Dephasing-induced jumps in non-Hermitian disordered lattices

Eleftherios N. Economou; Emmanouil T. Kokkinakis; Konstantinos G. Makris

REVIEW 2 major objections 5 minor 139 references

Dephasing-induced jumps in non-Hermitian disordered lattices

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Dephasing, normally a delocalizer, sharpens localization and causes abrupt jumps between distant sites in weakly disordered non-Hermitian lattices.

desk verdict A plausible and clean mechanism for dephasing-induced jumps, but the a priori prediction is tested on one realization only — fix that and it's solid. read the letter →

arxiv 2412.20306 v2 pith:CC3NRSES submitted 2024-12-29 cond-mat.dis-nn

Emmanouil T. Kokkinakis , Konstantinos G. Makris , Eleftherios N. Economou This is my paper · ORCID

classification cond-mat.dis-nn PACS 42.25.Dd71.23.-k

keywords dephasingAndersonlocalizationnon-Hermitiandisorderjumpsincoherentpropagatorphotoniclatticegainandlossinverseparticipationratio

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

Dephasing—randomizing the wave function's phase at fixed intervals—is usually thought of as a decoherence that destroys Anderson localization and restores diffusion. This paper argues that in one-dimensional non-Hermitian lattices with weak complex on-site disorder the opposite happens: rapid dephasing sharpens the localization of the eigenmodes and turns the evolution of the normalized wave function into a sequence of abrupt jumps between spatially distant regions. The authors show that the position and duration of each jump can be computed in advance from the eigenvalues and eigenvectors of the averaged incoherent propagator $S$, whose largest eigenvalue eventually dominates. This matters because it suggests that a normally destructive environmental noise can, in open systems with gain and loss, be used as a controllable switching mechanism rather than only as a source of decoherence.

What carries the argument

The load-bearing object is the incoherent propagator $S(l)$, the real symmetric non-negative matrix with entries $S_{kj}=|(e^{iHl})_{kj}|^2$; it advances the ensemble-averaged site probabilities by one dephasing period. Its eigen-equation $S |v_j\rangle = \beta_j |v_j\rangle$ supplies the straight-line projection formula $\ln|d_j| = \ln|d_{j0}| + (\ln \beta_j / l)\, z$ and the power-derivative formula $d\ln P/dz = \ln \beta_m / l$, where $m$ labels the dominant eigenvalue. Jumps occur at crossings of the dominant projections, and the fast-dephasing expansion $U \approx I + iHl - \tfrac12 H^2 l^2$ shows why: to second order in $l$, $S$ is nearly diagonal with diagonal entries $1 - 2 b_n l + (2 b_n^2 - 2)l^2$ set by the imaginary disorder, making the eigenstates approximate single lattice sites.

What would settle it

Run the full dephased dynamics for many independent random-phase sequences using the same weak-disorder realization (e.g., $N=50$, $W_R=W_I=1$, single-site excitation at $n=25$, $l=0.01$) and compare each jump's propagation distance with the crossings of the $\ln|d_j|$ lines computed from $S$; if the distances scatter widely instead of clustering at the predicted values, the single-realization predictability claim fails.

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Extended reading notes

Core claim

The central claim is that in the fully incoherent regime—phases randomized every propagation step $l$—the ensemble-averaged site probabilities evolve under the non-negative symmetric matrix $S_{kj}=|(e^{iHl})_{kj}|^2$, and the projections $d_j$ onto $S$'s eigenstates obey $\ln|d_j| = \ln|d_{j0}| + (\ln \beta_j / l)\, z$. Because the eigenstate with the largest eigenvalue $\beta_{\max}$ dominates after a finite distance $z_{\rm cr}$, the probability density switches sharply between regions of the lattice when one dominant projection overtakes another; the same crossing explains plateaus in $d\ln P/dz = \ln \beta_m / l$. Under weak disorder, coherent dynamics is smooth because eigenstates overlap, but dephasing makes $S$'s eigenstates increasingly single-site localized, so the jumps become abrupt; in the fast-dephasing limit $l\to 0$ the diagonal entries of $S$ are set by the imaginary (gain/loss) disorder alone, and all eigenstates become single-site localized. The paper verifies these predictions for single phase realizations and shows that the mean inverse participation ratio of $S$'s eigenstates tends to 1 below a critical dephasing period.

Load-bearing premise

The paper's predictions assume that one particular sequence of random phases behaves like the statistical average over all phase sequences, so that the eigenvalues and eigenvectors of $S$ accurately locate the jumps in a single experimental run.

Editorial extensions

If this is right

A single-channel excitation in a weakly disordered non-Hermitian lattice with $l < 10^{-4}$ will evolve as a sequence of hops between distant single sites, because all eigenstates of $S$ are then localized at one site.
The propagation distance at which the wave function jumps can be fixed by the disorder realization and dephasing rate, so dephasing becomes a tunable trigger for abrupt switching.
The long-term growth or decay rate of optical power is set by $\ln \beta_{\max} / l$ and saturates as $l$ shrinks, giving a bounded, predictable power response.
Imaginary on-site disorder (gain/loss) alone suffices to produce the localization and jumps, since the real potential does not enter $S$'s diagonal to second order in $l$.
The predictions apply to existing photonic discrete-time quantum walk experiments that can implement periodic random phase kicks and measure the normalized wave function.

Reading between the lines

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

Because Eqs. (9) and (12) are derived after averaging over phase ensembles, the a priori prediction of jump positions may hold only for the ensemble-averaged probability; whether a single laboratory run with one random phase sequence reproduces those exact jump distances is a separate test the paper leaves open by comparing to only one realization.
The mechanism, a non-negative symmetric propagator whose dominant eigenvector changes with parameters, is generic, so similar dephasing-induced jumps should appear in other non-Hermitian transport models where noise is equivalent to periodic phase randomization, not only the tight-binding chain studied here.
One testable extension: measure the jump-distance distribution over many phase realizations for a fixed disorder sample; a narrow distribution would validate single-shot predictability, while a broad distribution would confine the prediction to averaged dynamics.
The saturation of $d\ln P/dz$ and $\langle\mathrm{IPR}\rangle$ with decreasing $l$ suggests a crossover line $l_{\rm cr}(W)$ that could be mapped experimentally as a function of gain/loss disorder strength, providing a phase diagram for jumpy versus smooth incoherent transport.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The headline: this paper has a real idea — rapid dephasing localizes the eigenmodes of the incoherent propagator S in weakly disordered non-Hermitian lattices, producing abrupt jumps that are absent under coherent dynamics. That's a nice counterintuitive result, and the paper gives an elementary, internally consistent account: S is real symmetric, the dominant eigenvalue controls long-z dynamics, and the fast-dephasing expansion makes S nearly diagonal. No parameters are fitted; the jump positions come from the spectral decomposition of S. That is real value.

What's new: coherent Anderson jumps are known, and dephasing in non-Hermitian systems has been studied (Longhi's recent work especially), but the specific combination — weak complex disorder plus rapid dephasing localizes eigenmodes and makes jumps analytically predictable — is not in the cited literature. The paper credits the right prior work and doesn't oversell the novelty.

The soft spot is the one you'd expect: the comparison in Figs. 2 and 3 is between ensemble-averaged theory (Eqs. 9 and 12) and a single realization of random phases. The authors are honest about this — they state it explicitly in Sec. IV. But their central claim is that jump positions and durations can be predicted a priori for a given experiment. A single realization with no phase-ensemble statistics, no seed, and no code does not establish that. The interference cross terms that vanish under averaging may or may not shift the crossing points in a typical run. If the gaps between ln β_j are large relative to realization fluctuations, the prediction is robust; if not, it only holds on average. The paper doesn't show which regime it's in.

There's also a second, minor gap: the transition from S nearly diagonal to actual localized eigenmodes is sketched via a second-order expansion, and the saturation in Fig. 4 is shown for one potential realization. The disorder-averaged IPR in Fig. 5 covers that for localization, but not for the jump-prediction claim.

Bottom line: this is worth engaging with. The mechanism is plausible and the math is transparent. The single-realization issue is a load-bearing caveat but not a refutation — it should be fixed with a phase-ensemble statistical analysis, and probably it will hold. I'd send it to review. A serious referee can ask for the missing statistics without throwing out the idea.

Best, [You]

Referee Report

2 major / 5 minor

Summary. The paper studies a one-dimensional tight-binding lattice with complex on-site disorder under periodic random-phase dephasing. It argues that, in contrast to coherent weak-disorder dynamics, rapid dephasing makes the eigenmodes of the ensemble-averaged incoherent propagator S(l) highly localized and produces abrupt jumps between spatially distant regions. The principal analytic results are Eq. (9), ln|d_j| = ln|d_{j,0}| + (ln β_j / l) z, and Eq. (12), d ln P/dz = ln β_m / l, obtained from the spectral decomposition of S. The authors claim that jump positions and durations can be predicted a priori from the potential distribution and initial condition, and they compare these predictions with single-realization numerical simulations in Figs. 2 and 3. The dephasing-rate dependence of the power derivative and of the eigenmode IPR is analyzed in Figs. 4 and 5.

Significance. If the central claim holds, this is a useful counterpoint to the well-known dephasing-induced delocalization in Hermitian disordered systems: in non-Hermitian lattices, dephasing can instead sharpen localization and create abrupt dynamical transfers. The analytical framework is elementary and parameter-free: S is real and symmetric, its Perron-Frobenius eigenvalue controls the long-z evolution, and the fast-dephasing expansion in Eq. (B2) makes the eigenmodes nearly site-localized. The predictions in Eqs. (9) and (12) are falsifiable and can be checked by direct matrix computation. The main weakness is that the a priori prediction is compared only with a single random-phase realization, although Eqs. (9) and (12) are derived after statistical averaging; this gap is acknowledged in the text and needs to be addressed quantitatively.

major comments (2)

[Sec. IV, Eqs. (9) and (12); Figs. 2 and 3] The paper’s strongest claim, that jump positions and durations can be predicted a priori from the eigenvalues and eigenvectors of S, is supported only at the level of the ensemble-averaged probability, not for the single random-phase realization shown. The text states in Sec. IV that Eqs. (9) and (12) were derived under the assumption of statistical averaging, while the numerical wavefunction and d ln P/dz are computed for a single realization. A single realization evolves by a random product of phase matrices and unitary steps, and its instantaneous intensity contains interference cross terms that do not survive the averaging. The crossing of the averaged coefficients ln|d_j| therefore need not coincide with the crossing in any particular run. The authors should either provide phase-ensemble statistics (for example, distributions of jump positions, jump durations, and d ln P/dz over many phase realizations for the same disorder potential) or give a quantitative self-averaging or concentration argument showing that inter-realization fluctuations are small compared with the gaps between ln β_j. Without such evidence, the a priori prediction is established for the average, not for a single experimental run.
[Sec. IV, Fig. 2(b)] The quantity plotted in Fig. 2(b) is not fully specified. The projection coefficients d_j in Eq. (9) are defined for the unnormalized, ensemble-averaged probability vector |P(z)>, whereas the wavefunction in Fig. 2(a) is the normalized single-realization wavefunction |φ> = |ψ>/√P(z). Because normalization introduces a z-dependent prefactor and because the single-realization intensity is not the averaged probability, it is unclear whether the plotted ln|d_j| values are computed from the averaged P(z) or from the instantaneous single-realization intensity. The authors should state exactly which quantity is plotted and whether the comparison in Fig. 2(b) is intended to be literal or illustrative.

minor comments (5)

[Sec. IV, heading] The heading contains a typo: “Emergenge” should be “Emergence”.
[Eq. (10)] The inline label “(7)” in Eq. (10) is not a proper reference; the sentence should read “using Eq. (7)”.
[Sec. V, Fig. 5] The text invokes a critical dephasing period l_cr(W) but never defines it; the paper should specify the threshold criterion used for “complete localization” and should provide error bars or a statistical measure for the mean IPR over the 1000 realizations.
[Figs. 1–4] No random seed or code is provided for the single-realization simulations, so the numerical results are not reproducible; providing a seed or a small code repository would allow readers to verify the claimed agreement.
[Sec. II, Eq. (5)] The derivation of Eq. (5) assumes that the dephasing phases at different sites and steps are independent and uniformly distributed; this assumption should be stated explicitly in the main text rather than only in Appendix A.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the jump prediction follows exactly from the spectral decomposition of the incoherent propagator S, with no fitted parameters and no load-bearing self-citation.

full rationale

The central predictive relations, Eq. (9), ln|d_j| = ln|d_j,0| + (ln beta_j / l) z, and Eq. (12), d ln P / dz = ln beta_m / l, are derived directly from the eigenvalue equation S|v_j> = beta_j|v_j> and the ensemble-averaged evolution |P(al)> = S^a |P(0)> given in Eqs. (5)-(7) and Appendix A. No parameter is fitted to the jump positions or durations; beta_j and d_j,0 are the exact eigenvalues and initial projections of the fixed incoherent propagator S defined by the given Hamiltonian, dephasing period, and initial condition. The comparison in Figs. 2 and 3 of a single random-phase realization against these ensemble-averaged formulas is explicitly acknowledged as such in Sec. IV, and it is a statistical-support limitation rather than a circular step: the single-realization curves are not used to adjust the theoretical lines. Self-citations to the group's earlier coherent-jump work (Refs. 100-102 and 71) provide contextual background, but the present dephasing result does not depend on them, and the incoherent-propagation setup cites Longhi (Ref. 132) as external support. No derivation step reduces by construction to its own input, there is no fitted input renamed as a prediction, and no load-bearing self-citation, so the circularity score is 0.

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

No free parameters are fitted to the target result; WR, WI, l, and N are control parameters specified in the figure captions, with l swept to expose the fast-dephasing regime. The central result rests on standard dephasing modeling and on the assumption that ensemble-averaged incoherent propagation is representative of single realizations. That latter assumption is the least tested part of the paper.

assumptions (3)

domain assumption Each dephasing step multiplies every site amplitude by an independent random phase uniformly drawn from [0,2pi], with no amplitude noise or spatial correlations.
Invoked in Section II, Eq. (4). This is the standard Haken-Strobl-type dephasing model; the no-correlation assumption is what makes the averaged probability evolution take the simple S = |U|^2 form.
ad hoc to paper Ensemble-averaged probability evolution via S(l) is representative of a single random-phase realization.
Used for the a priori jump prediction in Section IV. The paper acknowledges that Eqs. (9) and (12) are derived by averaging over phases and then compares them to one realization, with no phase ensemble statistics.
domain assumption The second-order expansion U approximately I + iHl - (1/2)H^2 l^2 is accurate for the dephasing periods l used, including l up to the numerically determined critical values.
Appendix B Eq. (B1) derives the near-diagonal form of S in this limit. The quantitative claim that the mean IPR approaches 1 below l < 1e-4 relies on this expansion being valid in that regime.

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Cite this review

Pith. "Pith review of Dephasing-induced jumps in non-Hermitian disordered lattices." pith.science (2026). https://pith.science/paper/CC3NRSES

@misc{pith2026241220306,
  author       = {Pith},
  title        = {Pith review of: Dephasing-induced jumps in non-Hermitian disordered lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CC3NRSES}},
  note         = {Machine review of arXiv:2412.20306}
}

read the original abstract

Changes in the wavefunction's phase during propagation in a random Hermitian lattice, a process known as dephasing, results in diffusion rather than Anderson localization. However, when non-Hermiticity is introduced, the wave behavior changes drastically. In particular, we demonstrate that in weakly disordered non-Hermitian lattices, dephasing enhances eigenmode localization which results in abrupt jumps between spatially distant regions. These jumps, which are absent under purely coherent conditions, emerge from the interplay between complex disorder and dephasing.

Figures

Figures reproduced from arXiv: 2412.20306 by the authors.

**Figure 2.** FIG. 2. Single realization (same as the bottom row of Fig. 1) o [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Evolution of the derivative of the power logarithm [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: FIG. 5. Statistically averaged mean inverse participation [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

Discussion (0). Continue with ORCID to comment.

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