Pith. sign in

REVIEW 3 major objections 5 minor 3 cited by

Universal Driven Critical Dynamics near the Boundary

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read For cooling through an ordinary boundary transition, the boundary order parameter depends on the driving rate logarithmically rather than as the power law expected from the Kibble-Zurek mechanism.

desk verdict A genuinely new but under-supported anomaly: boundary FTS works for many protocols, but the log law for ordinary-transition cooling is not yet established beyond a two-parameter fit. read the letter →

arxiv 2509.10049 v1 pith:S6GCAP63 submitted 2025-09-12 cond-mat.stat-mech

Yu-Rong Shu , Shuai Yin This is my paper
classification cond-mat.stat-mech
keywords Kibble-Zurekmechanismfinite-timescalingboundarycriticalphenomenaordinarytransitionIsingmodellogarithmicsurfacedynamicsnonequilibrium
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

The paper extends finite-time scaling — the modern generalization of the Kibble-Zurek mechanism — to boundary critical phenomena in the 2D and 3D Ising models. It shows that most driven boundary universality classes are described by the same power-law scaling as the bulk, with boundary exponents substituted in. The central exception is cooling through the ordinary transition, where the boundary order parameter shrinks as a logarithm of the cooling rate instead of a power law, revealing a striking heating/cooling asymmetry. The authors also treat the special transition driven by surface-coupling ramps, where the initial critical state invalidates the standard adiabatic-impulse picture and a new scaling form involving the waiting time, or 'age', of the boundary is required.

What carries the argument

Boundary finite-time scaling (BFTS): a set of scaling forms that generalize bulk finite-time scaling by replacing the bulk order-parameter exponent β with the boundary exponent β_1. The key variable is the driving-induced length scale ξ_d ∝ v^{-1/r} with r = z + 1/ν, which controls both the temperature dependence and spatial crossover through the scaling variable x v^{1/r}. For the ordinary transition, the BFTS form fails for cooling because the exponent combination 2β_1/ν − (d−1) is non-negative, so surface fluctuations prevent domain formation and the driving-rate dependence becomes logarithmic.

What would settle it

Measure the squared boundary order parameter at the bulk critical point for 2D and 3D Ising cooling from the disordered phase at large driving rates; if M_s1^2 L^{d-1} scales as a power of v rather than as log(b v), or if the data do not collapse according to Eq. (13), the claimed abnormal logarithmic scaling is refuted.

Watch

Extended reading notes

Core claim

For cooling dynamics through an ordinary boundary transition, the squared boundary order parameter M_s1^2 obeys M_s1^2 = a L^{-(d-1)} log(b v) at the bulk critical point (Eq. 12), with a<0, b>0 constants. This is in contrast to the normal BFTS power-law form M_s1^2 ∝ L^{-(d-1)} v^{[2β_1/ν-(d-1)]/r} (Eq. 11), which fails for both 2D and 3D. The authors propose a generalized scaling form M_s1^2 = L^{-(d-1)}[a log(b v) + f_9(g v^{-1/(ν r)})] (Eq. 13) that collapses data away from the critical point. The breakdown is attributed to strong surface fluctuations that suppress formation of correlated domains at the boundary during cooling.

Load-bearing premise

The analysis assumes that boundary driving introduces no new dynamic critical exponent, so the bulk dynamic exponent z and the combination r = z + 1/ν fully determine the boundary scaling forms; if a separate surface relaxation time scale or a boundary-specific dynamic exponent exists, the predicted exponents and the log-vs-power-law discrimination would change.

Editorial extensions

If this is right

  • For ordinary-boundary cooling, the squared boundary order parameter at the critical point is predicted to obey M_s1^2 = a L^{-(d-1)} log(b v), so experiments should look for a logarithmic, not power-law, dependence on the ramp rate.
  • The sign of 2β_1/ν − (d−1) becomes a diagnostic: negative values allow the normal power-law BFTS; non-negative values signal the abnormal logarithmic regime.
  • Heating and cooling are not symmetric at boundaries: heating obeys BFTS in all boundary universality classes, while cooling fails for the ordinary class, implying that separate scaling forms must be used for opposite ramp directions.
  • For surface-coupling ramps across the special transition, increasing the coupling from the ordinary line requires an initial 'age' (waiting time) t_a; the proposed scaling form Eq. (23) ties the boundary order to t_a^{-2β_1^o/(ν z)} and v^{2(β_1−β_1^o)/(ν r_J)}.
  • For the extraordinary transition, the ordered boundary enforces a power-law decay M_sx^2 ∝ x^{-2β/ν} into the bulk, and the full scaling form Eq. (24) holds with the driving dependence entering through v x^r.

Reading between the lines

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

  • The logarithmic law may be generic in any boundary universality class where 2β_1/ν ≥ d−1, so testing another model (e.g., the O(N) model with an ordinary boundary) would probe universality of the mechanism.
  • The waiting-time scaling for increasing surface coupling is reminiscent of aging phenomena; one could extend the formalism to other tricritical points where the initial state is critical along a different direction.
  • If suppression of boundary domain formation during cooling is the true cause, spatially resolved measurements of M_sx^2 should show a depth profile: bulk power-law behavior at large x crossing over to the log behavior near the surface, with a crossover scale set by x v^{1/r}.
  • The same driving-rate correction q(v) = v(1 + c v^{-ω/r}) used for the special transition could improve collapse in the ordinary cooling case, suggesting that higher-order scaling corrections may be present but do not change the leading log law.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies driven critical dynamics near boundaries of 2D and 3D Ising models under linear temperature sweeps, and for the special transition also under surface-coupling sweeps. The authors propose a boundary finite-time scaling (BFTS) framework in which the bulk FTS forms are generalized by replacing the bulk order-parameter exponent beta with the boundary exponent beta_1. For heating in the ordinary transition, this yields scaling collapses in 2D and 3D, including a crossover full scaling form. The central new claim is that for cooling in the ordinary transition the normal BFTS form (11) fails and the boundary order parameter obeys an anomalous logarithmic law, Eq. (12), M^2_s1 = a L^{-(d-1)} log(bv). For the surface and special transitions, BFTS is reported for both heating and cooling, and for the special transition a generalized BFTS for aged initial states is proposed. The extraordinary transition is treated only through heating data. The abstract and summary, however, assert that cooling in the extraordinary transition is also BFTS, which is not demonstrated in the text.

Significance. Understanding driven critical dynamics near boundaries is a natural and important extension of the Kibble-Zurek mechanism, and this paper is the first systematic numerical exploration of it across ordinary, surface, special and extraordinary boundary classes. If the claimed logarithmic cooling law for the ordinary transition is correct, it would establish a qualitatively new boundary effect and imply that the standard FTS framework must be modified near boundaries. The paper makes productive use of published equilibrium boundary exponents and supports the ordinary/special/surface heating BFTS forms with data collapses over broad L and v ranges. The generalized BFTS for nonequilibrium initial states, Eq. (23), is also a useful conceptual addition. However, the paper contains no machine-checked proofs or code release; the main new quantitative claims rest entirely on numerical fits, and Eq. (12) is a falsifiable prediction that is not yet established by the evidence presented.

major comments (3)
  1. [Abstract; Sec. VII] The abstract and Sec. I state that cooling in the extraordinary transition is described by BFTS, but Sec. VII contains no cooling data: Fig. 15 and the text explicitly verify Eq. (24) only for heating (T0=Tb-2). The universal claim for ordinary/special/surface/extraordinary cooling is therefore unsupported for the extraordinary class. This is not merely presentational; the abstract's factual content is load-bearing. Either add cooling simulations (with a data collapse for M_s1^2 L^{d-1} vs vL^r) or restrict the claim to the cases actually simulated.
  2. [Eq. (12); Figs. 5(a1), 6(a1), 5(b2), 6(b2)] The central log law is established only by a two-parameter fit of M_s1^2 L^{-(d-1)} vs v at J_s=1 and T0=Tb+2. The fits are shown without statistical uncertainties or residuals, and no alternative forms (power law with small negative exponent, crossover to BFTS) are compared. Over roughly two decades a monotone function with slowly varying slope is consistent with several forms. The collapses in Figs. 5(b2)/6(b2) subtract exactly the fitted a log(bv) from the ordinate, so they do not independently validate the logarithmic form. In addition, the additive decomposition in Eq. (13) is assumed, not derived, and the constants a and b are not shown to be independent of J_s or of the initial temperature offset. At present Eq. (12) is an empirical curve, not a demonstrated universal scaling law.
  3. [Sec. II premise; Sec. VIII criterion] All BFTS forms rely on the premise that the bulk dynamic exponent z and r=z+1/nu determine boundary driven dynamics with no additional boundary time scale (Sec. II). This assumption is not tested. The criterion in Sec. VIII (2 beta_1/nu - (d-1) >= 0 for the abnormal log behavior) is a restatement of the sign of the power in Eq. (11) and does not by itself explain why a logarithm, rather than a power law with a different exponent, appears. A concrete check would be to measure the surface autocorrelation time directly, or to repeat the ordinary-cooling collapse with a free dynamic exponent and show that the bulk z gives a statistically acceptable collapse.
minor comments (5)
  1. [Sec. VI.A.1] Below Eq. (16), two references to Eq. (14) should be Eq. (16). The caption of Fig. 10(a2) also labels the vertical axis as M^2_sL L^{2 beta/nu}; the boundary quantity considered is M^2_s1 L^{2 beta_1/nu}.
  2. [Fig. 11(a1) caption] The quoted exponent, 2 beta_1/nu r, is inconsistent with the plotted quantity M^2_s1 L^{d-1}; the expected exponent is [2 beta_1/nu - (d-1)]/r.
  3. [Sec. VI.B.1] The sentence after Eq. (22) says 'for increasing J', but the paragraph and Fig. 13 describe decreasing J_s. Please correct this slip.
  4. [Eqs. (19) and (23)] The label f15 is used for two different scaling functions; renumber one of them to avoid ambiguity. Eq. (22) also reuses f12 from Eq. (16).
  5. [Sec. IV.A, opening paragraph] The Mermin-Wagner sentence refers to 'continuous symmetry breaking' in an Ising context where the spin symmetry is discrete. Please clarify that the 1D boundary of the 2D Ising model does not order independently for continuous-symmetry reasons and that any symmetry argument is not the same as the existing boundary criticality analysis.

Circularity Check

1 steps flagged · score 6.0 of 10

Logarithmic cooling law in ordinary transition is a two-parameter empirical fit; its 'confirmation' via Eq. (13) subtracts the same fitted form, so the central new claim is not independently validated.

  1. fitted input called prediction [Sec. IV B (Cooling dynamics), Eqs. (12)-(13), Figs. 5(a1), 5(b2), 6(a1), 6(b2)]
    "From Figs. 5(a1) and 6(a1), for large v, we find that the dependence of M^2_{sL}L^{(d-1)} on v satisfies a logarithmic function: M^2_{s1} ∝ aL^{-(d-1)} log(bv), (12) in which a and b are dimension-dependent constants ... After rescaling the data according to Eq. (13), the rescaled curves collapse well, confirming the validity of Eq. (13)."

    Eq. (13) is proposed immediately after a and b are fitted to M^2_{s1}L^{-(d-1)} vs v in Figs. 5(a1)/6(a1) (caption values a=-0.4539,b=0.0132 and a=-0.5026,b=0.0023). At g=0, Eq. (13) reduces to the fitted log law plus the constant f9(0); the collapse in Figs. 5(b2)/6(b2) is obtained by subtracting the very same fitted a log(bv) from the data. Hence the collapse does not independently confirm the logarithmic form; it is partly forced by the fit. No RG or microscopic derivation of the log is supplied, and no test of a,b under variation of J_s, initial offset, or alternative functional forms is reported, so the 'abnormal logarithmic scaling' is an empirical two-parameter fit presented as a discovered universal law.

full rationale

Most of the paper is self-contained and not circular: the BFTS forms for heating (Eqs. 8, 10, 16, 18, 22) and for cooling in surface, special, and extraordinary transitions (Eqs. 15, 19, 24) are tested by data collapse using exponents β, β1, ν, z taken from independent equilibrium literature. Those tests have substantive content and are not reduced to the paper's own inputs. The central anomalous claim, however, is Eq. (12): M^2_{s1} = a L^{-(d-1)} log(bv). This form is not derived from FTS or RG; it is read off the same simulation curves in Figs. 5(a1) and 6(a1), with a and b as fitted constants. The subsequent verification of Eq. (13) subtracts the fitted a log(bv) from the same observable, so the data collapse in Figs. 5(b2)/6(b2) cannot certify the logarithmic law independently. That is a partial circularity: a fitted input is presented as a discovered prediction, and its purported confirmation is partly by construction. The score is 6 rather than higher because the paper does contain substantial non-circular content (the BFTS framework and the special/surface/extraordinary collapses), and the log law, while weakly grounded, is at least a concrete empirical proposal that could be tested by future independent simulations. Self-citations appear (e.g., Refs. 25, 26, 32, 34, 104, 106) but are not load-bearing for the central ordinary-transition log claim; the issue is specifically the fit-to-validation overlap.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles or forces. The free parameters are fitting constants used to achieve data collapse and to parameterize the log law. The axioms are the standard FTS framework, the analogy-based substitution of boundary exponents, and the assumption that bulk dynamic exponents apply at boundaries. The ad hoc axioms are the log law and the waiting-time scaling form, both proposed from the data rather than derived.

free parameters (3)
  • a and b in log scaling M^2_s1 = a L^{-(d-1)} log(bv) = 2D: a = -0.4539, b = 0.0132; 3D: a = -0.5026, b = 0.0023
    Fitted to the large-v Monte Carlo data in Figs. 5(a1) and 6(a1). No theoretical prediction for their values or universality.
  • c and omega in finite-size correction y(L) = 1 + c L^{-omega} = 2D: c = -2.12, omega = 0.50; 3D: c = -3.39, omega = 0.50
    Used in Figs. 3(a2), 4(a2), 5(a2), 6(a2) to improve the data collapse. omega = 0.50 is chosen by hand, not derived; c is fitted to optimize collapse.
  • c and omega in driving-rate correction q(v) = v(1 + c v^{-omega/r}) = c = 0.05, omega = 1 (3D special transition)
    Introduced in Fig. 10(c2) and Fig. 11(c2) to collapse the crossover data for the special transition. Values are fitted to the data.
assumptions (5)
  • standard math Adiabatic-impulse scenario of the Kibble-Zurek mechanism, with a driving-induced time scale zeta_d ~ v^{-z/r} and length scale xi_d ~ v^{-1/r}.
    Invoked in Sec. III as the foundation of FTS, following Refs. [19, 20, 25, 26].
  • ad hoc to paper Normal generalization of bulk FTS to the boundary: the boundary order parameter obeys the bulk FTS form with the bulk exponent beta replaced by the boundary exponent beta_1.
    Stated in Sec. IV A before Eq. (8): 'This analogy suggests that the dynamic scaling properties at the boundary may be obtained by a similar substitution.' This assumption is the basis for all BFTS forms and is not derived.
  • domain assumption No new dynamic critical exponent is needed for boundary driven dynamics; the bulk dynamic exponent z governs the boundary time scale.
    Stated in Sec. II: 'For driven critical dynamics, there is no additional divergence in the time direction, and thus no new critical exponents need to be introduced.' Used in every BFTS form, e.g., Eqs. (8), (11), (16).
  • ad hoc to paper The logarithmic scaling form (Eq. 12) is the correct asymptotic form for cooling in the ordinary transition.
    Proposed in Sec. IV B based on the numerical observation that M^2_s1 decreases with v and the argument that the normal BFTS form fails. No RG or analytical derivation is given.
  • ad hoc to paper Generalized BFTS scaling form (Eq. 23) combining ordinary and special critical exponents with the waiting time t_a.
    Assumed in Sec. VI B 2: 'Accordingly, we can assume the scaling form as ...' The form is motivated by memory of the initial critical state but is not derived from a renormalization-group analysis.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Universal Driven Critical Dynamics near the Boundary." pith.science (2026). https://pith.science/paper/S6GCAP63

@misc{pith2026250910049,
  author       = {Pith},
  title        = {Pith review of: Universal Driven Critical Dynamics near the Boundary},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6GCAP63}},
  note         = {Machine review of arXiv:2509.10049}
}
read the original abstract

The celebrated Kibble-Zurek mechanism (KZM) describes the scaling of physical quantities when external parameters sweep through a critical point. Boundaries are ubiquitous in real systems, and critical behaviors near the boundary have attracted extensive research. Different boundary universality classes, including ordinary, special, extraordinary, and surface transitions, have been identified. However, the driven critical dynamics near boundaries remains unexplored. Here, we systematically investigate the driven critical dynamics in various boundary universality classes of the Ising model in both two and three dimensions, and discover a wealth of dynamic scaling behaviors. We find that for heating dynamics in all boundary universality classes, as well as for cooling dynamics in special, extraordinary, and surface transitions, the dynamic scaling behaviors of the order parameter can be described by a normal generalization of the KZM, called boundary finite-time scaling (BFTS). In contrast, for cooling dynamics in ordinary transition, we discover an abnormal logarithmic scaling on the driving rate. Moreover, for the special transition, in addition to temperature driving, we also consider the driven dynamics by driving the surface couplings. For increasing the surface coupling across the special transition point along the line of the ordinary transition, the prerequisite of the KZM, which requires that the correlation length/time in the initial state to be short-ranged, breaks down. We develop a generalized BFTS for a nonequilibrium initial state characterized by the waiting time, or the ``age'', of the boundary. Possible generalizations are also discussed.

Figures

Figures reproduced from arXiv: 2509.10049 by the authors.

Figure 1
Figure 1. FIG. 1. General phase diagram on the boundary of a system [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Driven dynamics of the order parameter in the bulk [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Heating dynamics of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Heating dynamics in 3D ordinary transition at [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Cooling dynamics in 2D ordinary transition at [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Cooling dynamics in 3D ordinary transition at [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Illustration of suppressed domain formation near the [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Cooling dynamics in 3D surface transition at [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Heating dynamics in 3D special transition at [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Cooling dynamics in 3D special transition at [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Driven dynamics in 3D special transition at [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Heating dynamics in 3D extraordinary transition at [PITH_FULL_IMAGE:figures/full_fig_p013_15.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Driven dynamics in 3D special transition at [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Anomalous Dynamical Scaling at Topological Quantum Criticality

    cond-mat.str-el 2025-12 unverdicted novelty 7.0 of 10

    Topological quantum critical points exhibit anomalous dynamical scaling in boundary dynamics and defect production due to edge modes, beyond conventional Kibble-Zurek scaling.

  2. Finite-time Scaling of the surface special transition in a 3D classical Heisenberg model

    cond-mat.stat-mech 2026-07 conditional novelty 6.5 of 10

    Driving from the extraordinary-log surface critical state yields the novel large-rate form Ms^{2} ∝ R^{(1+ u s)/rs} [log(L R^{1/rs})]^{-q} for the surface order parameter.

  3. Finite-time Scaling of the surface special transition in a 3D classical Heisenberg model

    cond-mat.stat-mech 2026-07 conditional novelty 6.0 of 10

    When driven from an extraordinary-log boundary critical state into the 3D Heisenberg surface special transition, the surface order parameter obeys M_s^2 ∝ R^{(1+η_s)/r_s} [log(LR^{1/r_s})]^{-q}, a new log-corrected fi...

Reference graph

Works this paper leans on

115 extracted references · 8 linked inside Pith · cited by 2 Pith papers

  1. [1]

    Heating dynamics To study the temperature-driven heating dynamics of the special transition, we fix the boundary couplingJsc = 1.50243 [92] and choose the initial temperature asT 0 = Tb −2, corresponding to the symmetry-breaking ordered phase. In this setup, thenormalgeneralization of the FTS forms leads to the BFTS form at the boundary with x= 1, M 2 s1(...

  2. [2]

    Cooling dynamics Given the similarity between the special transition and ordinary transition in heating dynamics, it is natural to ask whether the same applies to cooling. For the cooling 11 10−5 10−4 10−3 10−2 10−1 v 101 102 M 2 s1Ld−1 (a1) 3D special cooling ∝ v−[(d−1)−2β1/ν]/r∝ v−[(d−1)−2β1/ν]/r 101 103 105 vLr 10−1 100 M 2 s1L2β1/ν (a2) L = 20 L = 30 ...

  3. [3]

    The temperature is fixed atT b during the driving process

    Decreasing coupling Here, we consider the driven dynamics of the special transition point by linearly decreasingJ s fromJ s0 = Jsc + 1 across the critical valueJ sc. The temperature is fixed atT b during the driving process. In the initial state, the bulk is at the critical point, while the surface is in the ordered phase, sinceT b is below the surface tr...

  4. [4]

    In this case, the equilibrium states for the initial parameters withJ s0 < Jsc andT=T b are critical in both the bulk and the surface

    Increasing coupling Here we study the case for increasing coupling from Js < Jsc. In this case, the equilibrium states for the initial parameters withJ s0 < Jsc andT=T b are critical in both the bulk and the surface. In particular, the latter 10−5 10−4 10−3 10−2 10−1 v 2 4 6 M 2 s1 (a1) 3D special decrease Js ×10−1 ∝ v2β1/νrJ∝ v2β1/νrJ 100 102 104 vLrJ 10...

  5. [5]

    Topology of cosmic domains and strings,

    T. W. B. Kibble, “Topology of cosmic domains and strings,” Journal of Physics A: Mathematical and Gen- eral9, 1387 (1976)

    1976

  6. [6]

    Cosmological experiments in superfluid helium?

    W. H. Zurek, “Cosmological experiments in superfluid helium?” Nature317, 505–508 (1985)

    1985

  7. [7]

    Dynamics of a Quantum Phase Transition,

    W. H. Zurek, U. Dorner, and P. Zoller, “Dynamics of a Quantum Phase Transition,” Phys. Rev. Lett.95, 105701 (2005)

    2005

  8. [8]

    Dynamics of a Quantum Phase Tran- sition: Exact Solution of the Quantum Ising Model,

    J. Dziarmaga, “Dynamics of a Quantum Phase Tran- sition: Exact Solution of the Quantum Ising Model,” Phys. Rev. Lett.95, 245701 (2005)

    2005

Show all 115 references
  1. [9]

    Universal adiabatic dynamics in the vicinity of a quantum critical point,

    A. Polkovnikov, “Universal adiabatic dynamics in the vicinity of a quantum critical point,” Phys. Rev. B72, 161201 (2005)

    2005

  2. [10]

    Topological de- fects as relics of emergent continuous symmetry and Higgs condensation of disorder in ferroelectrics,

    S.-Z. Lin, X. Wang, Y. Kamiya, G.-W. Chern, F. Fan, D. Fan, B. Casas, Y. Liu, V. Kiryukhin, W. H. Zurek, C. D. Batista, and S.-W. Cheong, “Topological de- fects as relics of emergent continuous symmetry and Higgs condensation of disorder in ferroelectrics,” Nature Physics10, 9...

    2014

  3. [11]

    Kibble-Zurek univer- sality in a strongly interacting Fermi superfluid,

    B. Ko, J. W. Park, and Y. Shin, “Kibble-Zurek univer- sality in a strongly interacting Fermi superfluid,” Nature Physics15, 1227–1231 (2019)

    2019

  4. [12]

    Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg sim- ulator,

    A. Keesling, A. Omran, H. Levine, H. Bernien, H. Pich- ler, S. Choi, R. Samajdar, S. Schwartz, P. Silvi, S. Sachdev, P. Zoller, M. Endres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, “Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg sim- ulator,” N...

    2019

  5. [13]

    Kibble-Zurek Mechanism for Dynamical Ordering in a Driven Vortex System,

    S. Maegochi, K. Ienaga, and S. Okuma, “Kibble-Zurek Mechanism for Dynamical Ordering in a Driven Vortex System,” Phys. Rev. Lett.129, 227001 (2022)

    2022

  6. [14]

    Quantum phases of mat- ter on a 256-atom programmable quantum simulator,

    S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuleti´ c, and M. D. Lukin, “Quantum phases of mat- ter on a 256-atom programmable quantum simulator,” Nature595, 227–232 (2021)

    2021

  7. [15]

    Kibble–Zurek mechanism of Ising domains,

    K. Du, X. Fang, C. Won, C. De, F.-T. Huang, W. Xu, H. You, F. J. G´ omez-Ruiz, A. del Campo, and S.-W. Cheong, “Kibble–Zurek mechanism of Ising domains,” Nature Physics19, 1495–1501 (2023)

    2023

  8. [16]

    Observation of generalized Kibble-Zurek mechanism across a first-order quantum phase transition in a spinor condensate,

    L.-Y. Qiu, H.-Y. Liang, Y.-B. Yang, H.-X. Yang, T. Tian, Y. Xu, and L.-M. Duan, “Observation of generalized Kibble-Zurek mechanism across a first-order quantum phase transition in a spinor condensate,” Sci- ence Advances6, eaba7292 (2020)

    2020

  9. [17]

    Quantum optimization of maximum inde- pendent set using Rydberg atom arrays,

    S. Ebadi, A. Keesling, M. Cain, T. T. Wang, H. Levine, D. Bluvstein, G. Semeghini, A. Omran, J.-G. Liu, R. Samajdar, X.-Z. Luo, B. Nash, X. Gao, B. Barak, E. Farhi, S. Sachdev, N. Gemelke, L. Zhou, S. Choi, H. Pichler, S.-T. Wang, M. Greiner, V. Vuleti´ c, and M. D. Lukin, “Qu...

    2022

  10. [18]

    Universal scaling of the dynamic BKT transi- tion in quenched 2D Bose gases,

    S. Sunami, V. P. Singh, D. Garrick, A. Beregi, A. J. Barker, K. Luksch, E. Bentine, L. Mathey, and C. J. Foot, “Universal scaling of the dynamic BKT transi- tion in quenched 2D Bose gases,” Science382, 443–447 (2023)

    2023

  11. [19]

    Probing Critical Behavior of Long- Range Transverse-Field Ising Model through Quantum Kibble-Zurek Mechanism,

    B.-W. Li, Y.-K. Wu, Q.-X. Mei, R. Yao, W.-Q. Lian, M.- L. Cai, Y. Wang, B.-X. Qi, L. Yao, L. He, Z.-C. Zhou, 15 and L.-M. Duan, “Probing Critical Behavior of Long- Range Transverse-Field Ising Model through Quantum Kibble-Zurek Mechanism,” PRX Quantum4, 010302 (2023)

    2023

  12. [20]

    A snapshot of do- main evolution between topological vortex and stripe in ferroelectric hexagonal ErMnO3,

    J. Kang, Z. Gao, C. Guo, W. Zhu, H. Huang, J. Hong, S.-W. Cheong, and X. Wang, “A snapshot of do- main evolution between topological vortex and stripe in ferroelectric hexagonal ErMnO3,” Journal of Applied Physics133, 124102 (2023)

    2023

  13. [21]

    Universal Breakdown of Kibble-Zurek Scaling in Fast Quenches across a Phase Transition,

    H.-B. Zeng, C.-Y. Xia, and A. del Campo, “Universal Breakdown of Kibble-Zurek Scaling in Fast Quenches across a Phase Transition,” Phys. Rev. Lett.130, 060402 (2023)

    2023

  14. [22]

    Defects and their Time Scales in Quantum and Classical Annealing of the Two-Dimensional Ising Model,

    P. Weinberg, N. Xu, and A. W. Sandvik, “Defects and their Time Scales in Quantum and Classical Annealing of the Two-Dimensional Ising Model,” arXiv:2507.09273 (2025)

    2025 arXiv

  15. [23]

    Dynamic Monte Carlo renor- malization group determination of critical exponents with linearly changing temperature,

    F. Zhong and Z. Xu, “Dynamic Monte Carlo renor- malization group determination of critical exponents with linearly changing temperature,” Phys. Rev. B71, 132402 (2005)

    2005

  16. [24]

    Finite-time scaling via linear driving,

    S. Gong, F. Zhong, X. Huang, and S. Fan, “Finite-time scaling via linear driving,” New Journal of Physics12, 043036 (2010)

    2010

  17. [25]

    Probing criticality with linearly varying ex- ternal fields: Renormalization group theory of nonequi- librium critical dynamics under driving,

    F. Zhong, “Probing criticality with linearly varying ex- ternal fields: Renormalization group theory of nonequi- librium critical dynamics under driving,” Phys. Rev. E 73, 047102 (2006)

    2006

  18. [26]

    Determination of the dynamic and static critical exponents of the two-dimensional three-state Potts model using linearly varying temper- ature,

    S. Fan and F. Zhong, “Determination of the dynamic and static critical exponents of the two-dimensional three-state Potts model using linearly varying temper- ature,” Phys. Rev. E76, 041141 (2007)

    2007

  19. [27]

    Finite- time scaling via linear driving: Application to the two- dimensional Potts model,

    X. Huang, S. Gong, F. Zhong, and S. Fan, “Finite- time scaling via linear driving: Application to the two- dimensional Potts model,” Phys. Rev. E81, 041139 (2010)

    2010

  20. [28]

    Nonequilibrium quan- tum criticality in open systems: The dissipation rate as an additional indispensable scaling variable,

    S. Yin, P. Mai, and F. Zhong, “Nonequilibrium quan- tum criticality in open systems: The dissipation rate as an additional indispensable scaling variable,” Phys. Rev. B89, 094108 (2014)

    2014

  21. [29]

    Kibble- Zurek mechanism and finite-time scaling,

    Y. Huang, S. Yin, B. Feng, and F. Zhong, “Kibble- Zurek mechanism and finite-time scaling,” Phys. Rev. B90, 134108 (2014)

    2014

  22. [30]

    Theory of driven nonequilibrium critical phenomena,

    B. Feng, S. Yin, and F. Zhong, “Theory of driven nonequilibrium critical phenomena,” Phys. Rev. B94, 144103 (2016)

    2016

  23. [31]

    Critical dynamics of the two- dimensional random-bond Potts model with nonequi- librium Monte Carlo simulations,

    S. Fan and F. Zhong, “Critical dynamics of the two- dimensional random-bond Potts model with nonequi- librium Monte Carlo simulations,” Phys. Rev. E79, 011122 (2009)

    2009

  24. [32]

    Scaling of the entangle- ment spectrum in driven critical dynamics,

    Q. Hu, S. Yin, and F. Zhong, “Scaling of the entangle- ment spectrum in driven critical dynamics,” Phys. Rev. B91, 184109 (2015)

    2015

  25. [33]

    Scaling theory of entan- glement entropy in confinements near quantum critical points,

    X. Cao, Q. Hu, and F. Zhong, “Scaling theory of entan- glement entropy in confinements near quantum critical points,” Phys. Rev. B98, 245124 (2018)

    2018

  26. [34]

    Equilibration of topological defects near the deconfined quantum multicritical point,

    Y.-R. Shu, S.-K. Jian, A. W. Sandvik, and S. Yin, “Equilibration of topological defects near the deconfined quantum multicritical point,” Nature Communications 16, 3402 (2025)

    2025

  27. [35]

    Kibble-Zurek mechanism for a one-dimensional incarnation of a deconfined quantum critical point,

    R.-Z. Huang and S. Yin, “Kibble-Zurek mechanism for a one-dimensional incarnation of a deconfined quantum critical point,” Phys. Rev. Res.2, 023175 (2020)

    2020

  28. [36]

    Finite-time scaling beyond the Kibble-Zurek prereq- uisite in Dirac systems,

    Z. Zeng, Y.-K. Yu, Z.-X. Li, Z.-X. Li, and S. Yin, “Finite-time scaling beyond the Kibble-Zurek prereq- uisite in Dirac systems,” Nature Communications16, 6181 (2025)

    2025

  29. [37]

    Nonequilib- rium critical dynamics with emergent supersymmetry,

    Z. Zeng, Y.-K. Yu, Z.-X. Li, and S. Yin, “Nonequilib- rium critical dynamics with emergent supersymmetry,” Phys. Rev. B112, L060301 (2025)

    2025

  30. [38]

    Finite-time scal- ing with two characteristic time scales: Driven critical dynamics with emergent symmetry,

    Y.-R. Shu, L.-Y. Yang, and S. Yin, “Finite-time scal- ing with two characteristic time scales: Driven critical dynamics with emergent symmetry,” arXiv:2503.16796 (2025)

    2025 arXiv

  31. [39]

    Relaxation critical dynamics with emergent symmetry,

    Y.-R. Shu, T. Liao, and S. Yin, “Relaxation critical dynamics with emergent symmetry,” Phys. Rev. B110, 134306 (2024)

    2024

  32. [40]

    Driving driven lattice gases to identify their universality classes,

    Y. Li, Z. Zeng, and F. Zhong, “Driving driven lattice gases to identify their universality classes,” Phys. Rev. E100, 020105 (2019)

    2019

  33. [41]

    W. Wang, S. Liu, J. Li, S.-X. Zhang, and S. Yin, “,” arXiv: 2411.06648 (2024)

    2024 arXiv

  34. [42]

    Kibble- Zurek mechanism beyond adiabaticity: Finite-time scal- ing with critical initial slip,

    Y. Huang, S. Yin, Q. Hu, and F. Zhong, “Kibble- Zurek mechanism beyond adiabaticity: Finite-time scal- ing with critical initial slip,” Phys. Rev. B93, 024103 (2016)

    2016

  35. [43]

    Scaling in driven dy- namics starting in the vicinity of a quantum critical point,

    S. Yin, C.-Y. Lo, and P. Chen, “Scaling in driven dy- namics starting in the vicinity of a quantum critical point,” Phys. Rev. B94, 064302 (2016)

    2016

  36. [44]

    Dynamical non-ergodic scaling in continuous finite-order quan- tum phase transitions,

    S. Deng, G. Ortiz, and L. Viola, “Dynamical non-ergodic scaling in continuous finite-order quan- tum phase transitions,” Europhysics Letters84, 67008 (2009)

    2009

  37. [45]

    Kibble-Zurek problem: Universality and the scaling limit,

    A. Chandran, A. Erez, S. S. Gubser, and S. L. Sondhi, “Kibble-Zurek problem: Universality and the scaling limit,” Phys. Rev. B86, 064304 (2012)

    2012

  38. [46]

    Universal nonequilibrium quantum dynamics in imag- inary time,

    C. De Grandi, A. Polkovnikov, and A. W. Sandvik, “Universal nonequilibrium quantum dynamics in imag- inary time,” Phys. Rev. B84, 224303 (2011)

    2011

  39. [47]

    Nonequilibrium Dynamic Critical Scaling of the Quan- tum Ising Chain,

    M. Kolodrubetz, B. K. Clark, and D. A. Huse, “Nonequilibrium Dynamic Critical Scaling of the Quan- tum Ising Chain,” Phys. Rev. Lett.109, 015701 (2012)

    2012

  40. [48]

    Dy- namic scaling at classical phase transitions approached through nonequilibrium quenching,

    C.-W. Liu, A. Polkovnikov, and A. W. Sandvik, “Dy- namic scaling at classical phase transitions approached through nonequilibrium quenching,” Phys. Rev. B89, 054307 (2014)

    2014

  41. [49]

    Space and time renormalization in phase transition dy- namics,

    A. Francuz, J. Dziarmaga, B. Gardas, and W. H. Zurek, “Space and time renormalization in phase transition dy- namics,” Phys. Rev. B93, 075134 (2016)

    2016

  42. [50]

    Domb and J

    C. Domb and J. L. Lebowitz, eds.,Phase Transitions and Critical Phenomena, Vol. 8 (Academic Press, 1983)

    1983

  43. [51]

    Domb and J

    C. Domb and J. L. Lebowitz, eds.,Phase Transi- tions and Critical Phenomena, Vol. 10 (Academic Press, 1986)

    1986

  44. [52]

    The Theory of Boundary Critical Phe- nomena,

    H. W. Diehl, “The Theory of Boundary Critical Phe- nomena,” International Journal of Modern Physics B 11, 3503–3523 (1997)

    1997

  45. [53]

    Critical phenomena at perfect and non- perfect surfaces,

    M. Pleimling, “Critical phenomena at perfect and non- perfect surfaces,” Journal of Physics A: Mathematical and General37, R79 (2004)

    2004

  46. [54]

    Phase Transitions and Static Spin Correlations in Ising Models with Free Sur- faces,

    K. Binder and P. C. Hohenberg, “Phase Transitions and Static Spin Correlations in Ising Models with Free Sur- faces,” Phys. Rev. B6, 3461–3487 (1972)

    1972

  47. [55]

    Crossover Scaling and Critical Behavior at the “Surface-Bulk

    K. Binder and D. P. Landau, “Crossover Scaling and Critical Behavior at the “Surface-Bulk” Multicritical Point,” Phys. Rev. Lett.52, 318–321 (1984). 16

    1984

  48. [56]

    Monte Carlo study of surface phase transitions in the three-dimensional Ising model,

    D. P. Landau and K. Binder, “Monte Carlo study of surface phase transitions in the three-dimensional Ising model,” Phys. Rev. B41, 4633–4645 (1990)

    1990

  49. [57]

    Critical parameters for the d= 3 Ising model in a film geometry,

    C. Ruge and F. Wagner, “Critical parameters for the d= 3 Ising model in a film geometry,” Phys. Rev. B 52, 4209–4216 (1995)

    1995

  50. [58]

    Surface and bulk transitions in three-dimensional O(n) models,

    Y. Deng, H. W. J. Bl¨ ote, and M. P. Nightingale, “Surface and bulk transitions in three-dimensional O(n) models,” Phys. Rev. E72, 016128 (2005)

    2005

  51. [59]

    Bulk and surface phase transitions in the three-dimensionalO(4) spin model,

    Y. Deng, “Bulk and surface phase transitions in the three-dimensionalO(4) spin model,” Phys. Rev. E73, 056116 (2006)

    2006

  52. [60]

    Boundary criticality of the O(N) model ind= 3 critically revisited,

    M. A. Metlitski, “Boundary criticality of the O(N) model ind= 3 critically revisited,” SciPost Phys.12, 131 (2022)

    2022

  53. [61]

    Boundary Critical Behavior of the Three-Dimensional Heisenberg Universality Class,

    F. Parisen Toldin, “Boundary Critical Behavior of the Three-Dimensional Heisenberg Universality Class,” Phys. Rev. Lett.126, 135701 (2021)

    2021

  54. [62]

    Extraordinary-Log Surface Phase Transition in the Three-DimensionalXY Model,

    M. Hu, Y. Deng, and J.-P. Lv, “Extraordinary-Log Surface Phase Transition in the Three-DimensionalXY Model,” Phys. Rev. Lett.127, 120603 (2021)

    2021

  55. [63]

    Boundary bootstrap for the three-dimensional O(N) normal universality class,

    R. Hu and W. Li, “Boundary bootstrap for the three-dimensional O(N) normal universality class,” arXiv:2508.20854 (2025)

    2025 arXiv

  56. [64]

    Boundary Crit- icality of the 3D O(N) Model: From Normal to Extraor- dinary,

    F. Parisen Toldin and M. A. Metlitski, “Boundary Crit- icality of the 3D O(N) Model: From Normal to Extraor- dinary,” Phys. Rev. Lett.128, 215701 (2022)

    2022

  57. [65]

    The extraordinary bound- ary transition in the 3dO(N) model via conformal boot- strap,

    J. Padayasi, A. Krishnan, M. A. Metlitski, I. A. Gruzberg, and M. Meineri, “The extraordinary bound- ary transition in the 3dO(N) model via conformal boot- strap,” SciPost Phys.12, 190 (2022)

    2022

  58. [66]

    Classical-quantum cor- respondence of special and extraordinary-log criticality: Villain’s bridge,

    Y. Sun, J. Lyu, and J.-P. Lv, “Classical-quantum cor- respondence of special and extraordinary-log criticality: Villain’s bridge,” Phys. Rev. B106, 174516 (2022)

    2022

  59. [67]

    Surface criticality of the antiferromagnetic Potts model,

    L.-R. Zhang, C. Ding, Y. Deng, and L. Zhang, “Surface criticality of the antiferromagnetic Potts model,” Phys. Rev. B105, 224415 (2022)

    2022

  60. [68]

    Quantum extraordinary-log uni- versality of boundary critical behavior,

    Y. Sun and J.-P. Lv, “Quantum extraordinary-log uni- versality of boundary critical behavior,” Phys. Rev. B 106, 224502 (2022)

    2022

  61. [69]

    Boundary operator expan- sion and extraordinary phase transition in the tricritical O(N) model,

    X. Sun and S.-K. Jian, “Boundary operator expan- sion and extraordinary phase transition in the tricritical O(N) model,” SciPost Phys.18, 210 (2025)

    2025

  62. [70]

    Unconventional Surface Crit- ical Behavior Induced by a Quantum Phase Transi- tion from the Two-Dimensional Affleck-Kennedy-Lieb- Tasaki Phase to a N´ eel-Ordered Phase,

    L. Zhang and F. Wang, “Unconventional Surface Crit- ical Behavior Induced by a Quantum Phase Transi- tion from the Two-Dimensional Affleck-Kennedy-Lieb- Tasaki Phase to a N´ eel-Ordered Phase,” Phys. Rev. Lett.118, 087201 (2017)

    2017

  63. [71]

    Bipartite entanglement and surface criticality,

    Y. Zhu, Z. Liu, Z. Wang, Y.-C. Wang, and Z. Yan, “Bipartite entanglement and surface criticality,” arXiv:2508.07277 (2025)

    2025

  64. [72]

    Surface phase transitions in a (1 + 1)-dimensionalSU(2) 1 conformal field theory boundary coupled to a (2 + 1)-dimensionalZ 2 bulk,

    Z. Wang, S.-Q. Ning, Z. Liu, J. Rong, Y.-C. Wang, Z. Yan, and W. Guo, “Surface phase transitions in a (1 + 1)-dimensionalSU(2) 1 conformal field theory boundary coupled to a (2 + 1)-dimensionalZ 2 bulk,” Phys. Rev. B110, 115122 (2024)

    2024

  65. [73]

    Measuring the Boundary Gapless State and Crit- icality via Disorder Operator,

    Z. Liu, R.-Z. Huang, Y.-C. Wang, Z. Yan, and D.-X. Yao, “Measuring the Boundary Gapless State and Crit- icality via Disorder Operator,” Phys. Rev. Lett.132, 206502 (2024)

    2024

  66. [74]

    Edge modes of topological Mott insulators and deconfined quantum critical points,

    Y. Liu, T. Sato, D. Hou, Z. Wang, W. Guo, and F. F. Assaad, “Edge modes of topological Mott insulators and deconfined quantum critical points,” arXiv:2508.04455 (2025)

    2025 arXiv

  67. [75]

    Ex- traordinary transition at the edge of a correlated topo- logical insulator,

    F. P. Toldin, F. F. Assaad, and M. A. Metlitski, “Ex- traordinary transition at the edge of a correlated topo- logical insulator,” arXiv:2508.00999 (2025)

    2025 arXiv

  68. [76]

    Boundary criticality of topological quantum phase transitions in two-dimensional systems,

    X.-C. Wu, Y. Xu, H. Geng, C.-M. Jian, and C. Xu, “Boundary criticality of topological quantum phase transitions in two-dimensional systems,” Phys. Rev. B 101, 174406 (2020)

    2020

  69. [77]

    Gapless Symmetry-Protected Topological Order,

    T. Scaffidi, D. E. Parker, and R. Vasseur, “Gapless Symmetry-Protected Topological Order,” Phys. Rev. X 7, 041048 (2017)

    2017

  70. [78]

    Engineering Surface Critical Behavior of (2 + 1)-Dimensional O(3) Quantum Critical Points,

    C. Ding, L. Zhang, and W. Guo, “Engineering Surface Critical Behavior of (2 + 1)-Dimensional O(3) Quantum Critical Points,” Phys. Rev. Lett.120, 235701 (2018)

    2018

  71. [79]

    Nonordi- nary edge criticality of two-dimensional quantum criti- cal magnets,

    L. Weber, F. Parisen Toldin, and S. Wessel, “Nonordi- nary edge criticality of two-dimensional quantum criti- cal magnets,” Phys. Rev. B98, 140403 (2018)

    2018

  72. [80]

    Gapless Topological Phases and Symmetry- Enriched Quantum Criticality,

    R. Verresen, R. Thorngren, N. G. Jones, and F. Poll- mann, “Gapless Topological Phases and Symmetry- Enriched Quantum Criticality,” Phys. Rev. X11, 041059 (2021)

    2021

  73. [81]

    Surface critical behavior of coupled Haldane chains,

    W. Zhu, C. Ding, L. Zhang, and W. Guo, “Surface critical behavior of coupled Haldane chains,” Phys. Rev. B103, 024412 (2021)

    2021

  74. [82]

    Conformal Boundary Conditions of Symmetry-Enriched Quantum Critical Spin Chains,

    X.-J. Yu, R.-Z. Huang, H.-H. Song, L. Xu, C. Ding, and L. Zhang, “Conformal Boundary Conditions of Symmetry-Enriched Quantum Critical Spin Chains,” Phys. Rev. Lett.129, 210601 (2022)

    2022

  75. [83]

    The effects of surfaces on dynamic critical behavior,

    S. Dietrich and H. W. Diehl, “The effects of surfaces on dynamic critical behavior,” Zeitschrift f¨ ur Physik B Condensed Matter51, 343–354 (1983)

    1983

  76. [84]

    Monte Carlo Study of Crit- ical Relaxation near a Surface,

    M. Kikuchi and Y. Okabe, “Monte Carlo Study of Crit- ical Relaxation near a Surface,” Phys. Rev. Lett.55, 1220–1222 (1985)

    1985

  77. [85]

    Universality classes for the dynamic sur- face critical behavior of systems with relaxational dy- namics,

    H. W. Diehl, “Universality classes for the dynamic sur- face critical behavior of systems with relaxational dy- namics,” Phys. Rev. B49, 2846–2860 (1994)

    1994

  78. [86]

    Universal Short-Time Be- havior in Critical Dynamics near Surfaces,

    U. Ritschel and P. Czerner, “Universal Short-Time Be- havior in Critical Dynamics near Surfaces,” Phys. Rev. Lett.75, 3882–3885 (1995)

    1995

  79. [87]

    Nonequilibrium Critical Dy- namics at Surfaces: Cluster Dissolution and Nonalge- braic Correlations,

    M. Pleimling and F. Igl´ oi, “Nonequilibrium Critical Dy- namics at Surfaces: Cluster Dissolution and Nonalge- braic Correlations,” Phys. Rev. Lett.92, 145701 (2004)

    2004

  80. [88]

    Nonequilibrium critical dy- namics in inhomogeneous systems,

    M. Pleimling and F. Igl´ oi, “Nonequilibrium critical dy- namics in inhomogeneous systems,” Phys. Rev. B71, 094424 (2005)

    2005

  81. [89]

    Short-time critical dynamics at perfect and imperfect surfaces,

    S. Z. Lin and B. Zheng, “Short-time critical dynamics at perfect and imperfect surfaces,” Phys. Rev. E78, 011127 (2008)

    2008

  82. [90]

    Monte Carlo computation of correlation times of independent relax- ation modes at criticality,

    M. P. Nightingale and H. W. J. Bl¨ ote, “Monte Carlo computation of correlation times of independent relax- ation modes at criticality,” Phys. Rev. B62, 1089–1101 (2000)

    2000

  83. [91]

    Push- ing the limits of Monte Carlo simulations for the three- dimensional Ising model,

    A. M. Ferrenberg, J. Xu, and D. P. Landau, “Push- ing the limits of Monte Carlo simulations for the three- dimensional Ising model,” Phys. Rev. E97, 043301 (2018)

    2018

  84. [92]

    25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice,

    M. Campostrini, A. Pelissetto, P. Rossi, and E. Vi- cari, “25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice,” Phys. Rev. E65, 066127 (2002)

    2002

  85. [93]

    The lightcone bootstrap and the spectrum of the 3d Ising CFT,

    D. Simmons-Duffin, “The lightcone bootstrap and the spectrum of the 3d Ising CFT,” Journal of High Energy Physics2017, 86 (2017). 17

    2017

  86. [94]

    Dynamic critical exponentzof the three-dimensional Ising universality class: Monte Carlo simulations of the improved Blume-Capel model,

    M. Hasenbusch, “Dynamic critical exponentzof the three-dimensional Ising universality class: Monte Carlo simulations of the improved Blume-Capel model,” Phys. Rev. E101, 022126 (2020)

    2020

  87. [95]

    B. M. McCoy and T. T. Wu,The Two-Dimensional Ising Model(Harvard University Press, Cambridge, MA, 1973)

    1973

  88. [96]

    Monte Carlo study of surface criti- cal phenomena: The special point,

    M. Hasenbusch, “Monte Carlo study of surface criti- cal phenomena: The special point,” Phys. Rev. B84, 134405 (2011)

    2011

  89. [97]

    Thermodynamic Casimir force: A Monte Carlo study of the crossover between the ordi- nary and the normal surface universality class,

    M. Hasenbusch, “Thermodynamic Casimir force: A Monte Carlo study of the crossover between the ordi- nary and the normal surface universality class,” Phys. Rev. B83, 134425 (2011)

    2011

  90. [98]

    Binder and D

    K. Binder and D. W. Heermann,Monte Carlo Simula- tion in Statistical Physics(Springer Berlin, Heidelberg, 2010)

    2010

  91. [99]

    Theory of dy- namic critical phenomena,

    P. C. Hohenberg and B. I. Halperin, “Theory of dy- namic critical phenomena,” Rev. Mod. Phys.49, 435– 479 (1977)

    1977

  92. [100]

    Critical dynamics: a field- theoretical approach,

    R. Folk and G. Moser, “Critical dynamics: a field- theoretical approach,” Journal of Physics A: Mathemat- ical and General39, R207 (2006)

    2006

  93. [101]

    U. C. T¨ auber,Critical Dynamics: A Field Theory Ap- proach to Equilibrium and Non-Equilibrium Scaling Be- havior(Cambridge University Press, 2014)

    2014

  94. [102]

    Direct Observa- tion of the Proliferation of Ferroelectric Loop Domains and Vortex-Antivortex Pairs,

    S. C. Chae, N. Lee, Y. Horibe, M. Tanimura, S. Mori, B. Gao, S. Carr, and S.-W. Cheong, “Direct Observa- tion of the Proliferation of Ferroelectric Loop Domains and Vortex-Antivortex Pairs,” Phys. Rev. Lett.108, 167603 (2012)

    2012

  95. [103]

    Unconventional Continuous Structural Disorder at the Order-Disorder Phase Transition in the Hexago- nal Manganites,

    S. H. Skjærvø, Q. N. Meier, M. Feygenson, N. A. Spaldin, S. J. L. Billinge, E. S. Bozin, and S. M. Sel- bach, “Unconventional Continuous Structural Disorder at the Order-Disorder Phase Transition in the Hexago- nal Manganites,” Phys. Rev. X9, 031001 (2019)

    2019

  96. [104]

    Scaling Behavior and Beyond Equilibrium in the Hexagonal Manganites,

    S. M. Griffin, M. Lilienblum, K. T. Delaney, Y. Kuma- gai, M. Fiebig, and N. A. Spaldin, “Scaling Behavior and Beyond Equilibrium in the Hexagonal Manganites,” Phys. Rev. X2, 041022 (2012)

    2012

  97. [105]

    Global Formation of Topological Defects in the Multiferroic Hexagonal Manganites,

    Q. N. Meier, M. Lilienblum, S. M. Griffin, K. Con- der, E. Pomjakushina, Z. Yan, E. Bourret, D. Meier, F. Lichtenberg, E. K. H. Salje, N. A. Spaldin, M. Fiebig, and A. Cano, “Global Formation of Topological Defects in the Multiferroic Hexagonal Manganites,” Phys. Rev. X7, 0410...

    2017

  98. [106]

    Contin- uous N´ eel-VBS quantum phase transition in non-local one-dimensional systems with SO(3) symmetry,

    C.-M. Jian, Y. Xu, X.-C. Wu, and C. Xu, “Contin- uous N´ eel-VBS quantum phase transition in non-local one-dimensional systems with SO(3) symmetry,” Sci- Post Phys.10, 033 (2021)

    2021

  99. [107]

    Anisotropic perturba- tions in three-dimensional O(N)-symmetric vector mod- els,

    M. Hasenbusch and E. Vicari, “Anisotropic perturba- tions in three-dimensional O(N)-symmetric vector mod- els,” Phys. Rev. B84, 125136 (2011)

    2011

  100. [108]

    Driven Critical Dynamics in Tricitical Point,

    T.-L. Wang, Y.-F. Jiang, and S. Yin, “Driven Critical Dynamics in Tricitical Point,” arXiv:2505.12595 (2025)

    2025

  101. [109]

    Tricritical Kibble-Zurek Scaling in Rydberg Atom Ladders,

    H. Wang, X. Li, and C. Li, “Tricritical Kibble-Zurek Scaling in Rydberg Atom Ladders,” arXiv:2505.12979 (2025)

    2025

  102. [110]

    Scaling corrections in driven critical dynamics: Application to the two- dimensional dimerized quantum Heisenberg model,

    J.-W. Liu, S. Yin, and Y.-R. Shu, “Scaling corrections in driven critical dynamics: Application to the two- dimensional dimerized quantum Heisenberg model,” Chinese Physics B34, 057502 (2025)

    2025

  103. [111]

    Driven dynamics of localization phase transition in the Aubry-Andr´ e model with initial gapless extended states,

    X.-Y. Wang, W.-J. Yu, Y.-M. Sun, and L.-J. Zhai, “Driven dynamics of localization phase transition in the Aubry-Andr´ e model with initial gapless extended states,” arXiv:2509.06358 (2025)

    2025 arXiv

  104. [112]

    Edge Quantum Criticality and Emergent Supersymmetry in Topological Phases,

    Z.-X. Li, Y.-F. Jiang, and H. Yao, “Edge Quantum Criticality and Emergent Supersymmetry in Topological Phases,” Phys. Rev. Lett.119, 107202 (2017)

    2017

  105. [113]

    Emer- gent Space-Time Supersymmetry at the Boundary of a Topological Phase,

    T. Grover, D. N. Sheng, and A. Vishwanath, “Emer- gent Space-Time Supersymmetry at the Boundary of a Topological Phase,” Science344, 280–283 (2014)

    2014

  106. [114]

    Boundary criticality in two-dimensional interacting topological insulators,

    Y. Ge, H. Yao, and S.-K. Jian, “Boundary criticality in two-dimensional interacting topological insulators,” arXiv:2504.12600 (2025)

    2025 arXiv

  107. [115]

    Boundary criticality for the Gross-Neveu-Yukawa models,

    H. Jiang, Y. Ge, and S.-K. Jian, “Boundary criticality for the Gross-Neveu-Yukawa models,” arXiv:2503.13247 (2025)

    2025

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.