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An unusual phase transition in a non-Hermitian Su-Schrieffer-Heeger model

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Strong loss-gain sends every SSH chain to the same topological phase.

desk verdict A clean analytical band-structure study of a new non-Hermitian SSH variant; the core phase sequence is solid, but the abstract and the 'non-trivial semi-metal' label overreach. read the letter →

arxiv 2411.14328 v1 pith:UJBSN5ZY submitted 2024-11-21 quant-ph cond-mat.other

A Niveth , S Karthiga , M Senthilvelan This is my paper · ORCID
classification quant-phcond-mat.other
keywords non-HermitianSSHmodeltopologicalphasetransitionexceptionalpointsZaksemi-metallicedgestatesskineffectloss-gainstrength
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

This paper studies a non-Hermitian Su-Schrieffer-Heeger (SSH) chain in which alternating unit cells carry balanced loss and gain. It claims that strong non-Hermiticity $u$ drives every parameter regime into a non-trivial insulating phase: even if the chain is a trivial insulator in the Hermitian limit, raising the loss-gain strength forces a transition, through a gapless semi-metallic phase, to a non-trivial insulator. If the Hermitian limit is already non-trivial, the system leaves that phase, passes through a semi-metallic gapless region, and returns to a non-trivial insulator with different edge physics. The two non-trivial insulating phases are distinguished under open boundary conditions: one has two zero-energy edge modes, the other has one zero mode and one non-zero mode, and the model shows a weak non-Hermitian skin effect.

What carries the argument

The central object is the four-band momentum-space Hamiltonian of Eq. (5), built from alternating Hermitian dimers ($c,d$) and non-Hermitian dimers ($a,b$) with onsite potentials $\pm iu$. It obeys chiral symmetry $\Gamma = \tau_z \otimes \sigma_0$ and belongs to the BDI$^\dagger$ symmetry class. The argument is carried by the trajectories of two exceptional points, EP1 and EP2, whose collision points define the critical loss-gain strengths $u_{c1}$, $u_m$, and $u_{c2}$; the region between $u_{c1}$ and $u_{c2}$ is gapless (semi-metallic) because of band merging at these exceptional points. Topological phases are classified with the Zak phase of Eq. (16) evaluated on biorthonormal left and right eigenstates.

What would settle it

Compute a regularized topological invariant, such as a winding number along a path in $k$ that avoids the exceptional points, across the semi-metallic region: if the invariant does not remain pinned at the non-trivial value for all $u\in(u_{c1}, u_{c2})$, the claim that the semi-metal is non-trivial—and with it the distinctive non-trivial-to-non-trivial story—fails, even though the insulator-gapless-insulator sequence would remain.

Watch

Extended reading notes

Core claim

The central claim is that for loss-gain strength $u > u_{c2}$ the bulk always stabilizes in a non-trivial insulating phase, regardless of whether the Hermitian limit at $u=0$ is trivial or non-trivial. In the trivial case the transition runs trivial insulator $\to$ gapless semi-metal $\to$ non-trivial insulator; in the non-trivial case the transition runs non-trivial insulator $\to$ (claimed non-trivial) semi-metal $\to$ non-trivial insulator. Gap closing at exceptional points produces the intermediate semi-metallic region, and although gap closing usually signals a change in topology, here it mediates a return to the same topological phase. Under open boundaries the two non-trivial insulating phases differ: the low-$u$ phase hosts two zero-energy edge modes, the high-$u$ phase hosts a left zero mode and a right non-zero mode, with left modes more localized, indicating a weak non-Hermitian skin effect. The paper further reports that these edge states are robust to Gaussian disorder.

Load-bearing premise

The claim that the gapless semi-metallic region is itself non-trivial rests on a Zak phase computed where bands touch at exceptional points, where a quantized invariant is not well-defined; the paper notes the value fluctuates around $2\pi$.

Editorial extensions

If this is right

  • For $u > u_{c2}$, the bulk is always a non-trivial insulator for every dimerization angle $\theta$, so a sufficiently strong loss-gain term can override the designed trivial or non-trivial character of the Hermitian chain.
  • The gapless semi-metallic phase can mediate a transition between two topologically equivalent insulating phases, not only between topologically distinct ones, which runs against the usual reading of gap closing as a topology-change indicator.
  • Under open boundary conditions, the low-$u$ and high-$u$ non-trivial insulators are not identical: they carry different zero/non-zero edge-mode structures, and the model exhibits a weak non-Hermitian skin effect with left modes more localized than right modes.
  • Bulk-boundary correspondence holds in the insulating regimes but is only weakly preserved near the boundaries of the gapless region, where non-zero discrete modes appear.
  • The edge states, including those in the gapless region, remain stable under Gaussian disorder.

Reading between the lines

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

  • If the fluctuating Zak phase in the gapless region is replaced by an invariant that is well-defined despite the exceptional points, the 'non-trivial semi-metal' label may sharpen, but the robust part of the paper—strong loss-gain always ends in a non-trivial insulator—does not depend on that label.
  • The alternating Hermitian/non-Hermitian dimer pattern suggests a design principle: spatially periodic loss-gain can act as a topological 'reset' at strong coupling, which could be tested in other one-dimensional models such as the Kitaev chain or Aubry-André model.
  • The predicted asymmetry in inverse participation ratio between left and right edge modes is directly measurable in photonic waveguide arrays with engineered loss and gain; observing the asymmetry would verify the weak skin effect.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This paper studies a one-dimensional non-Hermitian Su-Schrieffer-Heeger (SSH) model with four sites per unit cell, in which two sites carry balanced loss and gain while the other two are Hermitian. The authors derive the four-band Bloch spectrum, identify two exceptional-point mechanisms, and obtain closed-form critical values u_c1, u_m, and u_c2 that separate insulating and semi-metallic intervals. Using a biorthonormal Zak phase, they conclude that for sufficiently strong loss-gain strength the system always enters a non-trivial insulating phase, even when the Hermitian limit is trivial, and that a non-trivial Hermitian insulator passes through a gapless semi-metallic phase before returning to a non-trivial insulating phase. They also analyze open-boundary spectra, edge modes, inverse participation ratios, and robustness to disorder.

Significance. If the phase diagram is correct, the model provides a simple analytically tractable example in which non-Hermiticity restores non-trivial topology in the strong-coupling limit and in which a topological gap closing is not accompanied by a change of the gapped-phase invariant. The closed-form expressions for the critical lines and the independent numerical confirmation are strengths, as is the OBC/IPR analysis that distinguishes the two non-trivial insulating regimes. However, the paper's most distinctive claim, that the gapless intermediate phase is itself 'non-trivial', is not supported by the presented invariant because the Zak phase is not quantized in the presence of exceptional points; in addition, the abstract contains a spectral statement contradicted by the authors' own band-structure results.

major comments (2)
  1. [Abstract; Sec. 3.1 after Eq. (12); caption of Fig. 5] The claim that the system 'supports only complex eigenspectra for all values of u ≠ 0' is contradicted by the authors' own band-structure analysis. In Sec. 3.1, for θ = π/4 and u = 1.25 (which lies between u_c1 and u_m), the text states that 'completely real energies exist only between EP 1 and EP 2', and Fig. 2(c) correspondingly shows extended intervals of real spectrum. The spectrum is therefore partially complex, not exclusively complex. This statement appears in the abstract and in the Fig. 5 caption and must be corrected, since it is presented as a headline result.
  2. [Sec. 3.2, Eq. (16), Fig. 4; Sec. 5] The topological label 'non-trivial' attached to the gapless semi-metallic phase is not established. In the interval u_c1 < u < u_c2 the bands touch at exceptional points (Eqs. (10) and (12)), where eigenstates coalesce and the biorthonormal frame used in Eq. (16) is not a continuous frame over the Brillouin zone. A Zak phase computed through such a region is gauge-dependent and is not a quantized invariant. The text itself concedes that Ω 'fluctuates around 2π' in this region. Fluctuating values cannot certify a non-trivial invariant. Since the abstract's central claim is that the transition is mediated by a 'non-trivial semi-metallic phase', this needs either a well-defined non-Hermitian invariant valid at exceptional points (for example, a winding of the complex-energy spectrum around the EPs) or revised wording that limits the claim to a gapless intermediate region whose topology is not classified by the Zak phase. The insulator-gapless-insulator sequence and the non-trivial labels of the two gapped phases are not in question.
minor comments (5)
  1. [Caption of Fig. 8] The caption says the OBC/PBC comparison is for θ = π/4, but the surrounding text and the displayed data correspond to θ = 3π/4.
  2. [Eq. (3)] The bracket structure in H_hop is inconsistent: the square bracket opens before the two bra-ket terms but closes after 'h.c.', leaving the parentheses unbalanced.
  3. [Sec. 1] The text refers to the 'Kiteav chain'; this should be 'Kitaev chain'.
  4. [Sec. 3.2] The manuscript should specify which bands are included in the sum in Eq. (16) and state the gauge and branch choices used for the multi-valued square roots in Eq. (9), so that the numerical Zak-phase values can be independently reproduced.
  5. [Sec. 4, Fig. 10] The claim that edge states in the gapless region are robust is not directly illustrated: Fig. 10 shows disorder robustness only for u = 0.5 and u = 3.5, which are gapped-phase values, not for a representative point inside the semi-metallic interval.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the band-structure critical lines and Zak-phase labels are computed from the model Hamiltonian without fitted inputs.

full rationale

The paper's central claim is that the considered non-Hermitian SSH model stabilizes a non-trivial insulating phase for strong loss-gain u, with the transition mediated by a gapless semi-metallic region. This is derived from the explicit 4x4 momentum-space Hamiltonian in Eq. (5): the eigenvalues in Eq. (9) yield the exceptional-point locations in Eqs. (10) and (12) and the critical values uc1, um, uc2 in Eqs. (11), (13), and (14) from gap-closing conditions. No parameter is fitted to reproduce the target phase diagram. The topological labels are obtained by numerically evaluating the Zak phase in Eq. (16) with the biorthonormal eigenstates of Eq. (17); the same formula is applied uniformly in gapped and gapless regions. Calling the gapless region 'non-trivial' because its Zak phase fluctuates around 2π is a definitional classification rather than a fitted input or an equation that equals its own conclusion, though the invariant's well-definedness at exceptional points is a legitimate correctness concern. Edge-state, IPR, and disorder-robustness calculations in Sec. 4 are additional independent diagnostics. The self-citations present (e.g., Ref. [14] for power oscillations) are background references and are not load-bearing for the phase-transition argument. The abstract's statement that the spectrum is complex for all u≠0 is inconsistent with the real-spectrum intervals shown in Fig. 2(c), but that is a factual/correctness issue, not circularity.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central claims rest on standard single-particle tight-binding and Zak-phase topology; no constants are fitted to reproduce the target phase diagram. The only hand-set model constant is delta=0.3, and the interpretation of the gapless region as non-trivial relies on an assumption about the Zak phase that the paper itself does not justify.

free parameters (1)
  • hopping modulation delta = 0.3
    Set to 0.3 for all numerical results; the phase diagram is not shown for other values, so the genericity of the claims is not demonstrated. This is a hand-set model parameter, not fitted to external data.
assumptions (3)
  • standard math The system is described by a single-particle tight-binding Hamiltonian and Bloch's theorem applies.
    Used to write the momentum-space Hamiltonian H(k) in Eq. (5) via Fourier transform.
  • domain assumption The Zak phase formula in Eq. (16) with biorthonormal eigenstates is a valid topological invariant for this non-Hermitian system.
    Invoked in Sec. 3.2 to classify phases; its validity in gapless regions is questionable because the bands touch at exceptional points.
  • domain assumption Bulk-boundary correspondence holds in the insulating phases away from the exceptional region.
    Used in Sec. 4 to relate OBC edge modes to PBC topology; the paper itself notes weak violation near the semi-metallic region.

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Pith. "Pith review of An unusual phase transition in a non-Hermitian Su-Schrieffer-Heeger model." pith.science (2026). https://pith.science/paper/UJBSN5ZY

@misc{pith2026241114328,
  author       = {Pith},
  title        = {Pith review of: An unusual phase transition in a non-Hermitian Su-Schrieffer-Heeger model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UJBSN5ZY}},
  note         = {Machine review of arXiv:2411.14328}
}
abstract

This article studies a non-Hermitian Su-Schrieffer-Heeger (SSH) model which has periodically staggered Hermitian and non-Hermitian dimers. The changes in topological phases of the considered chiral symmetric model with respect to the introduced non-Hermiticity are studied where we find that the system supports only complex eigenspectra for all values of $u \neq 0$ and it stabilizes only non-trivial insulating phase for higher loss-gain strength. Even if the system acts as a trivial insulator in the Hermitian limit, the increase in loss-gain strength induces phase transition to non-trivial insulating phase through a (gapless) semi-metallic phase. Interesting phenomenon is observed in the case where Hermitian system acts as a non-trivial insulator. In such a situation, the introduced non-Hermiticity neither leaves the non-trivial phase undisturbed nor induces switching to trivial phase. Rather, it shows transition from non-trivial insulating phase to the same where it is mediated by the stabilization of (non-trivial) semi-metallic phase. This unusual transition between the non-trivial insulating phases through non-trivial semi-metallic phase gives rise to a question regarding the topological states of the system under open boundary conditions. So, we analyze the possibility of stable edge states in these two non-trivial insulating phases and check the characteristic difference between them. In addition, we study the nature of topological states in the case of non-trivial gapless (semi-metallic) region.

Figures

Figures reproduced from arXiv: 2411.14328 by the authors.

Figure 1
Figure 1. Schematic diagram: (a) SSH model, (b) non-Hermitian SSH m [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. (a)-(e) show the band structures corresponding to th [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. (a) - (d) show the band structure for θ = π 2 case and (e) shows the trajectory of exceptional points EP1 (shown by yellowish green curve) and EP2 (shown by purple curve) for θ = π 2 . (f) - (i) show the band structure for θ = 3π 4 and (j) shows the trajectory of EP1 (shown by yellowish green curve) and EP2 (shown by purple curve) for θ = 3π 4 . Case (ii): Now considering the case of θ = π 2 (w1 = w2), the four-band… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Zak phase values for different cases. (a) and (b) are plot [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Phase diagram of Eq. (2): TR represents trivial insulating [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Real and Imaginary parts of z vs u in the open boundary conditions. (a) and (d) are plotted for the case θ = π 4 . (b) and (e) for θ = π 2 and (c) and (f) for θ = 3π 4 . Here, the dark red dots are used to distinguish the zero edge mode from the other eigenvalues. with…
Figure 7
Figure 7. Figure 7: Comparison of the band structures in the cases of OBC an [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Comparison of the band structures in the cases of OBC an [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: (a) - (d) Inverse participation ratio for different bulk and [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: Edge states observed in the lattice for open boundary c [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

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    Introduction In recent years, much attention has been paid to topological mate rials which support states that are robust against moderate levels of disorder and pe rturbation [1, 2, 3]. One such material that has been reported first in the literature is topo logical insulators. In this, besides bulk of the sample shows insulating behavior, their edge s (o...

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