Abstract
We propose a generalization of the SuSchriefferHeeger (SSH) model of the bipartite lattice, consisting of a periodic array of domain walls. The lowenergy description is governed by the superposition of localized states at each domain wall, forming an effective monoatomic chain at a larger scale. When the domain walls are dimerized, topologically protected edge states can appear, just like in the original SSH model. These new edge states are formed exclusively by solitonlike states and therefore, the new topological states are qualitatively different from the regular SSH edge states. They posses a much longer localization length and are more resistant to onsite disorder, in marked contrast to the standard SSH case.
Introduction
The last years have witnessed a growing interest on onedimensional models of nontrivial topological systems. This has been largely favored by the rapid advance of photonics and nanophotonics as an ideal playground to experimentally corroborate theoretical predictions^{1,2,3}. Different experimental setups, such as plasmonic nanoparticles and phononic lattices, have been used as practical realizations of onedimensional models^{4,5,6,7,8}. This emergence of novel experimental escenarios, in turn, has motivated further theoretical research, often on basic aspects or phenomena beyond the scope of common electronic systems, such as nonlinearities^{9,10} and Floquet insulators^{8,11,12}. The interest on onedimensional topologically protected modes is not only related to basic understanding, but to the practical implementation in the design of lowloss devices^{13,14}.
Perhaps, the most studied onedimensional model with a nontrivial topology is the SuSchriefferHeeger model of polyacetylene^{15} (SSH). It consists on a tightbinding model for the bipartite lattice, and displays solitonlike localized edge states at domain walls (i.e. stacking faults of the bipartite lattice). Recently, this model has been the subject of generalizations in order to observe new phenomena^{16,17}. Other onedimensional models, not restricted to the bipartite lattice, have been proposed and realized, showing novel edge states^{6,18,19,20}.
One of the main purposes behind the focus in simpler models as the SSH is to get insight on more complex systems or materials. For instance, depending on the termination, a nontrivial Zak phase in nanoribbons can arise, implying topologicallyprotected edge states^{21}. Similarly, it was shown that some edges states in 2D systems with a negligible spinorbit coupling for instance some terminations of black phosporus are indeed topologically protected^{22}. Also, the natural extension of the SSH model to two dimensions can host different topological phases, even with a zero Berry phase^{23}.
In this article we start by reproducing the results of the famous SSH model, Sec. 1, to set the notation and make its generalization easier. In Sec. 2, we introduce a new model, consisting on N interacting copies of the SSH model, originated from an array of domain walls. If the lattice of domain walls is dimerized a bipartite lattice of domain walls, we recover a SSHlike behavior formed by the superposition of localized modes. Naturally, this model has a nontrivial phase featuring edge states. In Sec. 3 the effect of disorder in the model is studied. Unlike the SSH model, the new edge modes are somewhat robust to onsite disorder.
The Bipartite Lattice
We start with a brief summary of the SSH model of polyacetylene^{15}, or more precisely, the tightbinding formulation of the bipartite lattice, see Fig. 1. We assume that the reader has some acquaintance with chiral symmetry, the SSH model and topological states of matter^{21,24}. In the optical context, the edge states from this model have been experimentally observed and theoretically explained in photonic superlattices^{1,25}.
In a finite bipartite lattice, the two different orderings (see, Fig. 1a,b) acquire a physical meaning. Localized edge states appears when the intercell coupling dominates. In this case, the edge states are sublattice polarized. It is tempting to call this phase ‘topological’, however in a finite chain the interaction between edge states is small but not negligible, resulting in a finite interaction energy. Nevertheless, the edge states remain sublattice polarized and they determine the lowenergy phenomena.
A stacking fault on the chain couplings results in a domain wall (DW), Fig. 1c, where a zeroenergy localized state appears (called a soliton in the context of polyacetylene). Often, domain walls come into pairs, Fig. 1d. If their distance is not too large, they interact weakly developing lowenergy modes. Therefore, the lowenergy physics is dominated by DW states, and we can ignore the bulk’s valence and conduction bands, Fig. 1e. A periodic set of DWs and its corresponding periodic lattice of DWs states leads to an effective monoatomic lattice, Fig. 1f. However, if the array of DWs is dimerized, the lowenergy states will be a bipartite lattice, see Fig. 1g,h. Such a lattice resembles the original SSH lattice and may have lowenergy, topologically protected edge states. In the remainder of this article we will delve on this possibility.
Free boundaries
The onedimensional bipartite lattice, see Fig. 1a,b, has the Hamiltonian
where t and v are the hopping amplitudes and each cell has two equal –but inequivalent– sites a, b (e.g. the sublattices). In spite of its simplicity, this system has a very rich physics. In the case of t < v, it has (almost) zeroenergy and sublattice polarized edge states solutions. In the limit N → ∞ the zeroenergy modes have a simple expression
where ψ_{L} is localized at the left edge of the chain, with width \(\varepsilon =\,\mathrm{log}(\frac{v}{t})\). The state ψ_{R} is localized at the right edge. α, β are normalization constants. For a finite chain size, N, the interaction between both edge states is small but finite \(\langle {\psi }_{L}{H}_{open}{\psi }_{R}\rangle \propto {e}^{\frac{N}{\varepsilon }}\), forming a bonding and antibonding pair: \(\frac{1}{\sqrt{2}}({\psi }_{L}\pm {\psi }_{R})\), see Fig. 2a,b. In Fig. 2 the finite size effects open a band gap nearly 100 times smaller than the band gap of the bulk system.
The bulklike states, have energies \(E=\pm \,\sqrt{{t}^{2}+{v}^{2}+2tv\,\cos (k)}\), see Fig. 2, the band gap (i.e. excluding edge states) is \(\mathrm{2}tv\gg 2\varepsilon \), validating our previous statement that bulk states are irrelevant for a lowenergy description.
Domain walls and periodic boundaries
Both bonding schemes of the bipartite lattice, see Fig. 1a,b can coexist next to each other, meeting on a lattice defect called Domain Wall (DW), see Fig. 1c. If each subsystem consist of N sites, after imposing periodic boundary conditions, i.e. by setting \({a}_{2N}^{\dagger }\equiv {a}_{0}^{\dagger }\), the whole lattice has two DWs, each with a localized state centered on it. The Hamiltonian of the full system is:
We can use the previous solutions ψ_{L}, ψ_{R}, Eqs (2 and 3) as an ansatz for the DW states at \(j=0,N\frac{1}{2}\):
where α, β are normalization constants. These solutions are valid for any value of t ≠ v. If v > t (or ε > 0), ψ_{a} is localized at j = 0 and ψ_{b} localized on \(j=N+\frac{1}{2}\). A negative value of ε just reverses the positions of the DW centers.
A Lattice of Periodic Domain Walls
Monospaced and Periodic Domain Walls
A bipartite lattice with M monospaced DWs –see Fig. 1f– has the following Hamiltonian
where h_{N}, h′_{N} are SSHlike Hamiltonians, but with different topological phase. The periodicity in H_{PDW} is two DWs, or 2N cells, but the spacing between domain walls is just N. This Hamiltonian is quite cumbersome, but if we focus on its lowenergy excitations, it can be greatly simplified, by just keeping the superposition of modes ψ_{a}, ψ_{b}, at each domain wall, see Eqs (5 and 6). Using them as a basis, the effective Hamiltonian is:
Each group of 2N sites is an effective ‘cell’ with two sublattices per cell (i.e. both DWs), just like in the standard bipartite lattice, but with just one single hopping t′. Therefore, the effective lowenergy excitations are just like those in a monoatomic chain with period N, see Figs 1f and 3a. The basis functions, located at the DWs, at sites 2mN, (2m + 1)N are
the functions \({\psi }_{a}^{\dagger },{\psi }_{b}^{\dagger }\) are based in Eqs (5 and 6). While Eq. (5) is already symmetrically centered around 2mN, we need to multiply Eq. (6) by \({e}^{\frac{1}{2}}\) to make it symmetrical around the DW at (2m + 1)N. The simple interpretation of this \(\frac{1}{2}\) factor is that the center of symmetry is in the middle of two adjacent b sites, see the lower inset from Fig. 2.
The interaction between two localized states is, approximately:
where the normalization constants α′, β′ are almost independent of N, \(\alpha ^{\prime} \approx 0.87(1+\frac{1}{2}{e}^{\frac{2N}{\varepsilon }})\). For large values of N the first term in the parenthesis dominates, but in some contexts like in optics the most common arrays consist of a limited number of waveguides.
The lowenergy states given by Eqs (7) or (10), are no longer topologically protected, even though they are locally sublattice polarized (on a scale of N sites), but on a larger scale (NM sites) the sublattices are mixed. Also, their energy is genuinely finite, forming an sband, Fig. 3a. This case was studied before in the continuum limit^{26} and the results agree with ours.
New topological states and the bipartite lattice of DWs
In this section we introduce a bipartite lattice of domain walls, starting by its Hamiltonian, and derive its lowenergy version, which is a new version of the SSH model, but with smaller hopping strenghts. As the SSH model, this new model has a phase with topologically protected edge states, see Fig. 3c. These new effective hoppings are strongly dependent on the distance between DWs, Fig. 3d. These predictions are confirmed by direct diagonalization of the full Hamiltonian, see Fig. 3b.
The distance between successive DWs can be dimerized, i.e. by setting the spacing from one DW to the next one as N cells to the right and N′ cells to the left. This changes slightly the Hamiltonian of a periodic lattice of DWs, from H_{PDW}, Eq. (7) to the Hamiltonian of a bipartite lattice of DWs (BDW hereafter):
where the limits in the sums of h_{N}′ \({h^{\prime} }_{N^{\prime} }\) must change accordingly. This produces two different hopping strengths between domain wall states:
where, for simplicity, we dropped the last term of Eq. (14), this approximation is valid if \(N,N^{\prime} \gg 1\) –see Fig. 3d.
The effective Hamiltonian for the BDW becomes a copy of the one for a bipartite lattice, but with a larger length scale and lower energies, see Fig. 3c,d:
Figure 3b shows the energy spectrum of the full Hamiltonian H_{BDW}. Inside the bulk gap there appears a band of the states at the DWs, and a new gap opens inside (reddish region), and in the middle of it two zeroenergy states appear. These states are built from DW states, localized at the edges of the system and are fully sublattice polarized (see inset in Fig. 3b).
The analogy with the regular SSH model^{24} in the periodic case is almost complete, in the limit of an infinitely long chain, M → ∞. The Fourier transform of the effective Hamiltonian, Eq. (15), gives
where h(k) is the kernel of the Hamiltonian. We can decompose h(k) as a linear combination of the Pauli matrices \(\vec{\sigma }\) = (σ_{x}, σ_{y}, σ_{z}),
with \(\vec{d}\) = (t′ + v′cos(k), v′sin(k), 0). Figure 3c shows the vector \(\overrightarrow{d}\) for some values of N, N′, and keeping the original hopping strengths constant. The geometric place of this vector is a circle and we can define a topological invariant related to it, the winding number or Chern number:
the geometrical interpretation of n_{c} is very simple, it counts how many times the curve \(\overrightarrow{d}\)(k) encircle the origin. The two possible values of n_{c} = {0, 1} defines the phase diagram of the system^{24}, see Fig. 3c,d. If N > N′ the system is in the ‘topological phase’, that is n_{c} = 1, conversely N < N′ is the trivial phase with n_{c} = 0. In the remaining case, N = N′, the curve \(\overrightarrow{d}\) touches the origin and no topological index can be defined: the system became metallic. Only the case with n_{c} = 1 has topologically protected edge states.
While \({H}_{BDW}^{eff}\) is useful to visualize the connection of the BDW and SSH models, it doesn’t capture other interesting complex phenomena, such as disorder. In order to explore this, we will employ the full Hamiltonian, Eq. (15) in the next section.
Disorder in the BDW Model
The resilience of topologically protected states in the SSH model to offdiagonal disorder is wellknown, as well as its weakness against disorder on the onsite energies^{27}. The band gap due to offdiagonal disorder is not shown since in the SSH model it is below our numerical precision. For the edges states of the BDW model, it lies below 10^{−4} for a disorder amplitude of δ = [0, 2t]. In smaller SSH chains the offdiagonal disorder is able to close the (residual) gap, but the chains considered in Fig. 4 are too large to show this effect.
The nonuniformity of onsite disorder directly breaks the chiral symmetry, destroying the topological protection. Figure 4a shows an almost linear band gap for the edge states of the SSH model.
The edge states of the BDW model are similar to their parent topological states, and they are –in principle– fragile to onsite disorder. However, the magnitude of the band gap due to onsite disorder is much smaller for the BDW edge states, see Fig. 4a. This can be partially explained as follows: While the SSH edge states are directly affected by the diagonal disorder, the BDW edge states are only affected by the averaged disorder over its characteristic length ε_{BDW} –which averages to zero for large ε_{BDW}. In the Figure, the asymptoticlike band gap is well below the band gap of the bands formed by the DWs states, which in turn is much smaller than the bandgap from the SSH model. Given t = 1, v = 1.5, the characteristic length of the SSH model is \({\varepsilon }_{SSH} \sim 2.5\) cells, or about 5 sites. Instead –for the same hopping strengths t, v– the characteristic length of the SSH model, see Eq. (17), is \({\varepsilon }_{BDW} \sim 1.9\) supercells or 42 sites. To test the relationship between band gap and ε, one would naively compare a SSH and BDW chains with the same localization length of the edges states, by using different t, v in each chain. But, that comparison is unfair: while the SSH edge states are sublattice polarized, the BDW edge states also are sublattice polarized on the lattice of DWs. Therefore, one could expect a similar behavior of SSH and BDW chains when \({\varepsilon }_{BDW} \sim 2{\varepsilon }_{SSH}\), this is achieved when t = 1, v = 1.1 in the SSH and t′ = 1, v′ = 1.5 in the BDW model. Figure 4b shows similar band gaps for both models when the previous condition is satisfied.
To understand the behavior of the BDW chain under onsite disorder, we show in Fig. 5 the averaged band gap when one of the hoppings, namely, v is varied, while keeping the other parameters fixed (t, δ). At moderate disorder (i.e. while the BDW increases linearly with disorder in Fig. 4a), δ = 0.5, both the SSH and the BDW models are similar: the band gap increases with v, which is to be expected since the localization lenght of edges states decreases with v. At each value of v the band gap of the BDW edge states is smaller than the gap from the SSH model: the BDW model has a band gap similar to the SSH with a smaller difference of the hoppings, t − v. Increasing the onsite disorder amplitude, δ = 1.0, Fig. 4b, the SSH bandgap is almost as twice as large for most values of v. But the BDW band gap doesn’t seem to increase appreciably: the resilience of BDW edge states to onsite disorder is a rather general feature and depends weakly of the original hopping amplitudes t, v. In contrast, the SSH model is fragile against disorder, especially when its hopping strengths are very different, that is, when i.e. ε_{SSH} comprises few sites.
To get a deeper insight on the effect of disorder on both, the SSH and the BDW models, we calculated the inverse participation ratio (IPR)^{28}:
where c_{n} is the wavefunction amplitude at site n. A completely localized state has IPR = 1, and a fully delocalized wave has IPR = 1/N, with N being the length of the chain. To introduce disorder into the SSH and BDW models, we add a random amount to the diagonal and/or the offdiagonal terms of the Hamiltonians. This random value is taken from a uniform random distribution of width δ. For diagonal disorder it is irrelevant whether the disorder averages to zero or not, since only differences in the onsite terms are meaningful. Figure 6 shows IPR for both, onsite and offdiagonal disorder (considering just nearestneighbors). The Blöch states of the SSH lattice become localized with disorder, regardless of whether it is onsite or offdiagonal disorder. The regular Blöch states of the BDW have a very similar behavior they are Blöch states too, but they are slighty more localized for any finite value of δ. This is a consequence of the ‘fragmentation’ of the extended states (valence and conduction bands) into smaller groups with smaller bandgaps –see the green region in Fig. 3b. In the BDW model there is another type of Blöch states, these formed by the interaction between DW states, forming a wavepacket, see the green region of Fig. 3a,b. The IPR of these states is very similar to the regular Blöch states, and for clarity they are not included in Fig. 6.
The BDW’s edges states (right panels in Fig. 6) show a sudden jump of the IPR doubling its value for tiny amounts of onsite disorder. When δ is of the order of the interaction between the edges states, they no longer form an bonding antibonding pair, \(\frac{1}{\sqrt{2}}({\psi }_{L}\pm {\psi }_{R})\), but instead they localize at the left or right edge. Due to the smallness of the energy involved in this process one can think of it as an artifact (i.e., the IPR should be ~0.18 for no disorder), but it shows that a very small onsite disorder can prevent the occurrence of charge fractionalization in the model^{29}.
After reaching \(\delta \sim tv\), the edge BDW states have a similar IPR, almost independent of the strenght of the onsite disorder δ: despite being localized by the chiral symmetry, the model allows a localization length of several sites, preventing a delocalization by onsite disorder. In contrast, the SSH edge states have an important delocalization due to the onsite disorder: the chiral symmetry already localized them to a few sites, and the breaking this symmetry overcomes the localization due to the disorder itself, the IPR decreases. This behavior, markedly different on both models, is consistent with the opening of a band gap by onsite disorder, Figs 4 and 5.
Finally, in regard to offdiagonal disorder, it increases the IPR of the bulk states of the SSH and BDW models. It also slightly increases the IPR of the edges states of both models. This is consistent with the absence of band gap due to offdiagonal disorder.
Conclusions
We have examinated a generalization of the SSH model of polyacetylene to an array of domain walls focusing on the corresponding localized states. Under a monospaced lattice of domain walls an sband appears in the middle of the bandgap, whose modes are formed by an extended superposition of the localized domain wall states.
If, instead, the domain walls form a bipartite lattice, a new bandgap appears in the middle of the extended domain wall states. This new configuration can host topologicallyprotected edge states, resembling the SSH model but on a much larger spatial scale and lower energies. A simple lowenergy description was given, including the phase diagram of the system’s topological invariant.
The modes derived from domain walls states have interesting properties related to onsite and offdiagonal disorder. While in some aspects such as the bandgap magnitude they are more resilient to onsite disorder than SSH edge states, an almost negligible amount of disorder suffices to localize the wavefunction on a single edge unlike the SSH edges states.
We believe that an understanding of the properties of lowdimensional systems in the presence of topological disorder (DWs) and local disorder (Anderson) is a necesary step towards the design of future robust lowloss devices. The effect of nonlinearity on the topological robustness of these systems is another subject of interest (under investigation) and will be reported elsewhere.
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Acknowledgements
This work was supported in part by Fondecyt Grants No. 1150806 and No. 1160177, Programa ICM Grant No. RC130001, the Center for the Development of Nanoscience and Nanotechnology CEDENNA FB0807 and CONICYT Doctoral fellowship grant #21151207.
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F.M. and M.I.M. conceived the idea and wrote the manuscript. F.M., F.P., J.M. and M.I.M. carried out the calculations and analysis.
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Munoz, F., Pinilla, F., Mella, J. et al. Topological properties of a bipartite lattice of domain wall states. Sci Rep 8, 17330 (2018). https://doi.org/10.1038/s41598018356516
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Keywords
 Periodic Domain Walls
 Bipartite Lattice
 DW States
 Topological Protection
 Edge States
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