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arxiv: 2604.05928 · v1 · submitted 2026-04-07 · ❄️ cond-mat.str-el · quant-ph

Quantum phases in the interacting generalized Su-Schrieffer-Heeger model

Jing-Hua Niu , Jia-Lin Liu , Ke Wang , Shan-Wen Tsai , Jin Zhang This is my paper

Pith reviewed 2026-05-10 18:46 UTC · model grok-4.3

classification ❄️ cond-mat.str-el quant-ph
keywords interacting Su-Schrieffer-Heeger modelsymmetry-protected topological phasesentanglement spectrumstring order parameterLuttinger liquidcharge-density wavegapless SPT phasequantum phase transitions
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The pith

Interactions transform the free-fermion SPT phases of a generalized Su-Schrieffer-Heeger model into distinct interacting topological phases that keep their diagnostic signatures.

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

The paper studies the quantum phases of a half-filled chain with intracell, nearest-neighbor, and next-nearest-neighbor hoppings plus an on-site interaction between sublattices. Without interactions the model shows one trivial phase and two symmetry-protected topological phases, identified by different winding numbers under periodic boundaries and by two-fold or four-fold entanglement-spectrum degeneracies under open boundaries. Adding interactions causes the topological phases to evolve into interacting versions that still display the same entanglement degeneracies, boundary modes, and nonzero string order parameters. Strong repulsive interactions insert a symmetry-breaking phase with uniform but unequal sublattice densities, while attractive interactions produce period-2 and period-4 charge-density waves, Luttinger liquids, a paired Luttinger liquid, and a gapless symmetry-protected topological phase with a gapless charge mode plus protected edge currents. These results matter because they show how topology can persist or change when electrons interact, a necessary step toward understanding real materials and tunable quantum systems.

Core claim

In the noninteracting limit the generalized Su-Schrieffer-Heeger model hosts one trivial phase and two SPT phases distinguished by winding numbers and entanglement-spectrum degeneracies. When interactions are turned on, these SPT phases become distinct interacting topological phases that continue to exhibit entanglement-spectrum degeneracy structures, boundary modes, and nonzero string order parameters. For strong repulsive interactions a symmetry-breaking phase with spatially uniform but unequal sublattice densities appears, while strong attractive interactions generate period-2 and period-4 charge-density-wave phases. At intermediate attractive interactions the competition between hopping,

What carries the argument

The generalized interacting Su-Schrieffer-Heeger model with intracell, nearest-neighbor, next-nearest-neighbor hoppings and on-site inter-sublattice interaction; the mechanism is the survival of topological diagnostics (entanglement-spectrum degeneracy, boundary modes, string order) under interactions together with the emergence of gapless phases classified by central charge.

Load-bearing premise

The numerical diagnostics such as entanglement-spectrum degeneracy and string order parameters continue to classify phases correctly even when interactions become strong enough to mix the original free-fermion bands.

What would settle it

A numerical computation that finds the string order parameter drops to zero or the entanglement spectrum loses its characteristic degeneracy inside the parameter region the paper identifies as an interacting topological phase would falsify the claim that the signatures are retained.

Figures

Figures reproduced from arXiv: 2604.05928 by Jia-Lin Liu, Jing-Hua Niu, Jin Zhang, Ke Wang, Shan-Wen Tsai.

Figure 1
Figure 1. Figure 1: FIG. 1. Phase diagram in the noninteracting limit with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Ground-state phase diagrams of the interacting (ex [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Ground-state phase diagrams of the interacting generalized SSH model in the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. String order parameters, edge modes, and entanglement spectra (ES) for three topologically distinct gapped phases. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows these order parameters along rep￾resentative cuts across symmetry-breaking and nearby symmetric gapped phases: CDW2 to trivial and SPT1 in Figs. 5(a) and (b), CDW4 to trivial and SPT2 in Figs. 5(c) and (d), and trivial to SP to SPT1 and trivial to SP to SPT2 in Figs. 5(e) and (f), respectively. For comparison with the SPT phases, we also plot SvN, the Schmidt gap ∆λ = λ1 − λ2, the SOPs, and the exces… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Single-particle and two-particle correlation functions [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. SOPs, neutral energy gap [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Entanglement entropy as a function of [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
read the original abstract

We investigate the quantum phases of a half-filled generalized interacting Su-Schrieffer-Heeger model with intracell, nearest-neighbor, and next-nearest-neighbor intercell hoppings, together with an on-site inter-sublattice interaction. In the noninteracting limit, the model hosts one topologically trivial phase and two symmetry-protected topological (SPT) phases, distinguished under periodic boundary conditions by different winding numbers and under open boundary conditions by two-fold and four-fold entanglement-spectrum degeneracies, respectively. When interactions are introduced, these free-fermion SPT phases evolve into distinct interacting topological phases that retain characteristic signatures such as entanglement-spectrum degeneracy structures, boundary modes, and nonzero string order parameters. For strong repulsive interactions, a symmetry-breaking phase with unequal but spatially uniform sublattice densities appears between the trivial and topological regimes. For strong attractive interactions, period-2 and period-4 charge-density-wave phases emerge from particle clustering. At intermediate attractive interactions, the competition between interaction-induced localization and hopping-induced delocalization gives rise to a Luttinger liquid phase, a paired Luttinger liquid phase, and a gapless symmetry-protected topological (gSPT) phase. The gSPT phase is characterized by a gapless charge mode together with symmetry-protected current-carrying edge states. We further characterize the gapless phases and the associated quantum phase transitions through central charges and critical exponents.

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

0 major / 4 minor

Summary. The manuscript investigates the quantum phases of the half-filled generalized interacting Su-Schrieffer-Heeger model with intracell, nearest-neighbor, and next-nearest-neighbor intercell hoppings together with an on-site inter-sublattice interaction. In the non-interacting limit it identifies one trivial phase and two SPT phases distinguished by winding numbers and entanglement-spectrum degeneracies (two-fold and four-fold). With finite interactions the free-fermion SPT phases evolve into interacting topological phases that retain entanglement-spectrum degeneracy, boundary modes, and nonzero string order parameters; strong repulsion produces a symmetry-breaking phase with uniform but unequal sublattice densities, while strong attraction yields period-2 and period-4 CDW phases. At intermediate attractive interactions the model realizes a Luttinger liquid, a paired Luttinger liquid, and a gapless SPT (gSPT) phase characterized by a gapless charge mode plus symmetry-protected current-carrying edge states; these gapless phases and their transitions are further analyzed via central charges and critical exponents.

Significance. If the numerical diagnostics remain reliable, the work supplies a comprehensive phase diagram for an extended 1D interacting fermionic chain that demonstrates the persistence of topological signatures under interactions and introduces a concrete realization of a gapless symmetry-protected topological phase. The multi-diagnostic approach (entanglement spectra, string order, central charges, boundary modes) and the explicit mapping from free-fermion to interacting regimes constitute a useful reference for the community studying interacting SPT physics in one dimension.

minor comments (4)
  1. The abstract states that the gSPT phase is 'characterized by a gapless charge mode together with symmetry-protected current-carrying edge states,' but the manuscript should explicitly show the edge-current operator or correlation function used to confirm the current-carrying property (rather than inferring it solely from symmetry protection).
  2. In the discussion of strong-repulsive interactions, the symmetry-breaking phase is described as having 'unequal but spatially uniform sublattice densities'; a brief plot or table of the density difference versus U would help readers assess the order-parameter magnitude and its saturation.
  3. The central-charge extractions for the Luttinger-liquid, paired-Luttinger-liquid, and gSPT phases rely on entanglement-entropy scaling; the manuscript should state the range of system sizes employed and the fitting window used to extract c, together with any finite-size extrapolation procedure.
  4. Notation for the three hopping amplitudes (t1, t2, t3) and the interaction U is introduced in the abstract but should be defined with explicit Hamiltonian terms in the first section of the main text for immediate readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and accurate summary of our work on the quantum phases of the half-filled generalized interacting Su-Schrieffer-Heeger model. We appreciate the positive evaluation of the significance of the results, including the comprehensive phase diagram, the persistence of topological signatures under interactions, and the concrete realization of the gapless SPT phase. The recommendation for minor revision is noted, and we are happy to incorporate any editorial improvements to enhance clarity or presentation.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's analysis relies on standard numerical diagnostics (entanglement-spectrum degeneracy, string order parameters, central charges, boundary modes) applied to the interacting generalized SSH model. These observables are independently established in the literature for classifying 1D fermionic phases and do not reduce to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. Phase boundaries and the evolution from free-fermion SPT phases to interacting counterparts are determined directly from these external benchmarks rather than by construction from the target quantities themselves. The derivation chain remains self-contained with no quoted steps exhibiting circular reduction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The phase diagram is obtained by scanning the model's hopping and interaction parameters; no new particles or forces are introduced.

free parameters (2)
  • interaction strength U
    Varied continuously to locate phase boundaries; values are not derived from first principles.
  • hopping amplitudes t1, t2, t3
    Model parameters scanned to realize different winding numbers and interaction regimes.
axioms (2)
  • domain assumption Half-filling constraint
    Imposed to enforce particle-hole symmetry and simplify the phase diagram.
  • standard math Periodic and open boundary conditions used for diagnostics
    Standard for distinguishing bulk topology from edge modes.

pith-pipeline@v0.9.0 · 5558 in / 1227 out tokens · 34061 ms · 2026-05-10T18:46:48.448908+00:00 · methodology

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  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
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    Relation between the paper passage and the cited Recognition theorem.

    We investigate the quantum phases of a half-filled generalized interacting Su-Schrieffer-Heeger model with intracell, nearest-neighbor, and next-nearest-neighbor intercell hoppings, together with an on-site inter-sublattice interaction... distinguished under periodic boundary conditions by different winding numbers and under open boundary conditions by two-fold and four-fold entanglement-spectrum degeneracies

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    the Luttinger parameter K is extracted from correlation functions... central charges and critical exponents

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Reference graph

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