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Spread complexity and entanglement entropy detect phase transitions and dynamical phases in a non-Hermitian extended SSH chain.

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2026-07-11 15:38 UTC pith:Z55KPYJW

load-bearing objection Solid free-fermion extension of Krylov-spread diagnostics to the extended non-Hermitian SSH: mode-resolved complexity and EE saturation are the real additions; analytic control is clean and scoped correctly. the 2 major comments →

arxiv 2607.04659 v1 pith:Z55KPYJW submitted 2026-07-06 quant-ph cond-mat.mes-hallcond-mat.str-el

Krylov complexity, mode-resolved complexity and entanglement entropy across phase transitions in the non-Hermitian extended Su-Schrieffer-Heeger model

Ling-Feng Zhang , Wing Chi Yu This is my paper
classification quant-ph cond-mat.mes-hallcond-mat.str-el
keywords Krylov complexityspread complexitymode-resolved complexitynon-Hermitian SSHexceptional pointsentanglement entropydynamical phasessaturation time
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies an extended Su-Schrieffer-Heeger chain that includes next-nearest-neighbor hoppings and a staggered imaginary potential. In the Hermitian limit the model already has several topological phases distinguished by winding numbers; weak non-Hermiticity splits each gap-closing point into a pair of exceptional points. The authors show that Krylov spread complexity, evaluated under two different dynamical protocols, and the long-time half-chain entanglement entropy both jump or change slope exactly where the number of exceptional points changes or where topological winding numbers jump. A mode-resolved version of the complexity further isolates the individual momenta that are responsible for those signatures and tracks how those critical momenta move when parameters are varied. Finally, when the spectrum is purely imaginary, the time at which complexity (and entanglement) saturates is controlled by the slowest-decaying mode; different locations of that mode define distinct dynamical phases that can be read off from either quantity.

Core claim

Krylov spread complexity of a unitary preparation of the non-Hermitian ground state, as well as the long-time average of spread complexity and of entanglement entropy under non-unitary evolution, locate both the topological phase boundaries of the Hermitian extended SSH model and the boundaries at which the number of exceptional points changes once non-Hermiticity is introduced; mode-resolved complexity identifies the responsible momenta, and the saturation time of both complexity and entanglement is fixed by the inverse of the slowest decay rate Γ(k*).

What carries the argument

Mode-resolved Krylov spread complexity C^s(k;t) obtained from the su(2)×su(2) coherent-state structure of each momentum pair, together with the saddle-point saturation time t* = 1/(2Γ(k*)) set by the slowest-decaying mode of a purely imaginary spectrum.

Load-bearing premise

Everything is derived for a free-fermion quadratic Hamiltonian that factors into independent su(2) algebras per momentum pair, starting from a product of lowest-weight reference states; interactions, disorder or a different initial state would remove the closed-form expressions.

What would settle it

Compute or measure the long-time half-chain entanglement entropy (or the unitary-preparation spread complexity) while sweeping a hopping parameter across a predicted exceptional-point boundary; if the first derivative remains smooth and no change appears in the saturation time of a purely imaginary spectrum, the claimed detection fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The manuscript studies the non-Hermitian extended SSH chain with next-nearest-neighbor hoppings and an imaginary staggered potential. It maps how exceptional points emerge in pairs from Hermitian gap-closing momenta near topological phase boundaries, then analyzes two analytically tractable dynamical protocols via Krylov spread complexity and entanglement entropy: (i) unitary preparation of the non-Hermitian ground state from a product of su(2) lowest-weight states, and (ii) non-unitary evolution under the Hamiltonian. Spread complexity (and its derivative) signals the transitions under protocol (i); long-time averages of complexity and half-chain entanglement entropy do so under protocol (ii). Mode-resolved complexity and associated fidelities identify the critical momenta and track their evolution. In the purely imaginary spectrum regime the authors derive a saddle-point saturation time t* = 1/(2Γ(k*)) controlled by the slowest decay mode and show that both complexity and entanglement entropy exhibit the same dynamical phases, with the critical mode now continuously tunable by the extra hoppings.

Significance. The work cleanly extends the Krylov-complexity diagnostics of Medina-Guerra et al. from the ordinary non-Hermitian SSH model to a richer free-fermion setting that hosts multiple winding numbers and continuously tunable critical modes. The closed-form su(2) coherent-state expressions for C_Ω and C(k;t), the Lambert-W and ε-independent saturation-time formulas, and the independent Gaussian-state entanglement-entropy numerics constitute concrete, reproducible advances. Mode-resolved complexity and the fidelity map supply a transparent momentum-space fingerprint of the transitions that is new relative to the existing literature. Because the model is experimentally relevant (photonic, acoustic, ultracold-atom platforms) and entanglement entropy is more accessible than Krylov complexity, the dynamical-phase characterization has clear observational value within free-fermion non-Hermitian physics.

major comments (2)
  1. Section V.A, Eqs. (49)–(53) and Fig. 8: the claim that first and second derivatives of t* signal a dynamical phase transition (t*_1 o t*_2) is well supported for the ordinary SSH limit, but for the extended model the continuous migration of k* produces only a crossover inside the t*_2 regime (green lines in Fig. 8). The manuscript should state more sharply which derivative discontinuities are true transitions versus smooth crossovers, and whether an order parameter (beyond the location of k*) can be defined that jumps across the t*_1/t*_2 boundary.
  2. Section III.B and Fig. 4(c,d): the authors correctly note that long-time averages ar C and ar S display intra-phase peaks/valleys that do not coincide with changes in n_EP, and therefore prefer protocol (i). This observation is load-bearing for the claim that C_Ω is the superior indicator; a short quantitative comparison (e.g., the magnitude of the spurious features relative to the true jumps) would strengthen the argument that the preference is not merely qualitative.
minor comments (5)
  1. Figure 5 caption and surrounding text: the notation C_Ω ≈ ar C versus C_Ω ≈ 1-ar C is introduced without an explicit definition of the numerical threshold used to draw the blue/red dashed lines; a sentence clarifying the criterion would improve reproducibility.
  2. Equation (2) and the anti-periodic boundary-condition choice: a brief remark on why anti-periodic rather than periodic conditions are preferred (avoidance of exact k=0,π for even L) would help readers who wish to reproduce the discrete spectra.
  3. Section IV, fidelity definitions (33)–(35): the small increment δ in t_c and the momentum discretization are not stated numerically; listing the values used for the color plots in Fig. 6 would remove ambiguity.
  4. Typographical: “UNIT AR Y” and “NON-UNIT AR Y” appear with spaces in section headings; “fidelity” is occasionally written “fidelities” inconsistently when referring to a single map.
  5. References: the recent experimental proposal for measuring Krylov complexity (Ref. [39]) is cited but not connected to the concrete free-fermion setting of the present work; a short sentence on possible photonic or cold-atom implementations would be useful.

Circularity Check

1 steps flagged

No significant circularity: diagnostics are computed from the free-fermion Hamiltonian; prior su(2) technology is reused but the extended-model spectra, mode maps and saturation formulas are independent calculations.

specific steps
  1. self citation load bearing [Sec. III opening paragraph and Sec. V.A (saturation formulas)]
    "In this section, we generalize the derivation in Ref. [22] to the extended SSH model. … The spread fidelity (cf. Eq. (36)) can be expanded as [21, 22] … t* = lim ε o0 t_ε / ln(1/ε) = 1/(2Γ(k*))."

    The analytic technology (su(2) normal ordering, coherent-state complexity, saddle-point saturation) is imported from the authors’ and collaborators’ prior ordinary-SSH papers. The import is not circular for the extended model: the new spectra, the continuous migration of k*, the mode-resolved maps and the fidelity diagnostics are calculated afresh from the extended Hamiltonian and are not forced by the cited results. The step is therefore only a minor self-citation, not a load-bearing definitional loop.

full rationale

The paper derives the Bloch Hamiltonian (Eqs. 5, 12), the su(2) imes su(2) generators (Eqs. 14–16), the coherent-state ground state and unitary preparation (Eqs. 20–25), the non-unitary evolution amplitudes (Eqs. 27–31), the mode-resolved fidelities (Eqs. 33–35), and the saddle-point saturation time t* = 1/(2Γ(k*)) (Eqs. 39–49) directly from the microscopic extended SSH Hamiltonian. These expressions are not fitted to the phase boundaries they later locate; the boundaries emerge as non-analyticities of C_Ω, of its derivatives, and of the fidelities. Self-citations to Medina-Guerra et al. [21,22] and to the authors’ own earlier works supply the ordinary-SSH baseline and the normal-ordering technology, but they are not load-bearing uniqueness theorems that force the present results. Entanglement-entropy numerics (Gaussian QR evolution) independently corroborate the same dynamical phases. The free-fermion assumption is correctly scoped and does not create a definitional loop. Score 1 reflects only the minor, non-load-bearing reuse of prior technology.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 2 invented entities

The central claims rest on the free-fermion quadratic structure that permits exact su(2) coherent-state evolution, on the definition of spread complexity for lowest-weight reference states, and on the saddle-point approximation for the slowest imaginary eigenvalue. No new physical particles or forces are postulated; the free parameters are only numerical choices used for illustration and finite-size numerics.

free parameters (3)
  • non-Hermiticity strength γ (illustrative values 0.12, 1.2, 4, 4.86)
    Chosen by hand to illustrate weak, moderate and strong non-Hermiticity regimes; not fitted to external data but control which spectral features appear.
  • scan line t_d = 1.21 and other fixed hoppings t_a = t_b = −1, J = 1
    Representative cuts through the phase diagram used for all derivative and mode-resolved plots; arbitrary but fixed for reproducibility of figures.
  • system size L = 100, time step Δt = 0.1, saturation thresholds ϵ = 10^{-15} (complexity) / 10^{-10} (EE)
    Numerical cut-offs for the entanglement-entropy evolution and for defining practical saturation times; affect quantitative t_ϵ but not the ϵ→0 analytic t*.
axioms (4)
  • domain assumption The Hamiltonian is quadratic free-fermion and can be block-diagonalized into independent su(2)×su(2) algebras labeled by ±k (Eqs. 13–16).
    Enables the exact coherent-state evolution and closed-form complexity; fails for interacting or disordered extensions.
  • standard math Spread complexity of a generalized coherent state of a lowest-weight su(2) reference state is given by |τ|^2/(1+|τ|^2) (or the analogous |A_+|^2 expression).
    Standard result for su(2) coherent states used throughout Sec. III; taken from the Krylov-complexity literature.
  • domain assumption In the purely imaginary spectrum the long-time average of mode complexity approaches 1/2 and the approach is dominated by the saddle of the slowest decay rate Γ(k*).
    Justifies the analytic t* = 1/(2Γ(k*)) and the comparison with entanglement entropy (Sec. V).
  • ad hoc to paper Anti-periodic boundary conditions and the Fourier convention of Eq. (2) are used so that the discrete momenta avoid k=0,π exactly when L is even.
    Technical choice that simplifies the mode set; results are expected to be insensitive in the L→∞ limit but are not re-derived for periodic BC.
invented entities (2)
  • mode-resolved (momentum-resolved) complexity C_Ω^s(k) and C^s(k;t) no independent evidence
    purpose: To keep the contribution of each momentum sector separate so that gap-closing or exceptional momenta can be identified and tracked.
    Defined by simply not summing over k in the usual spread-complexity formula; it is a diagnostic re-packaging rather than a new physical degree of freedom. Independent evidence is internal consistency with the known spectrum.
  • fidelities F_CΩ(t_c), F̃_CΩ(k) and the fidelity map F_MCΩ based on mode-resolved complexity no independent evidence
    purpose: To quantify abrupt changes in the momentum distribution of complexity across parameter or momentum space and thereby locate phase boundaries and characteristic momenta.
    Standard fidelity applied to the vector of mode complexities; useful visualization tool, not a new physical entity.

pith-pipeline@v1.1.0-grok45 · 27142 in / 3609 out tokens · 32411 ms · 2026-07-11T15:38:03.810139+00:00 · methodology

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read the original abstract

We investigate phase transitions in the extended Su-Schrieffer-Heeger (SSH) model with next-nearest-neighbor hoppings and an imaginary staggered chemical potential. In the presence of small non-Hermiticity, exceptional points emerge in pairs from the gap-closing momenta near the topological phase boundaries of the Hermitian limit. Utilizing the Krylov spread complexity and entanglement entropy, we analyze two dynamical protocols: (i) preparing the non-Hermitian ground state via a unitary transformation, and (ii) evolving the system under the non-Hermitian Hamiltonian. We show that the spread complexity, and long-time spread complexity as well as entanglement entropy can effectively signal phase transitions in the first and second protocols, respectively. To unravel the detailed structure of the transitions, we introduce the momentum-resolved complexity that identifies the characteristic modes and tracks their evolution with the driving parameter. In the regime where the system possesses a purely imaginary spectrum, we further identify dynamical phases based on the saturation behavior of the spread complexity. The entanglement entropy is also found to exhibit similar saturation behavior, thereby providing a more experimentally accessible probe of the dynamical phases.

Figures

Figures reproduced from arXiv: 2607.04659 by Ling-Feng Zhang, Wing Chi Yu.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic diagram of the extended SSH chain [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Distribution of exceptional points in the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) The spread complexity of the non-Hermitian ex [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a),(b) The long-time spread complexity on [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The top, middle, and bottom panels display the momentum-space distributions of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The top and middle panels display the fidelity of the mode-resolved complexity as a function of the parameter [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a),(b) The dynamics of the spread complexity and [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) The slowest decay mode [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗

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