Dynamical Phase diagram of the Quantum Ising model with Cluster Interaction Under Noisy and Noiseless Driven field

Alireza Akbari; Ehsan Nedaaee Oskoee; R. Jafari; Sasan Kheiri; S. Mahdavifar

REVIEW 3 major objections 5 minor 133 references

Dynamical Phase diagram of the Quantum Ising model with Cluster Interaction Under Noisy and Noiseless Driven field

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read In a cluster Ising chain, DQPTs survive a field ramp exactly when the ramp starts or ends between two critical fields, and Gaussian white noise cuts the survival velocity linearly with the square of the noise intensity.

desk verdict Solid noiseless DQPT phase diagram for the cluster Ising chain; the noisy phase diagram rides on an unproved p_k=1/2 criterion for mixed states. read the letter →

arxiv 2506.14372 v2 pith:KBD4SV23 submitted 2025-06-17 cond-mat.stat-mech quant-ph

Sasan Kheiri , R. Jafari , S. Mahdavifar , Ehsan Nedaaee Oskoee , Alireza Akbari This is my paper · ORCID

classification cond-mat.stat-mechquant-ph

keywords dynamicalquantumphasetransitionsclusterIsingmodeltransversefieldchainnoisyrampquenchGaussianwhitenoisecriticalsweepvelocitymulti-criticalmodesLandau-Zenertransition

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) survive in a transverse-field Ising chain with three-spin cluster interaction when the field is ramped linearly, with and without Gaussian white noise. Its central claim is a ramp-endpoint rule: DQPTs always occur if the starting or ending field value lies between two of the model's critical fields, even for a sudden quench, while otherwise there is a critical sweep velocity $v_c$ above which DQPTs disappear. It further claims that noise lowers $v_c$ according to $v_c^{(\xi)} = v_c^{(0)} + a\,\xi^2$, and that noise creates a multi-critical-modes region whose upper velocity $v_M$ scales as $\xi^2/m$. These results matter because they turn the movable gap-closing point of the cluster Ising model into a control knob for nonequilibrium criticality and give quantitative noise budgets for observing DQPTs in engineered quantum systems.

What carries the argument

The machinery is the exact mapping of the cluster Ising chain to independent momentum modes, each a two-level Landau-Zener problem with Hamiltonian $H_k(t)=A(k,t)\sigma^z+B(k)\sigma^x$, where $A=2(h(t)-J\cos k-J_3\cos 2k)$ and $B=2(J\sin k+J_3\sin 2k)$. In the noiseless case the rate function $g(t)=-(1/N)\ln|G(t)|^2$ is computed from the Loschmidt overlap, and its non-analyticities occur exactly at modes with excitation probability $p_k=1/2$; the Landau-Zener formula $p_k^{\mathrm{LZ}}=\exp(-2\pi\gamma^2/v)$ with $\gamma=B_k/2$ turns this into an analytic boundary $v_c$. For the noisy case the paper solves the exact master equation for the averaged density matrix, $\frac{d}{dt}\rho_k(t)=-i[H_k^{(0)}(t),\rho_k(t)]-\frac{\xi^2}{2}[H_1,[H_1,\rho_k(t)]]$, and reads $p_k$ from the excited-state population at the end of the ramp. The movable gap-closing mode $k_a=\arccos((J_3-h)/(4hJ_3))$ is what breaks the symmetry of the phase diagram and produces distinct ramp-up and ramp-down behavior.

What would settle it

Compute the true mixed-state rate function from the full noise master equation for the averaged density matrix and check whether its non-analyticities coincide with the $p_k = 1/2$ boundaries in the $(v, \xi)$ phase diagrams; any mismatch at strong noise would falsify the noise-modified DQPT phase diagram.

Watch

Extended reading notes

Core claim

The paper establishes a ramp-endpoint criterion for the one-dimensional cluster Ising model. After a linear ramp of the transverse field from $h_i=-10$ to $h_f$, DQPTs are guaranteed whenever the initial or final field lies between two of the gap-closing fields $h_c^{(a)}=-J_3$, $h_c^{(1)}=J_3-1$, and $h_c^{(2)}=J_3+1$; in that configuration the momentum-resolved excitation probability $p_k$ necessarily crosses $1/2$, so the dynamical free energy $g(t)$ develops non-analyticities at $t_n^*=(n+1/2)\pi/\epsilon_{k^*}^f$ for every $n$, including at the sudden-quench limit. When the ramp starts and ends outside such an interval, $p_k$ can stay above $1/2$ everywhere and DQPTs vanish above a critical sweep velocity computable from the Landau-Zener probability $p_k^{\mathrm{LZ}}=\exp(-2\pi\gamma_m^2/v)$. With Gaussian white noise, applying the same $p_k=1/2$ criterion to the averaged density matrix yields $v_c^{(\xi)}=a\,\xi^2+v_c^{(0)}$ and $v_M^{(\xi)}=\xi^2/m$, with all curves for different $J_3$ collapsing under the rescaled variables $v_c/\gamma_m^2$ versus $\xi^2/\gamma_m$ and $v_M/\gamma_m$ versus $\xi^2$.

Load-bearing premise

The load-bearing premise is that a mode with excitation probability one half still marks a dynamical quantum phase transition after the noise is averaged over, although that criterion was proven only for the noiseless pure-state rate function; if the transfer fails, the noisy phase diagram describes probabilities rather than true DQPTs.

Editorial extensions

If this is right

A ramp whose start or end sits between two critical fields is the clearly favorable regime: DQPTs remain present all the way down to the sudden-quench limit, so experiments looking for DQPTs should target that geometry.
A ramp that crosses critical fields but starts and ends outside such an interval has a finite sweep-velocity ceiling; driving faster than $v_c$ erases the non-analyticities completely.
Gaussian white noise lowers that ceiling as $v_c^{(\xi)} = v_c^{(0)} + a\xi^2$, so the noisier the drive, the slower the ramp must be to keep DQPTs.
For noise and sweep velocity of comparable size, the excitation probability locks to $1/2$ over a finite momentum window, which the paper identifies as a new multi-critical-modes phase; the window's upper velocity $v_M$ grows as $\xi^2/m$.
The collapse of all $J_3$ curves onto a single scaling curve shows that the noise-modified phase diagram is controlled by the single minimum-gap parameter $\gamma_m$.

Reading between the lines

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

If the $p_k=1/2$ condition transfers to the noise-averaged mixed state, then the multi-critical-modes region means DQPT critical times become continuous intervals rather than isolated instants, so experiments should see broadened, plateau-like cusps in the Loschmidt echo rather than sharp spikes.
The linear-in-$\xi^2$ laws suggest a practical calibration protocol: measure $v_c$ at two noise strengths to extract $a$ and $v_c^{(0)}$, then use the $\gamma_m$-scaling curves to predict the full phase diagram at other cluster couplings and ramp endpoints, a step the paper does not take.
The ramp-endpoint rule may be generic for integrable chains whose gap closes at a tunable interior momentum, such as extended XY or cluster-state models; whether the same criterion holds there is left open by this paper.
Only noise during the ramp is treated; adding dephasing after the ramp ends would test whether the critical times themselves survive once the drive stops, which is a separate question from the one answered here.

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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

Short version: the noiseless part is the real contribution, and it's mostly right. The noisy part is interesting but should not be taken at face value until the DQPT criterion for the averaged density matrix is actually justified.

What's new: putting the cluster interaction into the DQPT game gives you a gap-closing point that moves inside the BZ, and with it a phase diagram with one/two/three/four critical modes. The rule that DQPTs always occur when the ramp starts or ends between two critical points, and that otherwise there's a finite critical sweep velocity, is clearly stated and backed by the p_k plots and the LZ boundary curves. The noiseless numerics check out against the exact Landau-Zener formula—that's good evidence the machinery is right.

The noisy part: they solve a standard white-noise master equation, compute the excited-state population p_k at the end of the ramp, and draw the phase diagram from the condition p_k = 1/2. For a pure state that condition is equivalent to a zero of the Loschmidt amplitude and a nonanalyticity in the rate function. For the mixed state from Eq. (16), it is not. The dissipator dephases in the diabatic basis, so the final density matrix is not generally diagonal in the energy basis, and the interferometric or Uhlmann rate function depends on the full ρ, not just the population. The paper never computes a mixed-state rate function or a return amplitude. If the p_k=1/2 condition doesn't track a nonanalyticity of the right object, then v_c(ξ) ~ a ξ^2 + v_c(0), the MCM region, and the ξ^2 scaling for v_M are not statements about DQPTs. That's the soft spot, and it's load-bearing.

Secondary issues: the fits in Fig. 6 have no error bars, and no code is released, so the scaling laws are hard to verify. They also use the asymptotic LZ formula for final fields that aren't far above the critical point, though they note the numerics support that approximation.

Overall, the noiseless result is a solid addition to the DQPT literature, and the noisy result is a plausible conjecture that a good referee could help turn into a real result by asking for the missing mixed-state calculation. Worth sending to a serious referee. I'd cite the noiseless part; I'd hold off on citing the noisy scaling until it's backed by a proper mixed-state DQPT calculation.

Referee Report

3 major / 5 minor

Summary. The paper studies dynamical quantum phase transitions (DQPTs) in a one-dimensional transverse-field Ising chain with three-spin cluster interaction under a linear ramp, both without and with Gaussian white noise in the transverse field. For the noiseless case, the authors map the problem to independent Landau-Zener two-level systems, compute the mode-resolved excitation probability p_k, and use the standard condition p_k = 1/2 to locate DQPTs. They conclude that DQPTs always occur when the ramp starts or ends between two critical points (even for a sudden quench), while otherwise there is a critical sweep velocity v_c above which DQPTs disappear. For the noisy case, they solve the exact master equation for the averaged density matrix, extract p_k, and construct a noise-dependent dynamical phase diagram that contains a multi-critical-modes (MCMs) region. The headline quantitative claims are the scaling laws v_c(ξ) = a ξ^2 + v_c(0) and v_M(ξ) = ξ^2/m, together with a claimed universal collapse under appropriate rescalings.

Significance. If the results are valid, the paper extends DQPT phenomenology to a model with a movable gap-closing mode and provides concrete, falsifiable predictions for how white noise modifies the dynamical phase diagram. The noiseless part is a strength: the numerical transition probabilities are compared with the exact Landau-Zener solution, and the analytical boundaries for asymptotically large final fields are externally benchmarked. The noisy part rests on a master-equation simulation that is a standard tool, and the reported scaling laws are quantitative enough to be tested. However, the central noisy claim depends on an unproved transfer of the pure-state DQPT criterion to the averaged density matrix; this must be established before the noisy phase diagrams can be interpreted as DQPT boundaries.

major comments (3)

[Section V, Eq. (17)] The noisy DQPT criterion is assumed, not derived. In the noiseless case, DQPTs follow from nonanalyticities of g(t) in Eq. (2), which require the per-mode Loschmidt amplitude to vanish and hence p_k = 1/2 for a pure state. For the noisy problem, the authors identify DQPTs with p_k = 1/2 for the averaged density matrix ρ_k(t_f) obtained from Eq. (16), but they never compute a mixed-state return amplitude or rate function. Since the noise term in Eq. (16) is dephasing in the σ_z (diabatic) basis, ρ_k(t_f) is not guaranteed to be diagonal in the final eigenbasis {|α_k^f>, |β_k^f>}, so coherences can contribute to any physical return amplitude. The paper must either prove that p_k = 1/2 is necessary and/or sufficient for nonanalytic behavior of a chosen mixed-state DQPT quantifier, or the noisy phase diagrams in Fig. 6 should be re-framed as thresholds of the excitation probability rather than DQPT boundaries.
[Section IV.2 and Fig. 3(f)] The analytical phase boundaries are obtained from the asymptotic Landau-Zener formula Eq. (15), which is strictly valid for h_f much larger than the critical field. For h_f = 3/2 and J3 = 1/4, h_f = 1.5 is not asymptotically large compared with h_c^(2) = 1.25, yet Fig. 3(f) uses LZ-derived boundaries (the gray dashed-dotted curves). The text notes this limitation only in passing. Please quantify the error by comparing Eq. (15) with the exact finite-time expression Eq. (14) over the parameter ranges of Figs. 3 and 4, and state explicitly where the LZ approximation is used to draw v_c and v_T.
[Section V.B and Fig. 6(c)-(d)] The central quantitative claims v_c(ξ) = a ξ^2 + v_c(0) and v_M(ξ) = ξ^2/m are supported only by straight-line fits with no reported uncertainties, number of data points, or residual analysis. The statement that {a, m} scale as γ_m^{|α|} with |α| = 1 ± 0.001 (Appendix B) likewise rests on log-log fits without error bars. Since these scaling laws are the paper's headline noisy predictions, the fits should be documented with data ranges, fit functions, and confidence intervals, and the collapse in Fig. 6(e)-(f) should be tested with a quantitative collapse criterion.

minor comments (5)

[Throughout] There are numerous typographical errors, e.g., 'filed' for 'field' in Sec. IV.1 and the abstract's 'In other ways' should be 'Otherwise'; the manuscript needs a careful proofreading pass.
[Fig. 3(f)] The labels SCM, TwCMs, ThCMs, and FCMs are first used in the figure captions; please define these acronyms in the main text at their first occurrence.
[Eq. (17)] Eq. (17) defines p_k as an expectation value of ρ_k; please clarify in the text that this is the excited-state population of the averaged density matrix, not the amplitude |b_k|^2 of a pure state.
[References] Reference [36] is listed as 'M. Hey and J. C. Budich'; this should be 'M. Heyl and J. C. Budich'.
[Fig. 6(e)-(f)] The insets showing the v_M scaling are difficult to read; please enlarge them and label both axes, including the scaling variables and their units.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the phase diagrams and scaling laws are computed from the model Hamiltonian via standard Landau-Zener and exact noise master-equation inputs, not reduced to the paper's own definitions or fits.

full rationale

The noiseless derivation chain begins with the exact Bloch Hamiltonian (Eq. 11), maps it to the Landau-Zener problem (Appendix A), and uses the analytical LZ probability (Eq. 15) to obtain the critical sweep velocities v_c and v_FT, e.g. v_c=2πγ_m^2/ln(2). The DQPT condition p_k=1/2 is derived from the pure-state Loschmidt amplitude and rate function in Eqs. (1)-(2), which are external and standard. The phase boundaries in Figs. 3(f) and 4 are either continuity arguments about p_k or LZ-based analytical expressions, not fits renamed as predictions. The noisy section independently solves the exact noise master equation (Eq. 16) and defines p_k by Eq. (17); the reported v_c(ξ)=aξ^2+v_c(0) and v_M=ξ^2/m scalings are numerical outputs extracted from that solution, not quantities used to generate the same data. The only significant caveat is that Eq. (17) treats p_k=1/2 as the DQPT criterion for the averaged, mixed-state density matrix without computing a mixed-state rate function or return amplitude; that is a correctness and missing-support issue, not a circularity, because the paper does not define DQPT as p_k=1/2 by construction. Self-citations [93,94,108] supply the master-equation framework and earlier scaling findings, but the present numerical data are computed independently for a different model, so those citations are not load-bearing. No equation in the paper reduces by construction to a prior definition or to a fitted input.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The main fitted quantities are the slopes a and m and the exponent alpha of the empirical noise scaling laws. The axioms are standard exact-solution tools plus the two modeling assumptions that carry the noisy results: delta-correlated white noise with the exact master equation, and the p_k=1/2 DQPT criterion applied to the mixed state.

free parameters (3)

a = varies with J3; slope of v_c vs xi^2, Fig. 6(c)-(d)
Fitted slope of the linear relation v_c(xi) = a*xi^2 + v_c(0).
m = varies with J3; slope of v_M vs xi^2, Fig. 6(c)-(d)
Fitted from v_M(xi) = xi^2 / m.
alpha = 1 +/- 0.001
Exponent fitted from log-log plots of a and m versus gamma_m (Fig. 7), used for scaling collapse.

assumptions (6)

standard math Jordan-Wigner and Fourier transforms reduce the Hamiltonian to independent two-level systems (Eqs. 10-11).
Standard exact solution for the cluster Ising model; used throughout Sec. III.
domain assumption The system is initialized in the ground state at h_i = -10 with t_i approaching -infinity.
Sec. II A; required for the LZ transition probability formula.
domain assumption The noise is Gaussian white noise with delta-function correlations, and the exact noise master equation (Eq. 16) governs the averaged density matrix.
Sec. V; this is the model of noise used for all noisy results.
domain assumption The LZ transition probability (Eq. 14) and its asymptotic form (Eq. 15) are valid for the finite ramps considered, including h_f = 3/2 and h_f = 10.
Sec. IV; used to derive v_c analytically, though strictly valid only for h_f >> h_c.
ad hoc to paper DQPTs in the noisy case are identified by the condition p_k = 1/2 on the averaged density matrix (Eq. 17), without computing a mixed-state rate function.
Sec. V; this transfers the pure-state criterion to a mixed state and is not derived in the paper.
domain assumption The transition probability p_k is continuous in k, so it must cross 1/2 between modes where p_k=1 and modes where p_k=0.
Sec. IV.1-2; basis for the 'always DQPT' claim.

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Cite this review

Pith. "Pith review of Dynamical Phase diagram of the Quantum Ising model with Cluster Interaction Under Noisy and Noiseless Driven field." pith.science (2026). https://pith.science/paper/KBD4SV23

@misc{pith2026250614372,
  author       = {Pith},
  title        = {Pith review of: Dynamical Phase diagram of the Quantum Ising model with Cluster Interaction Under Noisy and Noiseless Driven field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KBD4SV23}},
  note         = {Machine review of arXiv:2506.14372}
}

read the original abstract

In most lattice models, gap closing typically occurs at high-symmetry points in the Brillouin zone. In the transverse field Ising model with cluster interaction, besides the gap closing at high-symmetry points, the gap closing at the quantum phase transition between paramagnetic and cluster phases of the model can be moved by tuning the strength of the cluster interaction. We take advantage of this property to examine the nonequilibrium dynamics of the model in the framework of dynamical quantum phase transitions (DQPTs) after a noiseless and noisy ramp of the transverse magnetic field. The numerical results show that DQPTs always happen if the starting or ending point of the quench field is restricted between two critical points. In other ways, there is always critical sweep velocity above which DQPTs disappear. Our finding reveals that noise modifies drastically the dynamical phase diagram of the model. We find that the critical sweep velocity decreases by enhancing the noise intensity and scales linearly with the square of noise intensity for weak and strong noise. Moreover, the region with multi-critical modes induced in the dynamical phase diagram by noise. The sweep velocity under which the system enters the multi-critical modes (MCMs) region increases by enhancing the noise and scales linearly with the square of noise intensity

Figures

Figures reproduced from arXiv: 2506.14372 by the authors.

**Figure 2.** FIG. 2. (Color online) The equilibrium phase diagram of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (Color plot) The transition probability [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. (Color online) The dynamical phase diagram of the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. (Color plot) The probability of excitations for a quench from [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. (Color plot) The phase diagram in the [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. (Color plot) The scaling of the slope of linear lines [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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(i):In the case thatJ 3 <1/2, the ramp field crosses the Ising type single critical pointh (1) c , atk c =π, and the prob- ability of excitationp k, depends on the value ofk

Quench across a single critical point (hf =−1/2) When time driven magnetic field crosses a single critical point, i.e,h f =−1/2, we consider two casesJ 3 <1/2 and J3 >1/2, respectively. (i):In the case thatJ 3 <1/2, the ramp field crosses the Ising type single critical pointh (1) c , atk c =π, and the prob- ability of excitationp k, depends on the value o...
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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

(i):In the case thatJ 3 <1/2, the ramp field crosses the Ising type single critical pointh (1) c , atk c =π, and the prob- ability of excitationp k, depends on the value ofk

Quench across a single critical point (hf =−1/2) When time driven magnetic field crosses a single critical point, i.e,h f =−1/2, we consider two casesJ 3 <1/2 and J3 >1/2, respectively. (i):In the case thatJ 3 <1/2, the ramp field crosses the Ising type single critical pointh (1) c , atk c =π, and the prob- ability of excitationp k, depends on the value o...

[2] [2]

Quench across the two critical points (hf = 3/2) For a quench crosses two critical points, i.e,h f = 3/2, we have considered once again two casesJ 3 <1/2 and 1/2< J3 <5/2, respectively. It should be mentioned that, forh f = 3/2 andJ 3 >5/2, the quench transverse field crosses only a single anisotropic transition point and the dynamics of the system is sim...

[3] [3]

Quench across the three critical points (hf = 10) To perform a quench across three critical points, the quench filed end should be larger thanh (2) c and the cluster interaction J3 >1/2. In such a case, the quench field is swept from one equilibrium paramagnetic phase to another one and passage through the critical pointsh (a) c ,h (1) c andh (2) c where ...

[4] [4]

3(f), the dynamical phase diagram of the model has been displayed inv−J 3 plane forh f = 1.5

Dynamical phase diagram In Fig. 3(f), the dynamical phase diagram of the model has been displayed inv−J 3 plane forh f = 1.5. The dy- namical phase diagram represents dynamics of the system for three cases of quench: crosses two critical points where quench field end does not confined between two critical points (J3 <1/2), across two critical points where...

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