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arxiv: 2606.09201 · v1 · pith:3EKPDFMPnew · submitted 2026-06-08 · ❄️ cond-mat.str-el · cond-mat.mes-hall· cond-mat.supr-con· quant-ph

Order parameters and ground-state phase diagram of the interacting topological Su-Schrieffer-Heeger model with extended-range hoppings

Tsz Hin Hui , Pedro D. Sacramento , Wing Chi Yu This is my paper

Pith reviewed 2026-06-27 15:10 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hallcond-mat.supr-conquant-ph
keywords interacting SSH modelextended-range hoppingstopological phasescharge-density-wave phasessuperconducting-like phasesorder parametersphase diagramSu-Schrieffer-Heeger model
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The pith

The interacting Su-Schrieffer-Heeger model with extended-range hoppings hosts topological phases plus two novel superconducting-like phases and five distinct charge-density-wave phases.

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

This paper examines the ground states of the Su-Schrieffer-Heeger chain when electron interactions and hoppings beyond nearest neighbors are both present. It constructs a phase diagram containing multiple topological phases together with two superconducting-like phases and five charge-density-wave phases that do not appear in simpler versions of the model. The work derives explicit order parameters for every phase and checks them against large-system numerical data, showing that the new phases arise specifically from the combination of imbalanced interactions and longer-range terms. In the interacting regime the order parameters allow bidirectional hoppings inside topological phases, whereas the non-interacting limit restricts them to unidirectional patterns. The simulations also confirm that entanglement entropy and fidelity correctly locate the boundaries between these phases.

Core claim

The central claim is that the interacting Su-Schrieffer-Heeger model with extended-range hoppings possesses a rich phase diagram that includes several topological phases, two novel superconducting-like phases, and five distinct charge-density-wave phases. The superconducting-like phases and two of the charge-density-wave phases are direct consequences of imbalanced interactions together with the extended hoppings. Order parameters are derived for each phase and are verified through large-system simulations, where they remain consistent with entanglement entropy and fidelity signatures of the transitions. Under interactions the derived order parameters permit non-unidirectional hoppings insid

What carries the argument

The order parameters derived separately for the topological, superconducting-like, and charge-density-wave phases, which serve as the diagnostic quantities that identify each phase in numerical simulations.

If this is right

  • The phase diagram contains topological phases coexisting with two novel superconducting-like phases and five distinct charge-density-wave phases.
  • The superconducting-like phases and two charge-density-wave phases appear only when interactions are imbalanced and hoppings extend beyond nearest neighbors.
  • Order parameters for all phases match the locations of phase transitions detected by entanglement entropy and fidelity.
  • Interactions allow non-unidirectional hoppings inside topological phases, unlike the unidirectional restriction seen without interactions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same order-parameter construction could be applied to variants with different interaction strengths or additional longer-range terms to predict further phases.
  • Experimental platforms such as optical lattices or quantum-dot arrays might realize the predicted phases by tuning interaction imbalance and hopping range.
  • The distinction between unidirectional and bidirectional hoppings under interactions suggests a route to engineer protected edge states that survive moderate disorder.

Load-bearing premise

Large-system numerical simulations correctly capture the true ground states without being distorted by finite-size effects or truncation artifacts that could mislabel the phases.

What would settle it

Observation in sufficiently large systems of one of the two novel superconducting-like phases or of the two charge-density-wave phases that appear only when both imbalanced interactions and extended-range hoppings are present would support the claim; their absence would falsify it.

Figures

Figures reproduced from arXiv: 2606.09201 by Pedro D. Sacramento, Tsz Hin Hui, Wing Chi Yu.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic diagram illustrating possible phases in var [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Ground-state expectation values of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The squared modulus of the weight [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The OPs for the possible phases in the interacting ESSH model with various [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The log-log plot of the pairing correlation function in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The OPs (top panel of each subplots), the half-block entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The OPs (top panel), half-block entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The OPs for the possible phases in the interacting SSH model with ( [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

Topological insulators have attracted numerous attentions recent years, where the Su-Schrieffer-Heeger (SSH) model is one of the most studied models. While the interacting version of it has been explored recently, the interplay between interactions and long-range hoppings merit further investigations. In this work, we uncover a rich phase diagram of the interacting SSH model with extended-range hoppings, in which it consists of several topological phases, two novel superconducting-like (SC-like) phases and five distinct charge-density-wave (CDW) phases. We substantiate that the SC-like and two CDW phases are direct consequences of imbalanced interactions and extended-range hoppings. We derive the order parameters (OPs) for each of the phases and verify them in large-system simulations, finding consistency with the entanglement entropy and the fidelity in capturing the phase transitions. In contrast to the non-interacting case where the favored hoppings are unidirectional in the topological phases, the derived OPs suggest non-unidirectional hoppings are possible under the influence of interactions.

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

1 major / 0 minor

Summary. The paper examines the interacting Su-Schrieffer-Heeger (SSH) model augmented by extended-range hoppings. It derives order parameters for multiple topological phases, two novel superconducting-like (SC-like) phases, and five distinct charge-density-wave (CDW) phases. A phase diagram is constructed from large-system numerical simulations whose results are cross-checked for consistency against entanglement entropy and fidelity. The central claim is that the SC-like phases and two of the CDW phases are direct consequences of interaction imbalance together with the extended hoppings; the work also notes that interactions permit non-unidirectional hoppings in topological phases, in contrast to the non-interacting limit.

Significance. If the numerical identification of the phases holds, the result is significant because it maps an extended interacting SSH model onto a rich phase diagram containing previously unreported SC-like and CDW orders whose microscopic origin is traced to imbalance and long-range terms. The explicit construction of order parameters together with their consistency checks against independent observables (entanglement entropy, fidelity) provides a concrete, falsifiable framework that can be tested in future DMRG or tensor-network studies of related models.

major comments (1)
  1. [Abstract and numerical-results sections] Abstract and the sections describing the numerical results: the manuscript states that order parameters are 'verified in large-system simulations' and 'consistent with entanglement entropy and fidelity' but supplies no values for Hamiltonian parameters (hopping ratios, interaction strengths), system sizes L, bond dimensions, or truncation-error thresholds. Because the attribution of the two SC-like and two CDW phases to imbalanced interactions plus extended hoppings rests entirely on these simulations, the absence of convergence data leaves open the possibility that finite-size drift or truncation artifacts have misclassified the phases.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of explicit numerical details. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and numerical-results sections] Abstract and the sections describing the numerical results: the manuscript states that order parameters are 'verified in large-system simulations' and 'consistent with entanglement entropy and fidelity' but supplies no values for Hamiltonian parameters (hopping ratios, interaction strengths), system sizes L, bond dimensions, or truncation-error thresholds. Because the attribution of the two SC-like and two CDW phases to imbalanced interactions plus extended hoppings rests entirely on these simulations, the absence of convergence data leaves open the possibility that finite-size drift or truncation artifacts have misclassified the phases.

    Authors: We agree that the manuscript as currently written does not supply the requested numerical parameters or convergence diagnostics in the abstract or main numerical-results sections, which is a legitimate concern for reproducibility and for confirming that the reported phases are free of finite-size or truncation artifacts. In the revised manuscript we will add explicit statements of the Hamiltonian parameters (hopping ratios and interaction strengths), the system sizes L used, the bond dimensions retained in the DMRG calculations, and the truncation-error thresholds. We will also include a brief discussion (or supplementary figures) demonstrating that the order parameters for the SC-like and relevant CDW phases remain stable upon increasing L and bond dimension, thereby supporting the attribution of these phases to interaction imbalance and extended hoppings. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper states that order parameters are derived for each phase and then verified via large-system simulations, with consistency checked against independent quantities (entanglement entropy and fidelity). No equations or steps are shown that reduce a claimed prediction or result to a fitted input, self-definition, or load-bearing self-citation by construction. The attribution of SC-like and CDW phases to imbalanced interactions plus extended hoppings is presented as a numerical finding rather than an algebraic identity, and the provided text contains no self-citation chains or ansatz smuggling that would force the central claims. The derivation chain is therefore self-contained against external numerical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on an unspecified numerical method whose assumptions are not stated.

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Reference graph

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    The ground state of the system in this limit is mainly contributed by the Fock state |c1 0c0 1c1 0c0 1c1 0c0 1c1 0c0 1⟩(17) and its global translation by one unit cell. Similar to the configuration in Eq. (12), these Fock states has a period of two unit cells, except that the wave “1100” now starts at site B while the one in Eq. (12) starts at site A. The...

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