An elementary rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model

Abstract

We generalize the Su-Schrieffer-Heeger (SSH) model with the inclusion of arbitrary long-range hopping amplitudes, providing a simple framework to investigate arbitrary adiabatic deformations that preserve the chiral symmetry upon the bulk energy bands with any arbitrary winding numbers. Using only elementary techniques, we obtain a mathematically rigorous and physically transparent proof of the bulk-boundary correspondence for the generalized SSH model. The multiplicity of robust zero-energy edge modes is shown to be identical to the winding number. On the other hand, nonzero-energy edge modes, if any, are shown to be unstable under adiabatic deformations and not related to the topological invariant. Furthermore, under deformations of small spatial disorder, the zero-energy edge modes remain robust. *

Long-range hopping and indexing assumption in one-dimensional topological insulators

“…This will allow for a comparison between that operator with the new one associated with a new chiral symmetry, see Section IV. Note that the extended SSH has been addressed in the past in several works [10][11][12][13][14][15][16][17][18][19][20][21], but as far as we know its indexation problem has never been discussed.…”

“…4(b)]. The latter implies in certain energy intervals the existence of two right (left) edge eigenstates in the infinite chain [states with decay in the right (left) direction] that may be combined in order to generate a single right (left) edge state with zero amplitudes at all the sites of a unit cell of the chain, so that OBC may be introduced at these sites [12]. Note that while in the case of the SSH model with OBC, a single condition selects the left edge state from the set of left edge-like states of the infinite chain and that condition is that of zero amplitude at a virtual site at the left end (that will also impose zero amplitude in the respective sublattice), when we have two left edge states in the infinite chain with the same energy we need two conditions (these will be the zeros of amplitude at the two virtual sites A and B to the left of the chain so that the resulting eigenstate of the infinite chain does not feel the OBC).…”

“…This will be expected in particular if a band inversion occurs in the band structure of the linear chain as the relative values of the hopping terms change. In order to illustrate the previous arguments, we consider in this paper the chiral symmetry protected * rdias@ua.pt topological SSH insulator and add to it long-range hopping terms [10][11][12][13][14][15][16][17][18][19][20][21] in such a way that the Hamiltonian eigenbasis remains the same and the usual SSH chiral operator remains a mapping operator between eigenstates of the Hamiltonian (allowing us to compare it with new chiral operators). This model is simply the extended SSH chain with next-nearest neighbor (NNN) hoppings.…”

Localized interface modes in one-dimensional hyperuniform acoustic materials

“…The emergence of the boundary modes is governed by the bulk-boundary correspondence, which is a relation between the eigenstates of the system within the bulk spectrum and the number of the supported interface modes. These modes are topologically protected, i.e., they are stable against adiabatic chiral symmetry preserving perturbations if the band gap remains open [22,8]. The topological state of the system can be controlled by the gap closing and reopening, which is driven by the geometrical parameters.…”

Experimentally Detecting Quantized Zak Phases without Chiral Symmetry in Photonic Lattices

“…In this scheme, a topologically nontrivial 1D system should possess chiral symmetry, and the bulk topological invariant is the winding number corresponding to a quantized Zak phase [43]. A paradigmatic example is provided by the famous Su-Schrieffer-Heeger (SSH) model of polyacetylene [2,4], where the bulk-boundary correspondence holds [4,44].…”