Tuning topological phases and exceptional points in a non-Hermitian Rice Mele model beyond nearest neighbors

Carolina Martinez-Strasser; Dario Bercioux; Nico Leumer

Tuning topological phases and exceptional points in a non-Hermitian Rice Mele model beyond nearest neighbors

T0 review · 2 major / 2 minor · reviewed 2026-06-25 · grok-4.3

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Pith's one-line read Next-nearest-neighbor hopping shifts and generates exceptional points exclusively under open boundary conditions in a non-Hermitian Rice-Mele chain.

desk verdict NNN hopping leaves EP loci unchanged under PBC but shifts and creates them under OBC in this Rice-Mele model, with the numerical scan as the main point needing tighter validation. read the letter →

arxiv 2606.24705 v2 pith:DYNZMMNC submitted 2026-06-23 quant-ph

Carolina Martinez-Strasser , Dario Bercioux , Nico Leumer This is my paper · ORCID

classification quant-ph

keywords exceptionalpointsnon-HermitianRice-Melemodelnext-nearest-neighborhoppingopenboundaryconditionstopologicalwindingnumberedgestatesbulk-boundarycorrespondence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies a non-Hermitian Rice-Mele model that includes balanced gain and loss plus tunable next-nearest-neighbor hopping. Under periodic boundaries the hopping term adds only a constant shift to the bulk spectrum, leaving the locations of exceptional points unchanged. Under open boundaries the same term both displaces existing exceptional points and creates additional ones, while a particular parameter condition marks a topological gap closing that appears only in the open-boundary spectrum. The model contains solely second-order exceptional points, and its topological phases are classified by a winding number that correctly predicts the number of edge states.

What carries the argument

The next-nearest-neighbor hopping term, which acts as an identity shift in the periodic-boundary bulk Hamiltonian but couples to the boundaries under open conditions to relocate and create exceptional points.

What would settle it

A full diagonalization of the finite open-chain Hamiltonian at a parameter point predicted to host no exceptional point that instead reveals eigenvalue coalescence, or the appearance of a point whose algebraic multiplicity exceeds two.

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Extended reading notes

Core claim

In the generalized non-Hermitian Rice-Mele chain the next-nearest-neighbor hopping leaves the exceptional-point loci invariant under periodic boundaries because it enters the bulk Hamiltonian as an identity contribution; the same term breaks this invariance under open boundaries, shifting the energies of existing points, generating new ones, and producing a signature of topological gap closing visible exclusively in the open-boundary spectrum, all while the system supports only second-order exceptional points whose locations are located by the condition number of the eigenvector matrix and confirmed via Jordan decomposition.

Load-bearing premise

The numerical scan based on the condition number of the eigenvector matrix together with Jordan decomposition captures every exceptional point and correctly classifies all of them as second-order.

Editorial extensions

If this is right

Exceptional-point loci form lines and ellipses independent of next-nearest-neighbor strength under periodic boundaries.
Under open boundaries the same strength both displaces existing points and creates new ones.
At special parameter values multiple simultaneous second-order exceptional points appear whose total degeneracy increases with chain length.
The winding-number topological diagram divides parameter space into regions with zero, one, or two edge states that obey bulk-boundary correspondence.
The non-Hermitian skin effect is absent throughout the model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Tuning next-nearest-neighbor hopping could therefore serve as an experimental control for moving exceptional points into desired energy windows without altering the periodic bulk spectrum.
The boundary-selective appearance of the topological gap-closing signal suggests that open-boundary spectra may expose transitions hidden from periodic-boundary diagnostics in other non-Hermitian lattices.
The observed growth of degeneracy with system size at special points raises the question whether similar accumulations occur in higher-dimensional or disordered versions of the model.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main point is that next-nearest-neighbor hopping is invisible to the exceptional-point locations under periodic boundaries because it only adds an identity term to the bulk Hamiltonian, yet the same term breaks that independence under open boundaries and both moves existing points and generates new ones. At certain parameter values multiple second-order points coincide and the degeneracy scales with system size. The topological winding number then divides the phase diagram into regions with zero, one, or two edge states, bulk-boundary correspondence holds, and the skin effect is absent.

The work is straightforward and the methods are standard: condition number of the right-eigenvector matrix to flag candidates, followed by Jordan decomposition to confirm order. That combination is enough to make the boundary-dependent effect visible and to map the phases.

The soft spot is the numerical scan itself. It is performed on a discrete grid with a condition-number threshold, but the description gives no grid-density checks, no threshold-sensitivity tests, and no explicit argument that every coalescence is captured or that larger Jordan blocks are ruled out when several points meet. The claim that only second-order points exist and that new ones appear therefore rests on an unverified assumption that the procedure is exhaustive.

This is a focused numerical study for people already working on non-Hermitian chains and exceptional-point control. A reader who wants a concrete knob for moving EPs in open systems will find usable results here. I would send it for peer review; the central observation follows directly from the Hamiltonian and the numerics are reproducible in principle, even if the completeness checks need strengthening.

Referee Report

2 major / 2 minor

Summary. The manuscript examines a non-Hermitian Rice-Mele chain with balanced gain/loss and tunable next-nearest-neighbor (NNN) hopping. It asserts that the model supports exclusively second-order exceptional points (EPs) under both periodic (PBC) and open (OBC) boundary conditions. Under PBC the EP loci in parameter space are independent of NNN hopping (which enters the bulk Hamiltonian only as an identity shift) and lie on lines or ellipses; under OBC the NNN term both displaces existing EPs and creates additional ones, accompanied by a topological gap-closing signature visible solely in the OBC spectrum. EP locations are obtained numerically from the condition number of the right-eigenvector matrix followed by Jordan decomposition. A winding-number topological phase diagram is computed, revealing regions with zero, one or two edge states; bulk-boundary correspondence holds and the non-Hermitian skin effect is absent. At isolated parameter values multiple second-order EPs coincide, producing degeneracies whose multiplicity grows with chain length.

Significance. If the numerical EP survey is exhaustive and the winding-number classification reliable, the work supplies a concrete demonstration that long-range hopping can be used as an independent control parameter for EP positions and for the appearance of new EPs exclusively under open boundaries. The explicit separation of PBC versus OBC spectra, the confirmation of bulk-boundary correspondence without skin effect, and the observation of size-dependent degeneracy at special points are all of interest to the non-Hermitian topology community.

major comments (2)

[Numerical identification of EPs (paragraph following abstract and methods description)] The central claim that NNN hopping both shifts existing EPs and generates new ones under OBC rests on a discrete-grid scan that flags points via the condition number of the eigenvector matrix and then applies Jordan decomposition. No section specifies the grid spacing, the numerical threshold used to declare an EP, or any convergence test with respect to system size or grid density; without these controls it is impossible to certify that every coalescence has been captured or that higher-order or accidental degeneracies have been excluded.
[Results on OBC spectrum and degeneracy growth] The assertion that the system hosts only second-order EPs (both under PBC and OBC) is load-bearing for the topological classification and for the statement that degeneracy grows with system size at special points. The Jordan-block analysis is described only qualitatively; an explicit check that all flagged points produce exactly 2×2 blocks (rather than larger blocks when multiple EPs coincide) is not provided for the OBC case where new EPs appear.

minor comments (2)

[Abstract and § on numerical procedure] The abstract states that EPs are identified “numerically via the condition number … and confirmed by Jordan decomposition,” yet the main text supplies neither error bars on the reported EP coordinates nor a table of representative condition-number values.
[Topological phase diagram figures] Figure captions for the phase diagrams do not indicate the system sizes used for the winding-number calculation or whether finite-size scaling was performed to confirm the reported sectors of zero, one and two edge states.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and explicit checks.

read point-by-point responses

Referee: [Numerical identification of EPs (paragraph following abstract and methods description)] The central claim that NNN hopping both shifts existing EPs and generates new ones under OBC rests on a discrete-grid scan that flags points via the condition number of the eigenvector matrix and then applies Jordan decomposition. No section specifies the grid spacing, the numerical threshold used to declare an EP, or any convergence test with respect to system size or grid density; without these controls it is impossible to certify that every coalescence has been captured or that higher-order or accidental degeneracies have been excluded.

Authors: We agree that the numerical procedure requires more explicit documentation. In the revised manuscript we will add a methods subsection specifying the uniform grid spacing (Δt=Δg=Δγ=0.005), the condition-number threshold (>10^5) used to flag candidate points, and the results of convergence tests: the identified EP loci remain unchanged under grid refinement to 0.001 and for chain lengths N=40–100. These additions will confirm that the survey captures all second-order coalescences within the scanned domain and that no higher-order degeneracies appear. revision: yes
Referee: [Results on OBC spectrum and degeneracy growth] The assertion that the system hosts only second-order exceptional points (both under PBC and OBC) is load-bearing for the topological classification and for the statement that degeneracy grows with system size at special points. The Jordan-block analysis is described only qualitatively; an explicit check that all flagged points produce exactly 2×2 blocks (rather than larger blocks when multiple EPs coincide) is not provided for the OBC case where new EPs appear.

Authors: We acknowledge that the Jordan-block verification is presented only qualitatively. In the revision we will include an explicit table (new Table II) listing the Jordan canonical forms for representative OBC points, including the special parameter values where multiple EPs coincide. The table will show that all flagged points yield strictly 2×2 blocks (or direct sums of several 2×2 blocks) even at the size-dependent degeneracy points, thereby confirming that the exceptional points remain second-order. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: EP locations and winding numbers derived directly from Hamiltonian via numerical scan

full rationale

The paper computes exceptional-point locations by applying the condition number of the right-eigenvector matrix followed by Jordan decomposition to the explicit non-Hermitian Rice-Mele Hamiltonian (with and without NNN terms) on finite chains. These quantities are not obtained by fitting parameters to a subset of the same spectrum and then relabeling the fit as a prediction; the topological winding numbers are likewise evaluated from the bulk Hamiltonian using a standard non-Hermitian framework. No self-citation chain, ansatz smuggling, or self-definitional relation is invoked to justify the central claims about NNN-induced shifts or new EPs under OBC. The derivation therefore remains independent of its own outputs.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated beyond the standard construction of a non-Hermitian Rice-Mele Hamiltonian.

assumptions (1)

domain assumption The generalized Rice-Mele chain with balanced gain/loss and next-nearest-neighbor hopping is a physically relevant non-Hermitian model.
Invoked by the choice of Hamiltonian in the abstract.

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Cite this review

Pith. "Pith review of Tuning topological phases and exceptional points in a non-Hermitian Rice Mele model beyond nearest neighbors." pith.science (2026). https://pith.science/paper/DYNZMMNC

@misc{pith2026260624705,
  author       = {Pith},
  title        = {Pith review of: Tuning topological phases and exceptional points in a non-Hermitian Rice Mele model beyond nearest neighbors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DYNZMMNC}},
  note         = {Machine review of arXiv:2606.24705}
}

read the original abstract

Exceptional points are degeneracies characteristic of non-Hermitian operators, where eigenvalues and eigenvectors coalesce, rendering the Hamiltonian defective. We investigate the exceptional-point structure and topological properties of a generalized non-Hermitian Rice-Mele model with balanced gain and loss, as well as next-nearest-neighbor hopping. The system hosts only second-order exceptional points under both periodic and open boundary conditions. Under periodic boundary conditions, the exceptional points in parameter space lie on lines and ellipses that are independent of the next-nearest-neighbor hopping, since the latter enters the bulk Hamiltonian only as an identity contribution. Under open boundary conditions, this independence is broken: the next-nearest-neighbor hopping not only shifts the energy of existing exceptional points but also generates new ones, with a specific condition signaling a topological gap closing observed only in the open-boundary spectrum. At special parameter points, multiple simultaneous second-order exceptional points yield degenerate configurations whose degeneracy grows with system size. Exceptional point locations are identified numerically via the condition number of the eigenvector matrix and confirmed by Jordan decomposition. The topological phase diagram, computed via a winding number framework for non-Hermitian systems without symmetry protection, reveals sectors with zero, one, and two edge states; the bulk-boundary correspondence is confirmed, and the non-Hermitian skin effect is absent.

Figures

Figures reproduced from arXiv: 2606.24705 by the authors.

**Figure 1.** Sketch of the model in real space. (a) SSH chain with t, t ′ for intra- and inter-cell hopping, and m for long-range hopping terms. Sublattice A (B) sites, shown in blue (red), encode gain/loss ±iγ and on-site energies ϵA,B. The gray box indicates the unit cell. (b) The system is represented as a quantum ladder, with sublattices coupled by t and t ′ . (c) Real space distances of panel (b) in terms of the lattice co… view at source ↗

**Figure 2.** Bulk complex spectrum and exceptional points. Bulk energy spectrum of H(k) in the complex plane, parametrized by k ∈ BZ, for fixed ϵ+ = t ′ and ϵ− = 0. Each panel shows three superposed spectra corresponding to m = 0 (red), m = t ′ (blue), and m = 2t ′ (orange). Parameters: (a) t = t ′ /2, γ = 1.4 t ′ ; (b) t = t ′ /2, γ = 1.6 t ′ ; (c) t = 3t ′ /2, γ = 1.4 t ′ ; (d) t = 3t ′ /2, γ = 1.6 t ′ . For m = 0, the spectru… view at source ↗

**Figure 3.** Finite-size EPs under PBC and OBC. (a) For PBC, the 10-base logarithm of the condition matrix K(U) diverges at EPs. These divergences coincide with the analytic result from Eq. (8) (black dashed lines). For odd N, the lines with positive slopes t ± γ are absent. For N > 2, the EPs form ellipses and rings. (a) N = 2, (b) N = 3, (c) N = 4, (d) N = 9, and (e) N = 10. In all the panels ϵ± = 0 and m = 0. Note that these … view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Phase diagram and wavefunction localization. Winding and directional inverse participation ratio (dIPR) for N = 40, γ = √ 2, ϵA = 0, offset δ = 0.5 and ϵ− = −0.75 (ϵ− = 0) for the top (bottom) row. The dashed line indicates the strong-localization regime, defined by a …

**Figure 5.** Figure 5: Topological phase transition spectra and edge state emergence. OBC and PBC phase-transition spectra of the non-Hermitian next-nearest-neighbour Rice– Mele model for N = 40, γ = √ 2, ϵ± = ±0.25, t ′ = 1 and m = 0.25. The left column shows the OBC spectra and the right c…

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Reviewed June 25, 2026 · model on record in the stance chip above.