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arxiv: 2605.00954 · v2 · submitted 2026-05-01 · 🪐 quant-ph

Multiple Bulk-Boundary Correspondences and Anomalous Modes in a Non-Hermitian Creutz Ladder

Xin Li , TongYi Li , Jingyu Peng , Yu Wang This is my paper

Pith reviewed 2026-05-11 00:54 UTC · model grok-4.3

classification 🪐 quant-ph
keywords non-Hermitian topologybulk-boundary correspondenceCreutz ladderskin effectPT symmetryanomalous modesnon-Bloch spectrum
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The pith

A non-Hermitian Creutz ladder with gain-loss and nonreciprocity supports multiple bulk-boundary correspondences through preserved symmetries.

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

The paper examines a non-Hermitian version of the Creutz ladder that includes both gain and loss terms as well as nonreciprocal couplings. It constructs several versions of the bulk-boundary correspondence that account for scale-free, normal, and anomalous skin modes along with topological zero-energy modes. Symmetries such as spatial inversion, PT symmetry, hidden chiral symmetry, and sublattice symmetry play key roles in characterizing phase transitions and enabling exact calculations of non-Bloch spectra. A hybrid spectral winding number is shown to capture the localization behavior of two unexpected bulk modes that appear alongside normal skin modes.

Core claim

In the non-Hermitian Creutz ladder, spatial inversion symmetry allows an average winding number to characterize the PT phase transition, while hidden chiral symmetry provides a Z2 invariant for topological transitions. Gain-loss breaks P symmetry but keeps PT, requiring only minor modifications to the BBC. Sublattice symmetry permits precise non-Bloch spectrum calculation, leading to a hybrid spectral winding that encodes localization information for two counterintuitive bulk modes coexisting with normal skin modes: one accumulating exponentially opposite to nonreciprocity, and another showing a surge of Bloch-wave states in nonreciprocal lattices.

What carries the argument

The hybrid spectral winding number, which encodes the localization or delocalization of counterintuitive bulk modes in the non-Hermitian Creutz ladder under sublattice symmetry.

If this is right

  • Multiple BBCs can be constructed for different types of skin modes and topological modes in non-Hermitian systems with specific symmetries.
  • The PT phase transition is characterized by an average winding number when P symmetry is present.
  • Topological phase transitions can be detected via a Z2 invariant guaranteed by hidden chiral symmetry.
  • Exact non-Bloch spectra calculations are enabled by sublattice symmetry, revealing anomalous bulk modes.

Where Pith is reading between the lines

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

  • If the hybrid winding works in this ladder, similar constructions might apply to other non-Hermitian lattice models with sublattice symmetry to predict unexpected localization behaviors.
  • Experimental realizations could test the opposite-direction accumulation of one bulk mode by measuring wavefunction profiles in photonic or cold-atom setups.
  • The coexistence of normal and anomalous modes suggests new ways to control transport in nonreciprocal systems.

Load-bearing premise

The system must preserve spatial inversion, PT, hidden chiral, and sublattice symmetries to enable the modified bulk-boundary correspondences and exact non-Bloch calculations.

What would settle it

Observation of the hybrid spectral winding number failing to predict the localization direction or presence of the counterintuitive bulk modes in a physical implementation of the non-Hermitian Creutz ladder would disprove the claim.

Figures

Figures reproduced from arXiv: 2605.00954 by Jingyu Peng, TongYi Li, Xin Li, Yu Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. (color on line) (a) Schematic for the non-Hermitian l view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (color on line) (a) view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (color on line) (a) Flipping the winding number of view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (color on line) PBC energy spectra view at source ↗
Figure 5
Figure 5. Figure 5: (b) and (c) show that the TI is indeed a reliable indicator even at moderate [ i.e., ∼ (101 )]. The TI corner point, at which the Bloch-wave-like extended states evolve into SF skin modes, coincides precisely with the PT phase transition point of H () in Eq. (2). IV. TOPOLOGICAL BBC In practice, topologically protected edge modes should re￾main outside the bulk energy band under OBC in the ther￾modynamic l… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (color on line) Numerical GBZ plus energy spectra (up view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (color on line) PBC energy spectra view at source ↗
read the original abstract

The synergy of non-Hermitian and topology renders the bulk-boundary correspondence (BBC) even more elusive. Here we study a non-Hermitian Creutz ladder that incorporates both gain-loss and nonreciprocity, and construct multiple BBCs involving scale-free, normal and anomalous skin modes, as well as topological zero-energy modes. In the presence of spatial inversion (P) symmetry, the parity-time (PT) phase transition is characterized by an average winding number,whereas a hidden chiral symmetry guarantees that topological phase transitions can be detected via a Z 2 invariant. The gain-loss breaks the P symmetry but preserves the combined PT symmetry, and then the previous BBC version only requires minor modifications. Intriguingly, sublattice symmetry enables the precise calculation of non-Bloch spectra, based on which a hybrid spectral winding can encode the localization (or delocalization) information of two counterintuitive bulk modes that coexist with normal skin modes. One type exhibits exponential boundary accumulation in the opposite direction to nonreciprocity. The other exemplifies a surge of Bloch-wave states in nonreciprocal lattices. These results reveal a series of unexpected phenomena governed by symmetry, thereby expanding our fundamental understanding of the BBC mechanism in non-Hermitian topological systems.

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 / 3 minor

Summary. The paper studies a non-Hermitian Creutz ladder incorporating gain-loss and nonreciprocity. It constructs multiple bulk-boundary correspondences involving scale-free, normal, and anomalous skin modes together with topological zero-energy modes. Symmetries (spatial inversion P, PT, hidden chiral, and sublattice) are used to define an average winding number for the PT transition, a Z2 invariant for topological transitions, and a hybrid spectral winding (enabled by exact non-Bloch spectra from sublattice symmetry) that encodes the localization properties of two counterintuitive bulk modes coexisting with normal skin modes—one accumulating exponentially opposite to the nonreciprocity direction and the other exhibiting a surge of Bloch-wave states.

Significance. If the derivations hold, the work meaningfully extends the understanding of bulk-boundary correspondence in non-Hermitian systems by demonstrating symmetry-protected multiple BBCs and exact non-Bloch calculations. The hybrid winding and identification of anomalous modes provide concrete, falsifiable predictions within the model. The exact calculability due to sublattice symmetry and the model-specific but internally consistent treatment of counterintuitive modes are clear strengths that could guide further studies in non-Hermitian topology.

major comments (2)
  1. §4 (non-Bloch spectra and hybrid winding): the central claim that the hybrid spectral winding encodes localization information for both counterintuitive bulk modes rests on the exact non-Bloch spectrum enabled by sublattice symmetry; the manuscript should supply the explicit generalized Brillouin zone or non-Bloch Hamiltonian form so that the winding's ability to distinguish opposite-to-nonreciprocity accumulation from the Bloch-wave surge can be independently verified.
  2. §3 (PT phase transition): the statement that the previous BBC version 'only requires minor modifications' when P is broken but PT is preserved needs an explicit side-by-side comparison of the average winding number before and after the modification, including how the invariant changes under the gain-loss term, to substantiate that the modification is indeed minor and does not alter the topological classification.
minor comments (3)
  1. The term 'scale-free' skin modes is used in the abstract and introduction without a concise definition or reference to its scaling behavior; adding one sentence in §2 would improve accessibility.
  2. Figure captions for the mode-localization plots should list the specific parameter values (gain-loss amplitude and nonreciprocity strength) used, to facilitate reproduction.
  3. Notation for the symmetry operators (P, PT, chiral, sublattice) should be introduced with explicit matrix forms in the model Hamiltonian section for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive comments, which help improve the clarity and verifiability of the manuscript. We address each major comment below and will incorporate the suggested additions in the revised version.

read point-by-point responses

  1. Referee: §4 (non-Bloch spectra and hybrid winding): the central claim that the hybrid spectral winding encodes localization information for both counterintuitive bulk modes rests on the exact non-Bloch spectrum enabled by sublattice symmetry; the manuscript should supply the explicit generalized Brillouin zone or non-Bloch Hamiltonian form so that the winding's ability to distinguish opposite-to-nonreciprocity accumulation from the Bloch-wave surge can be independently verified.

    Authors: We agree that providing the explicit form will enhance independent verification of the hybrid winding's role. In the revised manuscript, we will include the explicit generalized Brillouin zone (GBZ) and the corresponding non-Bloch Hamiltonian obtained from the sublattice symmetry. This addition will explicitly demonstrate how the hybrid spectral winding distinguishes the localization of the two counterintuitive bulk modes—one accumulating exponentially opposite to the nonreciprocity direction and the other showing a surge of Bloch-wave states—while coexisting with normal skin modes. The derivations remain unchanged. revision: yes

  2. Referee: §3 (PT phase transition): the statement that the previous BBC version 'only requires minor modifications' when P is broken but PT is preserved needs an explicit side-by-side comparison of the average winding number before and after the modification, including how the invariant changes under the gain-loss term, to substantiate that the modification is indeed minor and does not alter the topological classification.

    Authors: We thank the referee for this suggestion. In the revised manuscript, we will add an explicit side-by-side comparison of the average winding number with and without the gain-loss term. This will show that, under preserved PT symmetry, the topological classification encoded by the average winding number remains the same, with the gain-loss term shifting only the PT transition point without altering the invariant. The comparison will confirm that the BBC modification is indeed minor. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs multiple bulk-boundary correspondences (involving scale-free, normal, anomalous skin modes and topological zero modes) and a hybrid spectral winding directly from the model's imposed symmetries (P, PT, hidden chiral, sublattice) and non-Bloch spectral calculations. These steps follow standard symmetry-based definitions of invariants (average winding number, Z2 invariant) without reducing to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The phenomena are explicitly model-specific and internally consistent within the symmetry-protected setting, with no ansatz smuggling or renaming of known results as new derivations.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the specific non-Hermitian Creutz ladder Hamiltonian and the assumption that certain symmetries are present and preserved; these are standard domain assumptions in the field rather than new postulates.

free parameters (2)
  • gain-loss amplitude
    Strength of the imaginary on-site potential terms that break Hermiticity.
  • nonreciprocity strength
    Asymmetric hopping parameter that drives skin localization.
axioms (2)
  • standard math Linear algebra over complex numbers for non-Hermitian eigenvalue problems
    Used to obtain spectra and eigenmodes.
  • domain assumption Existence of PT symmetry, hidden chiral symmetry, and sublattice symmetry in the chosen parameter regime
    Invoked to define winding numbers, Z2 invariant, and non-Bloch spectrum.

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Works this paper leans on

73 extracted references · 73 canonical work pages · 1 internal anchor

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    can always be recast into an algebraic equation of A = /u1D6FD+ /u1D6FD− 1, explicitly /u1D453( /u1D6FD, /u1D438/u1D45C) = ( /u1D43DA − /u1D438/u1D45C) 2 + [ /u1D4612 0/u1D6FF2 − /u1D43D2/u1D7022 − /u1D4612 0]A 2 − 4( /u1D4612 0/u1D6FF2 − /u1D43D2/u1D7022) . (9) Here, the two roots of A can be expressed as A1, 2 = (− /u1D43D/u1D438/u1D45C∓ √ F )/ /u1D712,...

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    regardless of the phase boundary /u1D43D= /u1D4610. Surprisingly, the Z2 invariant D here can accurately predict the zero-energy edge modes, as the topolog- ical BBC is guaranteed by a hidden chiral symmetry, resultin g in det [H ( /u1D458)] = det[H ( /u1D70B− /u1D458)] (see App.‌ B). Note that the complex energies belonging to different bands can’t touch ...

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