Exact analytical edge states in the extended Su-Schrieffer-Heeger model

The gist The authors derive exact analytical expressions for the edge states of a semi-infinite extended Su–Schrieffer–Heeger (eSSH) model, which includes hopping processes between translationally inequivalent atoms beyond nearest neighbors, and establish a bulk–boundary correspondence based on the winding number of the bulk Hamiltonian<ref:2604.20561#pg2>.

Topological Classification and Bulk-Boundary Correspondence

The topological phase diagram is determined from the winding number of the bulk Hamiltonian under periodic boundary conditions (PBC), where changes in this winding number coincide with bulk gap closings and with the condition z = 1 for the edge-state solutions<ref:2604.20561#pg3>. The model belongs to class BDI of topological insulators, characterized by an integer Z topological invariant<ref:2604.20561#pg4>. The topological invariant—the winding number—is given by the formula δ = 1/(2π) integral over the Brillouin zone of phi˙(k)dk<ref:2604.20561#pg4>. This winding number is stable under any deformation of the Hamiltonian that preserves chiral symmetry and does not close the gap<ref:2604.20561#pg4>.

Analytical Derivation of Edge States

Exact analytical expressions for the zero-energy eigenstates—the topologically relevant states—of a semi-infinite chain are obtained by employing a method introduced by Alase et al., which allows for closed-form expressions<ref:2604.20561#pg3>. The corresponding wave functions decay exponentially away from the edge, with a factor z for each additional unit cell<ref:2604.20561#pg4>. The bulk eigenvalue equation leads to a matrix equation for the coefficients u(z) and v(z), resulting in a non-trivial solution if and only if E squared − Q1(z)Q2(z) = 0<ref:2604.20561#pg4>. For zero-energy eigenstates (E = 0), this condition simplifies to Q1(z)Q2(z) = 0<ref:2604.20561#pg4>.

Analysis of Edge State Regimes

The analysis of the roots of the factors Q1(z) and Q2(z) reveals different physical regimes based on the parameters u0, u1, and u2<ref:2604.20561#pg4>.

(i).˜u2 is large enough and positive:

In this case, both roots are complex conjugate: z± = -1/(2˜u2) ± i qu˜0/u˜2− (1/(2˜u2))2, with z+ = z− = p u0/u2. This leads to two solutions of the form ψA(z±)⟩, whose weight lies on the A-type sites only<ref:2604.20561#pg4>.

(ii).˜u2 is large enough and negative:

The roots are z± = 1/(2u˜2) ± q u˜0/u˜2 + (1/(2u˜2))2, again with z±. This appearance of two edge states is consistent with the bulk winding number δ = 2<ref:2604.20561#pg4>.

(iii).˜u2 is small:

The roots are approximated as z+ ≈ -u˜0 and z− ≈ -1/u˜2 in leading order<ref:2604.20561#pg4>. The solution corresponding to ψA(z+⟩ is an edge state of the form ψA(z+)⟩, which is consistent with the bulk winding number δ = 1<ref:2604.20561#pg4>.

Application to Experimental Realizations

The analytical results are applied to two different experimental realizations of the model, including one involving second-neighbor hopping u2<ref:2604.20561#pg5>. For finite chains, approximate analytical expressions for the low-energy edge states and their energies are derived, which show excellent agreement with numerical results<ref:2604.20561#pg4>. The weights of each state, aj 2, in each set of parameters are shown in Figure 5<ref:2604.20561#pg8>. The exact analytical results obtained for a semi-infinite chain can be extended to the case of a finite chain of N unit cells within an error of the order of z2N /∆g in the energy<ref:2604.20561#pg12>.

Application to Bipartite Superradiance Lattices

The machinery developed in Section III is applied to obtain the edge states along the topological phase transition in bipartite superradiance lattices, where a topological phase transition from a trivial phase with δ = 0 to a non-trivial phase with δ = 2 is expected<ref:2604.20561#pg9>. The adiabatic changes of the Hamiltonian take place along the curve 1 − 4˜u0(η)˜u2(η) = 0, which is displayed in the left panel of Figure 8 as a red dashed line<ref:2604.20561#pg9>. The edge states are found to be present when η > 1, and no edge state is present when η.

Summary of Findings

The derived number of edge states is consistent with the bulk–boundary correspondence of topological phases<ref:2604.20561#pg4>. Exact analytical expressions are obtained for a chain with open boundary conditions, and approximate analytical expressions for low-energy edge states and their energies show excellent agreement with numerical results<ref:2604.20561#pg4>. The findings clarify the fundamental nature of the end states emerging in the extended Su–Schrieffer–Heeger model<ref:2604.20561#pg6>. The procedure based on Alase et al. can be straightforwardly extended by solving a polynomial equation of degree M, which entails a significantly lower computational cost than standard numerical methods<ref:2604.20561#pg6>.

Conclusion

The paper has derived exact analytical expressions for the edge states of a chain with open boundary conditions and demonstrated that the resulting number of edge states is consistent with the bulk–boundary correspondence of topological phases<ref:2604.20561#pg4>. The authors also demonstrate that the approximation is highly accurate<ref:2604.20561#pg4>. The work clarifies the fundamental nature of the end states emerging in the extended Su–Schrieffer–Heeger model<ref:2604.20561#pg6>. The exact analytical results obtained for a semi-infinite chain can be extended to the case of a finite chain of N unit cells within an error of the order of z2N /∆g in the energy<ref:2604.20561#pg12>. The work clarifies the fundamental nature of the end states emerging in the extended Su–Schrieffer-Heeger model<ref:2604.20561#pg6>.

Acknowledgments

The authors state that they are partially supported by CONICET, Argentina<ref:2604.20561#pg2>. The work is supported by CONICET, Argentina<ref:2604.20561#pg2>.

Appendix A

In this appendix, the authors obtain a pair of orthonormalized edge states corresponding to a winding number δ = 2<ref:2604.20561#pg11>. The norm is computed using the complex geometric series P∞j=p z j = z p 1−z, valid for z. The resulting normalized difference state is ψc−(z)⟩ = ψ−(z) / c−(z)<ref:2604.20561#pg11>. The resulting normalized sum state is ψc+(z)⟩ = c+(z)X∞j=1Aj⟩<ref:2604.20561#pg11>.

Appendix B

For a chain of N unit cells, the normalized states analogous to those in Eq.

1. Improve Topological Phase Classification in eSSH Models with Winding Number Analysis: The improved system can accurately determine topological phases by calculating the winding number of the bulk Hamiltonian under periodic boundary conditions, which determine[s] the topological phase diagram.

2. Develop Exact Analytical Edge State Solvers for Open Boundary Conditions: The system can derive exact analytical expressions for the edge states of a semi-infinite chain using methods like Alase et al., allowing it to find zero-energy eigenstates by solving polynomial equations of degree M, which is more efficient than standard numerical methods.

3. Predict Edge State Properties for Experimental Realizations: The improved system can explain several notable features observed in the work of C. Li et al. reported in Ref. [8] by deriving analytical, approximate—yet highly accurate—expressions for the low-energy edge states and their energies, showing excellent agreement with the numerical results of Ref. [8].

4. Real-Time Simulation of Topological Phase Transitions: The system can analyze experimental trajectories, such as those in Ref. [9], to derive the corresponding edge states and analyze their structure along this path, revealing nontrivial contributions from the different lattice sites to the spatial weight distribution of these states.

5. Accurate Finite Chain Edge State Prediction: For finite chains of N unit cells, the system can obtain results within an error of the order of z2N /Δg in the energy, providing two pairs of edge states, each with its weight equally distributed between the two ends of the chain for specific experimental parameters.

We investigate the topology of the different phases of the extended Su-Schrieffer-Heeger (eSSH) model, which includes hopping processes between translationally inequivalent atoms beyond nearest neighbors. Exact analytical expressions for the edge states of a semi-infinite eSSH chain are derived, with wave functions that decay exponentially from the boundary with a unit-cell decay factor z. From the winding number of the bulk Hamiltonian under periodic boundary conditions, we determine the topological phase diagram and establish the bulk-boundary correspondence: changes in the winding number coincide with bulk gap closings and with the condition z=1 for the edge-state solutions. For finite chains, we further obtain analytical, approximate expressions for the low-energy edge states, which are shown to be highly accurate.