pith. sign in

arxiv: 2605.23425 · v1 · pith:CJ3KIHYLnew · submitted 2026-05-22 · ❄️ cond-mat.soft

Nonreciprocal surface tension: anisotropy-induced defect motility and organization

Laya Parkavousi , Suropriya Saha This is my paper

Pith reviewed 2026-05-25 02:55 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords nonreciprocal surface tensiondefect motilityCahn-Hilliard modelKardar-Parisi-Zhang universalitymosaic wavestarget patternsanisotropic fluctuationsphase slip
0
0 comments X

The pith

Nonreciprocal surface tension alone produces cycling target patterns and mosaic waves in conserved scalar fields.

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

The paper establishes that within the Nonreciprocal Cahn-Hilliard framework, interfacial nonreciprocity by itself generates defects that repeatedly form system-spanning target patterns, lose stability, self-destruct, and restart from a chaotic state. When bulk and interfacial terms combine in a specific manner, a mosaic-wave state emerges in which traveling waves stay coherent inside domains separated by lines of motile dislocations that function as phase slips. Large-scale fluctuations in this state are controlled by nonlinear terms that place the dynamics in the anisotropic Kardar-Parisi-Zhang universality class, where the sign of the anisotropy determines the character of the out-of-equilibrium behavior. A reader would care because the result shows how a single interfacial property can organize defects and select universal scaling without extra ingredients.

Core claim

Interfacial nonreciprocity transforms defect dynamics in conserved scalar fields. Nonreciprocal surface tension alone produces intermittently stable defects: system-spanning target patterns form, lose stability, self-destruct, and nucleate again from a defect-chaotic state. When bulk and interfacial contributions interplay in a particular way, the system forms a distinct mosaic-wave state: traveling waves remain coherent within finite domains demarcated by linear arrangements of motile dislocations, which act as lines of phase slip. Mosaic-waves exhibit scale-free fluctuations at length scales much larger than the average wavelength of the traveling patterns. The nonlinearities governing the

What carries the argument

The Goldstone-mode dynamics extracted from the Nonreciprocal Cahn-Hilliard model, whose nonlinear terms belong to the anisotropic Kardar-Parisi-Zhang class.

If this is right

  • Target patterns repeatedly form, destabilize, and reform from chaos when only nonreciprocal surface tension is present.
  • A mosaic-wave state appears when bulk and interfacial contributions satisfy a specific interplay, with motile dislocations acting as phase-slip lines.
  • Scale-free fluctuations occur at scales far larger than the pattern wavelength.
  • The sign of the nonlinear anisotropy selects the character of the out-of-equilibrium dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result implies that nonreciprocity at interfaces could serve as a minimal control knob for defect organization across other conserved active systems.
  • The mosaic state may permit experimental tuning of coherent wave domains by adjusting the relative strength of bulk versus interfacial nonreciprocity.
  • Numerical checks of the scaling exponents under controlled sign changes of the anisotropy term would provide a direct test of the KPZ mapping.

Load-bearing premise

The assumption that the Goldstone-mode dynamics constructed from the Nonreciprocal Cahn-Hilliard model have nonlinearities that exactly match those of the anisotropic KPZ class.

What would settle it

A direct measurement or simulation of the roughness and dynamic exponents in the mosaic-wave state that fails to match the known values for the anisotropic KPZ class with the predicted dependence on the sign of the nonlinear anisotropy.

Figures

Figures reproduced from arXiv: 2605.23425 by Laya Parkavousi, Suropriya Saha.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the periodic rise and fall in the magnitude of J, averaged over the system size and plotted over a long enough time interval such that the system under￾goes 20 or more cycles. Mosaic-waves (MW).— For a given α, and a suffi￾ciently large value of β with the same sign, the sys￾tem evolves into a mosaic structure consisting of domains within which the layered structure is preserved, see [PITH_FULL_IMAG… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Figure showing the value of [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
read the original abstract

We show that interfacial nonreciprocity transforms defect dynamics in conserved scalar fields within the framework of the Nonreciprocal Cahn-Hilliard model. Nonreciprocal surface tension alone produces intermittently stable defects: system-spanning target patterns form, lose stability, self-destruct, and nucleate again from a defect-chaotic state. When bulk and interfacial contributions interplay in a particular way, the system forms a distinct mosaic-wave state: traveling waves remain coherent within finite domains demarcated by linear arrangements of motile dislocations, which act as lines of phase slip. Mosaic-waves exhibit scale-free fluctuations at length scales much larger than the average wavelength of the traveling patterns. To explain the wide range of emergent dynamics, we construct the dynamics of the Goldstone-mode. The nonlinearities governing its large-scale fluctuations belong to the anisotropic Kardar-Parisi-Zhang universality class, with the sign of the nonlinear anisotropy controlling the nature of the out-of-equilibrium dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines the Nonreciprocal Cahn-Hilliard model and shows that nonreciprocal surface tension alone drives intermittently stable defects, producing cycling system-spanning target patterns that form, destabilize, and renucleate from a chaotic state. When bulk and interfacial terms balance appropriately, a mosaic-wave state emerges in which coherent traveling waves are separated by lines of motile dislocations acting as phase slips. The authors construct an effective dynamics for the Goldstone mode whose nonlinearities are asserted to belong to the anisotropic Kardar-Parisi-Zhang universality class, with the sign of the nonlinear anisotropy dictating the character of the out-of-equilibrium fluctuations.

Significance. If the Goldstone-mode reduction is shown to be controlled and the coefficient matching rigorous, the work supplies a concrete mechanism by which interfacial nonreciprocity organizes defects in conserved scalar fields and places the resulting large-scale dynamics in a known universality class. This would link microscopic nonreciprocity to macroscopic pattern selection and fluctuation statistics, offering testable predictions for active or driven soft-matter systems. The attempt to derive an effective anisotropic KPZ description from a microscopic model is a positive feature.

major comments (2)
  1. [Goldstone-mode construction] Goldstone-mode construction (final paragraph of abstract and associated derivation section): the claim that the nonlinearities exactly reproduce the anisotropic KPZ class requires explicit computation of the effective coefficients and demonstration that higher-order or nonreciprocal terms are irrelevant under the chosen scaling; without these steps the universality assignment remains unverified and is load-bearing for the central explanation of defect motility and mosaic-wave fluctuations.
  2. [mosaic-wave state] § on mosaic-wave state: the assertion that dislocations act as lines of phase slip and demarcate coherent domains needs quantitative support (e.g., measured phase-slip rate versus dislocation density or a direct comparison of the derived effective equation against the observed wavelength selection); the current description is qualitative and central to distinguishing this state from the target-pattern regime.
minor comments (2)
  1. Notation for the nonreciprocal surface tension term should be introduced with an explicit equation number at first use to allow readers to trace its contribution through the subsequent reduction.
  2. Figure captions for the defect trajectories and mosaic patterns should include the parameter values (e.g., nonreciprocity strength, bulk vs. interfacial ratio) used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The two major comments identify areas where the presentation of the Goldstone-mode reduction and the characterization of the mosaic-wave state can be strengthened. We address each point below and will revise the manuscript to incorporate the requested clarifications and quantitative support.

read point-by-point responses

  1. Referee: [Goldstone-mode construction] Goldstone-mode construction (final paragraph of abstract and associated derivation section): the claim that the nonlinearities exactly reproduce the anisotropic KPZ class requires explicit computation of the effective coefficients and demonstration that higher-order or nonreciprocal terms are irrelevant under the chosen scaling; without these steps the universality assignment remains unverified and is load-bearing for the central explanation of defect motility and mosaic-wave fluctuations.

    Authors: We agree that the universality assignment is central and benefits from more explicit verification. The derivation section already performs the projection onto the Goldstone mode and obtains the leading nonlinear terms that match the anisotropic KPZ equation, with the anisotropy coefficient fixed by the nonreciprocity parameter. A brief power-counting argument is given to indicate irrelevance of higher-order terms under the anisotropic scaling. Nevertheless, to make the coefficient matching fully transparent, we will add an appendix containing the explicit perturbative expansion, the resulting coefficient values, and a short renormalization-group sketch confirming the irrelevance of additional nonreciprocal contributions. This revision will be made. revision: yes

  2. Referee: [mosaic-wave state] § on mosaic-wave state: the assertion that dislocations act as lines of phase slip and demarcate coherent domains needs quantitative support (e.g., measured phase-slip rate versus dislocation density or a direct comparison of the derived effective equation against the observed wavelength selection); the current description is qualitative and central to distinguishing this state from the target-pattern regime.

    Authors: We accept that the current description of the mosaic-wave state remains largely qualitative. In the revised manuscript we will include direct measurements from the simulations: the phase-slip rate extracted from the time series of the order-parameter field plotted against local dislocation density, together with a comparison of the wavelength selected by the effective Goldstone-mode equation against the dominant wavelength measured in the mosaic-wave regime. These quantitative diagnostics will be added to the section describing the mosaic-wave state and will help distinguish it from the target-pattern regime. revision: yes

Circularity Check

0 steps flagged

No circularity: Goldstone-mode construction presented as independent derivation from Nonreciprocal Cahn-Hilliard model

full rationale

The abstract and provided text state that the authors construct the Goldstone-mode dynamics from the Nonreciprocal Cahn-Hilliard model and then identify its nonlinearities as belonging to the anisotropic KPZ class. No quoted equations, self-citations, or reductions show the KPZ assignment as forced by definition, fitted parameters renamed as predictions, or load-bearing self-citation chains. The derivation chain is presented as explanatory and self-contained against the model equations; no specific reduction to inputs by construction is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides insufficient detail to identify specific free parameters or invented entities; the central addition is nonreciprocal surface tension within an established model framework.

axioms (1)
  • domain assumption The Nonreciprocal Cahn-Hilliard model is the appropriate framework for describing conserved scalar fields with nonreciprocity.
    The entire analysis is conducted within this model as stated in the abstract.

pith-pipeline@v0.9.0 · 5692 in / 1462 out tokens · 38975 ms · 2026-05-25T02:55:34.227098+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages · 3 internal anchors

  1. [1]

    (3) Γ = 1 represents an isotropic system, implications of which were discussed in [8], while deviations from unity show that phase fluctuations at large length-scales are anisotropic. The degree of anisotropy is determined jointly byβ, andq 2 0, the latter of which is a dynamical quantity, as opposed to the former, which is a fixed parameter.q 0 exhibits ...

    work page

  2. [2]

    Ramaswamy, The mechanics and statistics of active matter, Annu

    S. Ramaswamy, The mechanics and statistics of active matter, Annu. Rev. Condens. Matter Phys.1, 323 (2010)

    work page 2010

  3. [3]

    Soto and R

    R. Soto and R. Golestanian, Self-assembly of catalytically active colloidal molecules: Tailoring activity through sur- face chemistry, Phys. Rev. Lett.112, 068301 (2014)

    work page 2014

  4. [4]

    S. Saha, S. Ramaswamy, and R. Golestanian, Pairing, waltzing and scattering of chemotactic active colloids, New Journal of Physics21, 063006 (2019)

    work page 2019

  5. [5]

    Shankar, A

    S. Shankar, A. Souslov, M. J. Bowick, M. C. Marchetti, and V. Vitelli, Topological active matter, Nature Reviews Physics4, 380 (2022)

    work page 2022

  6. [6]

    loving hate, brawling love

    R. Golestanian, Non-reciprocal active-matter: A tale of “loving hate, brawling love” across the scales, Euro- physics News55, 12 (2024)

    work page 2024

  7. [7]

    Fruchart and V

    M. Fruchart and V. Vitelli, Nonreciprocal many-body physics, arXiv preprint arXiv:2602.11111 (2026)

    work page arXiv 2026

  8. [8]

    S. Saha, J. Agudo-Canalejo, and R. Golestanian, Scalar Active Mixtures: The Nonreciprocal Cahn-Hilliard Model, Phys. Rev. X10, 041009 (2020)

    work page 2020

  9. [9]

    Pisegna, S

    G. Pisegna, S. Saha, and R. Golestanian, Emergent polar order in nonpolar mixtures with nonreciprocal interac- tions, Proceedings of the National Academy of Sciences 121, e2407705121 (2024)

    work page 2024

  10. [10]

    L. P. Dadhichi, J. Kethapelli, R. Chajwa, S. Ramaswamy, and A. Maitra, Nonmutual torques and the unimportance of motility for long-range order in two-dimensional flocks, Phys. Rev. E101, 052601 (2020)

    work page 2020

  11. [11]

    S. A. M. Loos, S. H. L. Klapp, and T. Martynec, Long-range order and directional defect propagation in the nonreciprocal XY model with vision cone inter- actions, Physical Review Letters130, 198301 (2023), arXiv:2206.10519 [cond-mat.stat-mech]

    work page arXiv 2023

  12. [12]

    Dinelli, J

    A. Dinelli, J. O’Byrne, A. Curatolo, Y. Zhao, P. Sollich, and J. Tailleur, Non-reciprocity across scales in active mixtures, Nat. Commun.14, 7035 (2023)

    work page 2023

  13. [13]

    Y. Du, Y. Cao, W. Liu, and Y. Li, Collective behavior of the non-reciprocal vision cone XY lattice-gas model, Newton2, 100291 (2026)

    work page 2026

  14. [14]

    Bandini, D

    G. Bandini, D. Venturelli, S. A. M. Loos, A. Jeli´ c, and 6 A. Gambassi, The XY model with vision cone: Non- reciprocal vs. reciprocal interactions, Journal of Statis- tical Mechanics: Theory and Experiment2025, 053205 (2025), arXiv:2412.19297 [cond-mat.stat-mech]

    work page arXiv 2025

  15. [15]

    Ouazan-Reboul, J

    V. Ouazan-Reboul, J. Agudo-Canalejo, and R. Golesta- nian, Self-organization of primitive metabolic cycles due to non-reciprocal interactions, Nature Communications 14, 4496 (2023)

    work page 2023

  16. [16]

    Y. Duan, J. Agudo-Canalejo, R. Golestanian, and B. Mahault, Dynamical pattern formation without self- attraction in quorum-sensing active matter: The inter- play between nonreciprocity and motility, Phys. Rev. Lett.131, 148301 (2023)

    work page 2023

  17. [17]

    Brauns and M

    F. Brauns and M. C. Marchetti, Nonreciprocal pattern formation of conserved fields, Phys. Rev. X14, 021014 (2024)

    work page 2024

  18. [18]

    J. Chen, X. Lei, Y. Xiang, M. Duan, X. Peng, and H. Zhang, Emergent chirality and hyperuniformity in an active mixture with nonreciprocal interactions, Physical Review Letters132, 118301 (2024)

    work page 2024

  19. [19]

    Guillet, A

    S. Guillet, A. Poncet, M. Le Blay, W. T. Irvine, V. Vitelli, and D. Bartolo, Melting of nonreciprocal solids: How dislocations propel and fission in flowing crystals, Pro- ceedings of the National Academy of Sciences122, e2412993122 (2025)

    work page 2025

  20. [20]

    T. H. Tan, A. Mietke, J. Li, Y. Chen, H. Higinbotham, P. J. Foster, S. Gokhale, J. Dunkel, and N. Fakhri, Odd dynamics of living chiral crystals, Nature607, 287 (2022)

    work page 2022

  21. [21]

    Y.-C. Chao, S. Gokhale, L. Lin, A. Hastewell, A. Bacanu, Y. Chen, J. Li, J. Liu, H. Lee, J. Dunkel, et al., Selective excitation of work-generating cycles in non-reciprocal liv- ing solids, Nature Physics , 1 (2026)

    work page 2026

  22. [22]

    R. Kant, R. K. Gupta, H. Soni, A. K. Sood, and S. Ra- maswamy, Bulk condensation by an active interface, Phys. Rev. Lett.133, 208301 (2024)

    work page 2024

  23. [23]

    Berezney, S

    J. Berezney, S. Ray, I. Kolvin, F. Brauns, S. Chen, M. Bowick, S. Fraden, V. Vitelli, and Z. Dogic, Active assembly and non-reciprocal dynamics of elastic mem- branes, Nature Physics , 1 (2026)

    work page 2026

  24. [24]

    F. A. Lavergne, H. Wendehenne, T. B¨ auerle, and C. Bechinger, Group formation and cohesion of active particles with visual perception–dependent motility, Sci- ence364, 70 (2019)

    work page 2019

  25. [25]

    Fruchart, R

    M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature592, 363 (2021)

    work page 2021

  26. [26]

    Z. You, A. Baskaran, and M. C. Marchetti, Nonreciproc- ity as a generic route to traveling states, PNAS117, 19767 (2020)

    work page 2020

  27. [27]

    Frohoff-H¨ ulsmann and U

    T. Frohoff-H¨ ulsmann and U. Thiele, Local- ized states in coupled Cahn–Hilliard equations, IMA Journal of Applied Mathematics86, 924 (2021), https://academic.oup.com/imamat/article- pdf/86/5/924/40744935/hxab026.pdf

    work page 2021

  28. [28]

    K. Ding, C. Fang, and G. Ma, Non-hermitian topol- ogy and exceptional-point geometries, Nature Reviews Physics4, 745 (2022)

    work page 2022

  29. [29]

    Saha and R

    S. Saha and R. Golestanian, Effervescence in a binary mixture with nonlinear non-reciprocal interactions, Na- ture Communications16, 7310 (2025)

    work page 2025

  30. [30]

    Compositional disorder in a multicomponent non-reciprocal mixture: stability and patterns

    L. Parkavousi and S. Saha, Compositional disorder in a multicomponent non-reciprocal mixture: stability and patterns (2025), arXiv:2505.02532 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  31. [31]

    Tucci, R

    G. Tucci, R. Golestanian, and S. Saha, Nonreciprocal col- lective dynamics in a mixture of phoretic janus colloids, arXiv preprint arXiv:2402.09279 (2024)

    work page arXiv 2024

  32. [32]

    Sahoo, R

    B. Sahoo, R. Mandal, and P. Sollich, Nonreciprocal model b: The role of mobilities and nonreciprocal in- terfacial forces (2025), arXiv:2508.02814 [cond-mat.soft]

    work page arXiv 2025

  33. [33]

    Frohoff-H¨ ulsmann and U

    T. Frohoff-H¨ ulsmann and U. Thiele, Nonreciprocal cahn- hilliard model emerges as a universal amplitude equation, Phys. Rev. Lett.131, 107201 (2023)

    work page 2023

  34. [34]

    Brauns, J

    F. Brauns, J. Halatek, and E. Frey, Phase-space geome- try of mass-conserving reaction-diffusion dynamics, Phys. Rev. X10, 041036 (2020)

    work page 2020

  35. [35]

    Popli, A

    P. Popli, A. Maitra, and S. Ramaswamy, Ordering and defect cloaking in nonreciprocal lattice XY models, Phys- ical Review Letters135, 088303 (2025)

    work page 2025

  36. [36]

    Rouzaire and D

    Y. Rouzaire and D. Levis, Dynamics of topological de- fects in the noisy Kuramoto model in two dimensions, Frontiers in Physics10, 976515 (2022)

    work page 2022

  37. [37]

    Rouzaire, D

    Y. Rouzaire, D. J. G. Pearce, I. Pagonabarraga, and D. Levis, Nonreciprocal interactions reshape topological defect annihilation, Physical Review Letters134, 167101 (2025), arXiv:2401.12637 [cond-mat.soft]

    work page arXiv 2025

  38. [38]

    C. Myin, S. Saha, and B. Mahault, Breakdown of emer- gent chiral order and defect chaos in nonreciprocal flocks (2026), arXiv:2605.05448 [cond-mat.soft]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  39. [39]

    C.-U. Woo, J. D. Noh, and H. Rieger, Extensive spatio-temporal chaos in non-reciprocal flocking (2026), arXiv:2604.06462 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  40. [40]

    D. E. Wolf, Kinetic roughening of vicinal surfaces, Phys- ical Review Letters67, 1783 (1991)

    work page 1991

  41. [41]

    L. M. Sieberer, G. Wachtel, E. Altman, and S. Diehl, Lattice duality for the compact kardar-parisi-zhang equa- tion, Physical Review B94, 104521 (2016)

    work page 2016

  42. [42]

    Wachtel, L

    G. Wachtel, L. M. Sieberer, S. Diehl, and E. Altman, Electrodynamic duality and vortex unbinding in driven- dissipative condensates, Physical Review B94, 104520 (2016)

    work page 2016

  43. [43]

    Rana and R

    N. Rana and R. Golestanian, Defect solutions of the non- reciprocal cahn-hilliard model: Spirals and targets, Phys- ical Review Letters133, 078301 (2024)

    work page 2024

  44. [44]

    Rana and R

    N. Rana and R. Golestanian, Defect interactions in the non-reciprocal cahn–hilliard model, New Journal of Physics26, 123008 (2024)

    work page 2024

  45. [45]

    Daviet, C

    R. Daviet, C. P. Zelle, A. Asadollahi, and S. Diehl, Kardar-parisi-zhang scaling in time-crystalline matter, Physical Review Letters135, 047101 (2025)

    work page 2025

  46. [46]

    Chen and J

    L. Chen and J. Toner, Universality for moving stripes: A hydrodynamic theory of polar active smectics, Phys. Rev. Lett.111, 088701 (2013)

    work page 2013

  47. [47]

    Vercesi, Q

    F. Vercesi, Q. Fontaine, S. Ravets, J. Bloch, M. Richard, L. Canet, and A. Minguzzi, Phase diagram of one- dimensional driven-dissipative exciton-polariton conden- sates, Physical Review Research5, 043062 (2023)

    work page 2023

  48. [48]

    Maitra, M

    A. Maitra, M. Lenz, and R. Voituriez, Chiral active hex- atics: Giant number fluctuations, waves, and destruction of order, Physical Review Letters125, 238005 (2020)

    work page 2020

  49. [49]

    Maitra, Activity unmasks chirality in liquid- crystalline matter, Annual Review of Condensed Matter Physics16, 275 (2025)

    A. Maitra, Activity unmasks chirality in liquid- crystalline matter, Annual Review of Condensed Matter Physics16, 275 (2025)

    work page 2025

  50. [50]

    Kardar, G

    M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic scaling of growing interfaces, Physical Review Letters56, 889 (1986)

    work page 1986

  51. [51]

    Widmann, S

    S. Widmann, S. Dam, J. D¨ ureth, C. G. Mayer, R. Daviet, C. P. Zelle, D. Laibacher, M. Emmerling, M. Kamp, 7 S. Diehl, et al., Observation of kardar-parisi-zhang uni- versal scaling in two dimensions, Science392, 221 (2026)

    work page 2026

  52. [52]

    L. M. Sieberer and E. Altman, Topological defects in anisotropic driven open systems, Phys. Rev. Lett.121, 085704 (2018)

    work page 2018

  53. [53]

    Altman, L

    E. Altman, L. M. Sieberer, L. Chen, S. Diehl, and J. Toner, Two-dimensional superfluidity of exciton po- laritons requires strong anisotropy, Phys. Rev. X5, 011017 (2015)

    work page 2015

  54. [54]

    we also include further details of the states and information on the supplemental movies

    See supplemental material contains details of the stability analysis of the equations of motion for vanishing average number densities of the two species, and of the numerical procedure followed to solve them including the nonlin- earities. we also include further details of the states and information on the supplemental movies

    work page

  55. [55]

    M. C. Cross and Y. Tu, Defect-mediated turbulence in finite systems, Physical Review Letters75, 834 (1995)

    work page 1995

  56. [56]

    U. C. T¨ auber and E. Frey, Universality classes in the anisotropic kardar-parisi-zhang model, EPL (Euro- physics Letters)59, 655 (2002)

    work page 2002

  57. [57]

    Zamora, N

    A. Zamora, N. Lad, and M. H. Szymanska, Vortex dy- namics in a compact kardar-parisi-zhang system, Phys. Rev. Lett.125, 265701 (2020)

    work page 2020

  58. [58]

    V. L. Berezinskii, Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group i. classical systems, Soviet Physics JETP32, 493 (1971)

    work page 1971

  59. [59]

    J. M. Kosterlitz, The critical properties of the two- dimensional xy model, Journal of Physics C: Solid State Physics7, 1046 (1974)

    work page 1974

  60. [60]

    J. M. Kosterlitz and D. J. Thouless, Ordering, metasta- bility and phase transitions in two-dimensional systems, Journal of Physics C: Solid State Physics6, 1181 (1973)

    work page 1973

  61. [61]

    J. M. Kosterlitz, Kosterlitz–thouless physics: A review of key issues, Reports on Progress in Physics79, 026001 (2016)

    work page 2016

  62. [62]

    Chatterjee, N

    R. Chatterjee, N. Rana, R. A. Simha, P. Perlekar, and S. Ramaswamy, Inertia drives a flocking phase transition in viscous active fluids, Physical Review X11, 031063 (2021)

    work page 2021

  63. [63]

    J. Denk, S. Kretschmer, J. Halatek, C. Hartl, P. Schwille, and E. Frey, MinE conformational switching confers ro- bustness on self-organized Min protein patterns, Proceed- ings of the National Academy of Sciences of the United States of America115, 4553 (2018)

    work page 2018

  64. [64]

    Wettmann, M

    L. Wettmann, M. Bonny, and K. Kruse, Effects of ge- ometry and topography on Min-protein dynamics, PLOS ONE13, e0203050 (2018)

    work page 2018

  65. [65]

    T. H. Tan, J. Liu, P. W. Miller, M. Tekant, J. Dunkel, and N. Fakhri, Topological turbulence in the membrane of a living cell, Nature Physics16, 657 (2020). End Matter APPENDIX A We consider the NRCH equation for the complex field ϕ(r, t) =ϕ 1 +iϕ 2 in Eq. (1), incorporating diffusive, nonlinear, and higher-order gradient terms with non- reciprocal parame...

    work page 2020

  66. [66]

    At linear order, the amplitude fluctuationδρis stable and rapidly relaxes

    Small perturbations are intro- duced viaρ=ρ 0 +ϵ δρandθ=q 0x−Ωt+ϵ δθ, and the dynamics are expanded order by order inϵ. At linear order, the amplitude fluctuationδρis stable and rapidly relaxes. It becomes slaved to gradients of the phase field, yielding the quasi-static relation δρ≈ − q0 ρ0 ∂xδθ+c∇ 2δθ.(4) This reduces the problem to an effective phase-o...

    work page

  67. [67]

    (10) In summary, the NRCH model reduces to an anisotropic KPZ equation for the phase field. The non-reciprocal parametersαandβcontrol drift, anisotropic diffusion, and nonlinear growth, establishing a direct connection between microscopic nonreciprocity and macroscopic sur- face dynamics. APPENDIX B 8 °0.50.00.5qx °0.50°0.250.000.250.50qy (a) °0.50.00.5qx...

    work page