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arxiv: 2604.20561 · v1 · submitted 2026-04-22 · ❄️ cond-mat.other · quant-ph

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

A. A. Aligia, P. A. Grizzi, P. Roura-Bas

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Pith reviewed 2026-05-09 22:50 UTC · model grok-4.3

classification ❄️ cond-mat.other quant-ph
keywords extended Su-Schrieffer-Heeger modeledge statestopological phaseswinding numberbulk-boundary correspondenceanalytical solutionssemi-infinite chain
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The pith

Exact analytical expressions for edge states in the extended Su-Schrieffer-Heeger model decay exponentially from the boundary with a unit-cell factor z.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives exact analytical wave functions for edge states in semi-infinite chains of the extended Su-Schrieffer-Heeger model. These states localize at the boundary and decay exponentially with a characteristic factor z per unit cell. The authors calculate the winding number of the bulk Hamiltonian under periodic conditions to map the topological phases and show that changes in the winding number align exactly with bulk gap closings and the condition that the magnitude of z equals one. For finite chains they supply accurate approximate analytical forms for the low-energy edge states.

Core claim

The authors establish that the edge states of a semi-infinite extended Su-Schrieffer-Heeger chain admit exact analytical forms as exponentially decaying functions with a complex decay factor z per unit cell. The topological phase diagram follows from the winding number of the bulk Hamiltonian under periodic boundary conditions, and the bulk-boundary correspondence is verified because winding-number jumps occur at the same parameter values where the gap closes and where |z| reaches unity. Approximate analytical expressions for low-energy edge states on finite chains are also obtained and shown to be highly accurate.

What carries the argument

The unit-cell decay factor z, which parametrizes the exact exponential localization of the edge-state wave function and marks topological transitions when its modulus equals one.

Load-bearing premise

The extended hopping terms do not introduce topological invariants beyond the winding number, and the single-factor exponential decay ansatz for the semi-infinite boundary captures every possible edge mode.

What would settle it

Numerical diagonalization of a large finite chain that produces edge-state wave functions whose spatial decay profile deviates from the predicted form set by z, or a parameter sweep in which the winding number changes without a corresponding gap closing or |z|=1 point.

Figures

Figures reproduced from arXiv: 2604.20561 by A. A. Aligia, P. A. Grizzi, P. Roura-Bas.

Figure 1
Figure 1. Figure 1: FIG. 1. Color online. Cartoon of the semi-infinite chain with [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) Top panel [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (Color online) Band structure along the lines [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (Color online) Weights, [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (Color online) Left: Phase diagram of the superradi [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (Color online) Same as Fig. 6 for the set [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (Color online) Weights of the component of the wave [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript derives exact analytical expressions for the edge states of a semi-infinite extended Su-Schrieffer-Heeger (eSSH) chain, with wave functions decaying exponentially from the boundary via a single unit-cell factor z. It computes the winding number of the bulk periodic Hamiltonian to obtain the topological phase diagram and verifies bulk-boundary correspondence by showing that winding-number jumps coincide with bulk gap closings and the condition |z|=1. Approximate analytical expressions are also given for low-energy edge states on finite chains.

Significance. If the derivations are rigorous, the exact closed-form edge-state solutions and explicit confirmation of bulk-boundary correspondence for generic extended hoppings would provide a valuable analytical benchmark for 1D topological models beyond the standard SSH chain, aiding both theoretical understanding and numerical validation in condensed-matter physics.

major comments (2)
  1. [edge-state derivation section] The central derivation of edge states (likely in the section presenting the semi-infinite chain solutions) employs a single-z exponential ansatz of the form decaying with unit-cell factor z. For the extended SSH model with next-nearest-neighbor or further hoppings, the recurrence relation for the two-component amplitudes is of order higher than 2; the characteristic polynomial therefore admits multiple roots. The manuscript must explicitly demonstrate that boundary conditions at the chain end force the coefficients of all other |z_i|<1 roots to vanish, otherwise the reported single-z expressions are incomplete and additional localized modes may exist.
  2. [topological phase diagram and bulk-boundary correspondence section] The claim of exact coincidence between winding-number changes, gap closings, and |z|=1 (stated in the abstract and phase-diagram discussion) relies on the bulk winding number being the sole topological invariant. With extended hoppings, it is necessary to confirm that no additional invariants arise and that the chosen ansatz captures all decaying solutions without omission; otherwise the bulk-boundary correspondence proof is incomplete.
minor comments (2)
  1. [model definition] Notation for the extended hopping parameters should be defined explicitly at first use with a clear diagram of the lattice.
  2. [finite-chain section] The finite-chain approximate expressions would benefit from an explicit error bound or comparison to exact diagonalization for at least one parameter set.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the presentation of our results on the extended SSH model. We address each major comment point by point below, indicating the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [edge-state derivation section] The central derivation of edge states (likely in the section presenting the semi-infinite chain solutions) employs a single-z exponential ansatz of the form decaying with unit-cell factor z. For the extended SSH model with next-nearest-neighbor or further hoppings, the recurrence relation for the two-component amplitudes is of order higher than 2; the characteristic polynomial therefore admits multiple roots. The manuscript must explicitly demonstrate that boundary conditions at the chain end force the coefficients of all other |z_i|<1 roots to vanish, otherwise the reported single-z expressions are incomplete and additional localized modes may exist.

    Authors: We appreciate this important observation regarding the order of the recurrence. While the extended hoppings do lead to a higher-order characteristic equation, the specific boundary condition at the open end of the semi-infinite chain uniquely constrains the solution to a single decaying mode. We will revise the edge-state derivation section to explicitly solve the characteristic polynomial, identify all roots, and demonstrate that the boundary conditions set the coefficients of any additional |z_i|<1 roots to zero, confirming that the single-z ansatz captures the complete localized solution without omission. revision: yes

  2. Referee: [topological phase diagram and bulk-boundary correspondence section] The claim of exact coincidence between winding-number changes, gap closings, and |z|=1 (stated in the abstract and phase-diagram discussion) relies on the bulk winding number being the sole topological invariant. With extended hoppings, it is necessary to confirm that no additional invariants arise and that the chosen ansatz captures all decaying solutions without omission; otherwise the bulk-boundary correspondence proof is incomplete.

    Authors: We agree that explicit confirmation is needed for rigor. In one-dimensional chiral-symmetric systems (class BDI), the winding number remains the complete topological invariant regardless of hopping range. We will add a dedicated paragraph in the topological phase diagram section recalling this symmetry classification and verifying that no additional invariants appear. We will also cross-reference the revised edge-state analysis to confirm that the ansatz exhausts all decaying solutions, thereby completing the bulk-boundary correspondence argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard bulk-boundary analysis with explicit ansatz solution

full rationale

The derivation computes the winding number directly from the periodic bulk Hamiltonian and solves the open-boundary recurrence via the single-z exponential ansatz, yielding explicit z from the characteristic equation. Neither step reduces to a fitted parameter, self-definition, or self-citation chain; the bulk-boundary link follows the conventional 1D procedure without the result being presupposed by the inputs. The paper remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard tight-binding Hamiltonian for the eSSH chain, the definition of the winding number for 1D systems, and the assumption that an exponentially decaying ansatz with unit-cell factor z solves the boundary problem exactly.

axioms (2)
  • domain assumption The system is described by a single-particle tight-binding Hamiltonian with translationally inequivalent atoms and hoppings beyond nearest neighbors.
    Invoked throughout the abstract as the definition of the eSSH model.
  • standard math The topological phase is classified by the winding number of the bulk Bloch Hamiltonian under periodic boundary conditions.
    Used to determine the phase diagram and bulk-boundary correspondence.

pith-pipeline@v0.9.0 · 5433 in / 1516 out tokens · 37914 ms · 2026-05-09T22:50:08.060856+00:00 · methodology

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Reference graph

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