rest of the bands are identical due to chiral symmetry.