Stability of dynamical quantum phase transitions in quenched topological insulators: From multiband to disordered systems

Christian B. Mendl; Jan Carl Budich

REVIEW 3 major objections 4 minor 48 references

Stability of dynamical quantum phase transitions in quenched topological insulators: From multiband to disordered systems

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper establishes that in quenched multiband topological insulators the Fisher zeros underlying dynamical quantum phase transitions generically form a (d-1)-dimensional set in momentum-time space, and that a quench into a three-band…

desk verdict Solid multiband analysis and a useful counterexample; the disorder-stability claim is an honest but unproven extrapolation from finite-supercell numerics. read the letter →

arxiv 1909.01402 v2 pith:SQAUIEOB submitted 2019-09-03 cond-mat.quant-gas cond-mat.mes-hallcond-mat.stat-mechquant-ph

Christian B. Mendl , Jan Carl Budich This is my paper · ORCID

classification cond-mat.quant-gascond-mat.mes-hallcond-mat.stat-mechquant-ph

keywords dynamicalquantumphasetransitionsFisherzerosLoschmidtamplitudequenchdynamicstopologicalinsulatorsdisorderedKitaevchainHofstadtermodelPancharatnamgeometric

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 asks when dynamical quantum phase transitions (DQPTs) — the cusps in the return probability of a quantum system after a quench — occur in multi-band topological insulators and in disordered wires. It shows that in systems with more than two bands the zeros of the Loschmidt amplitude (Fisher zeros) that produce those cusps generically form a set of dimension one less than the spatial dimension in momentum-time space, and that the familiar two-band guarantee fails: a quench into a three-band Chern insulator with band Chern numbers $(1,0,-1)$ can produce no Fisher zeros at any time if the initial filled band leans on the central trivial band. To approach disorder, the paper enlarges a periodic supercell and presents numerical evidence that rate-function cusps persist up to significant disorder strength as the supercell grows. The authors state that an analytical proof for the truly aperiodic disordered case remains open.

What carries the argument

The workhorse is the factorized Loschmidt amplitude $G(t)=\prod_k G_k(t)$ whose zeros are Fisher zeros; DQPTs are the resulting cusps of the rate function $g(t)=-\frac{1}{N}\log|G(t)|^2$. The argument runs on two criteria: the triangle-inequality overlap condition (5), which requires that no single band overlap exceed $1/2$ for a Fisher zero to be possible, and the dynamical exclusion condition (6), which forbids zeros when all post-quench phases lie on a minor arc of the unit circle. Dimensional counting uses ergodicity of the phases $e^{-iE_{k,\alpha}t}$ on the $n$-torus, yielding the generic $(d-1)$-dimensional Fisher-zero manifold. The Pancharatnam geometrical phase picture ties each Fisher zero to a phase vortex in momentum-time space. For disorder, the supercell representation promotes disorder coefficients into orbital indices of a $2\ell \times 2\ell$ Bloch Hamiltonian, making the Loschmidt amplitude real-analytic in the noise; this continuity is what lets the cusps survive finite disorder.

What would settle it

Compute the rate function and its time derivative for a genuinely aperiodic disordered Kitaev chain (no supercell repetition) at disorder strength $\Delta\mu_{\max}=2$ for system sizes $L=100$, $400$, $1600$: if the cusp near the critical time rounds off and disappears as $L$ grows, the supercell extrapolation would be falsified. Alternatively, test the three-band $(1,0,-1)$ counterexample with initial-state overlap exactly $1/\sqrt{2}+\varepsilon$ at all momenta; any Fisher zero at finite time would falsify the no-DQPT claim.

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Extended reading notes

Core claim

The central discovery is that the existence of DQPTs is decided by where the Fisher zeros of the momentum-resolved Loschmidt amplitude $G_k(t) = \sum_{\alpha} |\langle u_{k,\alpha}|\psi_k\rangle|^2 e^{-iE_{k,\alpha}t}$ can be located, which is controlled by two independent conditions: the static overlap condition $|\langle u_{k,\alpha}|\psi_k\rangle|^2 \le 1/2$ for all $\alpha$, and a dynamical condition on the post-quench eigenvalues. For more than two bands, the overlap condition holds in an entire $d$-dimensional momentum region, so both real and imaginary parts of $G_k(t)$ must be tuned to zero, making the Fisher-zero set generically $(d-1)$-dimensional in momentum-time space; for two bands the admissible region is $(d-1)$-dimensional but zeros are guaranteed there, so the set is again $(d-1)$-dimensional. A three-band counterexample with Chern numbers $(1,0,-1)$ shows that a topologically nontrivial final Hamiltonian does not force DQPTs: if the initial filled band has overlap $> 1/\sqrt{2}$ with the trivial middle band everywhere, the overlap condition is violated everywhere and no Fisher zero exists at any time. For disordered systems, real-analytic dependence of $G_k(t)$ on the disorder amplitudes implies that Fisher-zero vortex-antivortex pairs move continuously, so cusps cannot vanish discontinuously for finite supercell size; extensive Kitaev-chain numerics indicate they survive the large-supercell limit up to substantial disorder.

Load-bearing premise

The conclusion that DQPTs survive true randomness rests on the unproven leap that, for very large supercell period $\ell$, a periodic disorder realization behaves like an aperiodic random chain, so that the limit of large system size does not destroy the cusps.

Editorial extensions

If this is right

In $d$ spatial dimensions, generic multiband quenches with $n>2$ bands have Fisher zeros on a $(d-1)$-dimensional manifold in momentum-time space: isolated points in one dimension, curves in two dimensions.
The two-band result that a change of Chern number across the quench guarantees DQPTs does not extend to multiband systems; the $(1,0,-1)$ counterexample has a topological final Hamiltonian and yet no Fisher zeros.
For any finite supercell size $\ell$, Fisher zeros and rate-function cusps persist as continuous deformations under increasing disorder; they cannot disappear discontinuously.
Numerical evidence on the disordered Kitaev chain indicates that the cusps survive up to disorder strength comparable to the hopping and gap parameters as $\ell \to \infty$, i.e., approaching the truly disordered limit.
The variance of the rate function across disorder realizations decays as $\sim 1/\ell$, indicating self-averaging of the return probability in large supercells.

Reading between the lines

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

A testable extension would be to compute the dynamical topological order parameter for the $(1,0,-1)$ counterexample; the absence of Fisher zeros suggests the order parameter may also vanish, sharpening 'topology change implies DQPT' into 'only changes in the topology of fully occupied bands matter'.
The supercell approach implicitly assumes that rare disorder configurations do not generate Fisher zeros in the thermodynamic limit; studying the extreme-value statistics of $\min_k |G_k(t)|$ across many realizations would probe this assumption.
Because the paper assumes a single filled band throughout, the criterion (5) and the counterexample both need re-derivation for partially filled bands; the determinant form in Eq. (15) suggests the factorization structure, and with it the clean dichotomy, may break down there.
If the $1/\ell$ variance scaling holds generally, DQPT rate functions would be self-averaging observables, making them robust diagnostics in experiments with disordered optical lattices and wires.

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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

First thing to know: the multiband half of this paper is the stronger half, and the disorder half is an honest numerical extrapolation rather than a proof. The authors make a real contribution by clarifying when topology-induced DQPTs actually occur: they give a dimensional-counting argument that in n>2 bands Fisher zeros generically form a (d-1)-dimensional set in momentum-time space, and they build a three-band Chern insulator counterexample where the post-quench Hamiltonian has Chern numbers (1,0,-1) but no Fisher zeros appear because the initial state is concentrated in the trivial middle band. The Hofstadter simulations illustrate the criteria and the vortex dynamics cleanly. That part is solid.

The disorder part is more fragile. The idea of approaching disorder by enlarging a supercell with random onsite potentials is reasonable, and the numerics for the Kitaev chain show cusps persisting up to significant disorder strength for l up to 100. But the stress-test concern is legitimate: as l→∞, each l system is a different periodic chain, not a converging sequence to a single disordered chain. The authors themselves say the order-of-limits problem makes an analytical proof 'elusive,' and the variance argument uses the zero-disorder factorization (16) only approximately. So the headline 'stability against disorder' is supported by numerics but not proven. They are transparent about this, which earns credit, but it would be better if they released code or data so the figures could be independently checked.

Minor points: the dimensional counting is heuristic, not a rigorous theorem, though it is plausible and backed by the two-band exact result. The counterexample is conditional on a >1/√2 overlap assumption at all momenta; no explicit initial Hamiltonian is constructed, so it is more an argument that such quenches can avoid DQPTs than a demonstration with a concrete lattice model.

Who is this for? People working on DQPTs in topological or disordered systems, especially experimental groups in cold atoms or photonics who want to know when quenches will show cusps. The multiband criteria and counterexample are directly useful. I'd send it to peer review—it deserves a serious referee despite the soft disorder extrapolation, because the multiband analysis is new and the methods are transparent.

Referee Report

3 major / 4 minor

Summary. The paper studies dynamical quantum phase transitions (DQPTs) in multiband and disordered free-fermion systems. It derives necessary and sufficient conditions for Fisher zeros of the Loschmidt amplitude, uses a dimensional/ergodicity argument to claim that for more than two bands the Fisher zeros generically form a (d-1)-dimensional submanifold in momentum-time space, and constructs a three-band Chern-insulator scenario in which no DQPTs occur despite nontrivial band Chern numbers. The predictions are illustrated by Hofstadter-model simulations. For disordered systems, the paper introduces a supercell approach in which disorder is made periodic with period l, and presents numerical evidence that rate-function cusps persist up to significant disorder strength as l is increased, while explicitly acknowledging an order-of-limits problem for the truly disordered limit.

Significance. If its claims hold, the paper extends the topology-DQPT connection beyond two-band translation-invariant models and provides a dimension-dependent classification of Fisher-zero sets. The exact inequalities (5) and (6) and the explicit n=2 analysis are clean and useful, and the Hofstadter numerics support the dimensional picture. The disorder-stability result, however, is supported only by finite-period supercell numerics, and the paper itself concedes the lack of an analytical proof for the truly disordered case; as presented, the disorder claim is a numerical extrapolation rather than a demonstrated property. The multiband counterexample is conditional on an unproven existence assumption. These gaps are significant but addressable within the manuscript's scope.

major comments (3)

[Sec. IV B and Sec. V] The central claim that DQPTs survive up to significant disorder strength in the truly disordered thermodynamic limit is not established by the presented numerics. All calculations in Figs. 3-5 are for periodic supercells of finite period l, and the paper itself states in Sec. V that a non-trivial order of limits problem renders an analytical proof for the truly disordered case elusive. Each fixed l corresponds to a different translationally invariant system (one random l-site pattern repeated infinitely), and the rate function g_l(t) in Eq. (21) is defined through a supercell Brillouin-zone integral that has no direct counterpart in the disordered thermodynamic limit. The statement in Sec. IV A that 'for sufficiently large l, the system resembles a disordered system' is a heuristic, not a controlled approximation. To support the headline conclusion, the authors should provide direct real-space simulations of a disordered Kitaev chain without supercell periodicity, with finite-size scaling, or else substantially soften the claim in the title and abstract.
[Sec. II D] The proposed counterexample is conditional on the existence of an initial Hamiltonian whose lowest band has overlap >1/sqrt(2) with the central band of the post-quench Hamiltonian at all momenta, but no such Hamiltonian is constructed or even argued to exist. Since the text says 'we construct a basic counter-example', the authors should provide an explicit initial Hamiltonian H_i(k) realizing such a Bloch state as its occupied band, or at least explain why such a state is guaranteed to be representable as the lowest band of a gapped trivial Hamiltonian. Without that, the no-DQPT conclusion is a statement about an assumed state rather than a demonstrated counterexample to the general claim.
[Sec. II C] The claim that Fisher zeros generically form a (d-1)-dimensional manifold for n>2 rests on a codimension/ergodicity heuristic rather than a proof. The argument that the real and imaginary parts of G_k(t) can be independently tuned to zero assumes transversality of the zero set and rational independence of the band energies on the admissible region; neither condition is stated or verified, even for the Hofstadter example. The authors should formulate this as a precise conjecture with explicit non-resonance conditions, and clearly separate the rigorous n=2 result from the generic-n expectation.

minor comments (4)

[Sec. II D] There is a typo: 'DQOTs' in the sentence 'no Fisher zeros or DQOTs occur at any time' should read 'DQPTs'.
[Eq. (21)] The definition of the rate function in Eq. (21) uses log|G^l_k(t)|, whereas the rate function defined in the introduction and used in Eq. (8) is based on log|G|^2. This introduces a factor of 2 inconsistency; please make the definitions uniform.
[Sec. IV B, Fig. 5] The claimed ~1/l scaling of the variance is justified by invoking Eq. (16), which is exact only at zero disorder, and the text acknowledges this as an approximation. The figure caption should label the scaling as conjectural, since the independence assumption is not proven.
[Sec. IV A] The statement 'For sufficiently large l, the system resembles a disordered system' is not quantified. Please specify in what sense (e.g., local observables, finite-order correlation functions) the resemblance is expected to hold, or replace it with a more precise statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's central claims follow from explicit Fisher-zero criteria, a self-contained dimensional-counting argument, and direct numerical simulations; no fitted parameter or self-citation is renamed as a prediction.

full rationale

The derivation chain is self-contained. The multiband analysis starts from the exact factorized Loschmidt amplitude (Eq. (4)) and uses the necessary Fisher-zero condition (5) and the sufficient exclusion condition (6). The counterexample in Sec. II D applies condition (5) directly: an initial band with overlap >1/sqrt2 with one post-quench band violates the necessary condition at every momentum, so no Fisher zero can occur; this is a logical application, not a definitional rewording. The dimensional-counting claim in Sec. II C is an independent genericity argument (two real equations in d+1 variables, with ergodicity on the n-torus) and does not depend on any fitted parameter; the agreement with the authors' earlier mixed-state work [41] is corroboration, not input. The Hofstadter numerics are explicit simulations of the stated Hamiltonian. The disordered Kitaev chain study is likewise an honest numerical extrapolation: Sec. IV A states the periodic-supercell approximation ('For sufficiently large l, the system resembles a disordered system') and Sec. V explicitly flags the 'non-trivial order of limits problem' that prevents an analytical proof for the truly disordered case. That is an acknowledged limitation, not a circular reduction; the survival of rate-function cusps is observed in direct simulations, not obtained by fitting a parameter and then predicting the same quantity. The self-citations (Refs. [16] and [41]) supply the standard DQPT/geometric-phase framework and are not load-bearing in the new results: no uniqueness theorem or ansatz is imported from the authors' own prior work as the sole justification. Therefore the paper does not exhibit self-definition, fitted-input-as-prediction, or self-citation-based forcing.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the listed domain assumptions, especially single-filled-band, translation invariance, ergodicity, and the supercell approximation to true disorder. No new particles, forces, or entities are introduced. The numerical parameters are hand-chosen but not fitted to external data.

free parameters (1)

Kitaev chain numerical parameters (t=1, mu_bar=6, Delta=1)
Hand-chosen illustrative parameters for the disordered Kitaev quench simulations; the authors state that other disorder types give qualitatively similar results, so the central claims are not tuned to these specific values, but the numerical operating point is chosen by hand.

assumptions (4)

domain assumption The initial state is a filled lowest Bloch band, so the Loschmidt amplitude factorizes into single-occupied-mode overlaps.
Used throughout Sec. II, especially Eqs. (3) and (4); the authors acknowledge in Sec. V that deriving strict criteria for multiple occupied bands remains future work.
domain assumption Lattice translation invariance holds in the clean multiband models, giving momentum conservation and factorization of G(t) over k.
Sec. II A assumes translation invariance to write G(t) as a product over lattice momenta; this excludes non-translation-invariant clean systems.
domain assumption The post-quench Bloch phases are ergodic on the n-torus, requiring rationally independent energies E_{k,alpha}.
Used in the Sec. II C dimensional counting of Fisher-zero manifolds; degeneracies or resonances would change the codimension of the zero set.
domain assumption For large supercell period l, a periodic disorder realization is representative of a truly disordered system.
Sec. IV A states this explicitly; the extrapolation across the l-to-infinity order-of-limits gap is the basis for the disorder-stability claim and is not proven analytically.

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Cite this review

Pith. "Pith review of Stability of dynamical quantum phase transitions in quenched topological insulators: From multiband to disordered systems." pith.science (2026). https://pith.science/paper/SQAUIEOB

@misc{pith2026190901402,
  author       = {Pith},
  title        = {Pith review of: Stability of dynamical quantum phase transitions in quenched topological insulators: From multiband to disordered systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SQAUIEOB}},
  note         = {Machine review of arXiv:1909.01402}
}

read the original abstract

Dynamical quantum phase transitions (DQPTs) represent a counterpart in non-equilibrium quantum time evolution of thermal phase transitions at equilibrium, where real time becomes analogous to a control parameter such as temperature. In quenched quantum systems, recently the occurrence of DQPTs has been demonstrated, both with theory and experiment, to be intimately connected to changes of topological properties. Here, we contribute to broadening the systematic understanding of this relation between topology and DQPTs to multi-orbital and disordered systems. Specifically, we provide a detailed ergodicity analysis to derive criteria for DQPTs in all spatial dimensions, and construct basic counter-examples to the occurrence of DQPTs in multi-band topological insulator models. As a numerical case study illustrating our results, we report on microscopic simulations of the quench dynamics in the Harper-Hofstadter model. Furthermore, going gradually from multi-band to disordered systems, we approach random disorder by increasing the (super) unit cell within which random perturbations are switched on adiabatically. This leads to an intriguing order of limits problem which we address by extensive numerical calculations on quenched one-dimensional topological insulators and superconductors with disorder.

Figures

Figures reproduced from arXiv: 1909.01402 by the authors.

**Figure 2.** FIG. 2. Pancharatnam geometrical phase and rate function [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Pancharatnam geometrical phase for the disordered Kitaev chain with period [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Rate function and its derivative around the first criti [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Histogram and corresponding variance for random [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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