pith. sign in

arxiv: 2511.10115 · v3 · pith:2AYCWK26new · submitted 2025-11-13 · ❄️ cond-mat.quant-gas · nlin.PS· nucl-th· physics.atm-clus

Generalized Gross-Pitaevskii Equation for 2D Bosons with Attractive Interactions

Micha{\l} Suchorowski , Fabian Brauneis , Hans-Werner Hammer , Micha{\l} Tomza , Artem G. Volosniev This is my paper

Pith reviewed 2026-05-17 22:35 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas nlin.PSnucl-thphysics.atm-clus
keywords Gross-Pitaevskii equationquantum anomalytwo-dimensional Bose gasquantum dropletsattractive interactionsbreathing modesvortex statesquench dynamics
0
0 comments X

The pith

A generalized Gross-Pitaevskii equation with logarithmic density dependence captures the quantum anomaly in two-dimensional attractive Bose systems.

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

The paper introduces a modified mean-field equation for two-dimensional bosons with attractive interactions. Its central innovation replaces the usual constant coupling with one that depends logarithmically on local density. This modification explicitly breaks the scale invariance of the standard Gross-Pitaevskii equation and thereby incorporates the effects of the quantum anomaly. The resulting framework is then applied to compute universal bound states in free space, breathing modes after quenches in traps, and the existence of excited states that include vortices.

Core claim

By replacing the constant interaction strength in the Gross-Pitaevskii equation with a logarithmically density-dependent coupling, the authors obtain a nonlinear equation whose solutions automatically encode the quantum anomaly of two-dimensional attractive bosons, allowing direct calculation of universal droplet states, breathing frequencies, and vortex-carrying excitations without additional beyond-mean-field terms.

What carries the argument

The logarithmic density dependence of the coupling constant, which encodes the breaking of scale invariance due to the quantum anomaly.

If this is right

  • Universal bound states known as quantum droplets become calculable in free space.
  • Breathing modes and quench dynamics in trapped systems can be analyzed directly.
  • Universal excited states, including those with vortices, are predicted to exist.
  • The equation supplies a foundation for studying both static and non-equilibrium properties of finite 2D attractive systems.

Where Pith is reading between the lines

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

  • The model may reduce the computational cost of simulating 2D Bose droplets compared with full many-body methods.
  • Vortex-carrying excited states could be more readily observed than ground-state droplets in current ultracold-atom experiments.
  • The same logarithmic coupling might be tested in related 2D systems that exhibit scale anomalies, such as anyonic gases.

Load-bearing premise

The specific logarithmic form of the density-dependent coupling accurately encodes the quantum anomaly without additional beyond-mean-field corrections or higher-order terms.

What would settle it

Measure the breathing-mode frequencies of a quenched, trapped two-dimensional attractive Bose gas and check whether they match the parameter-free predictions of the generalized equation.

Figures

Figures reproduced from arXiv: 2511.10115 by Artem G. Volosniev, Fabian Brauneis, Hans-Werner Hammer, Micha{\l} Suchorowski, Micha{\l} Tomza.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Energy, ln [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The density profile in the transition between the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Breathing mode frequencies Ω as a function of interac [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We introduce a generalized Gross-Pitaevskii equation that provides a nonlinear framework for studying two-dimensional (2D) attractive Bose systems. Its defining feature is the logarithmic density dependence of the coupling constant, which breaks the scale invariance inherent in the standard mean-field equations. This framework allows straightforward calculations of the system properties arising from the quantum anomaly. As a first illustration, we study universal bound states in free space, commonly referred to as quantum droplets. Then, we analyze breathing modes and quench dynamics in trapped systems, paving the way for a systematic exploration of non-equilibrium phenomena in 2D attractive Bose systems. Finally, we predict the existence of universal excited states, including vortex configurations, which may be more accessible to experimental investigation than the ground state. Our results provide a robust theoretical foundation for studying both static and dynamical properties of finite systems, and offer guidance for the design of future experiments.

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 paper introduces a generalized Gross-Pitaevskii equation for 2D bosons with attractive interactions whose coupling constant acquires a logarithmic dependence on local density. This term is introduced to break the scale invariance of the standard mean-field theory and thereby incorporate the 2D quantum anomaly. The framework is then used to compute universal bound states (quantum droplets) in free space, breathing modes and quench dynamics in trapped geometries, and the existence of universal excited states including vortices.

Significance. If the specific logarithmic form is shown to be the leading and sufficient renormalization effect, the approach supplies a computationally tractable model for both static and dynamical properties of finite 2D attractive Bose systems and could guide experimental searches for quantum droplets and vortex states.

major comments (2)
  1. [§2] §2 (The Model): The manuscript states that the logarithmic density dependence of g(n) fully encodes the quantum anomaly, yet provides no explicit derivation or controlled expansion demonstrating that higher-order density-dependent corrections remain negligible at the densities and length scales relevant to the droplet and vortex calculations; this assumption is load-bearing for all universality and dynamical claims.
  2. [§4] §4 (Quench dynamics and breathing modes): The reported frequencies and stability windows are obtained from the pure logarithmic g(n); if residual beyond-mean-field terms appear at the trap densities used in the numerics, the predicted oscillation periods and post-quench evolution would be quantitatively altered.
minor comments (2)
  1. [§2] Notation for the logarithmic term is introduced without an explicit reference to the 2D scattering length or ultraviolet cutoff; adding this relation would clarify the connection to microscopic parameters.
  2. [Figs. 2-4] Figure captions for the droplet density profiles and vortex states should state the precise value of the logarithmic coefficient and the numerical grid parameters employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major comment below and indicate the revisions we will make to strengthen the presentation of the model and its range of validity.

read point-by-point responses

  1. Referee: [§2] §2 (The Model): The manuscript states that the logarithmic density dependence of g(n) fully encodes the quantum anomaly, yet provides no explicit derivation or controlled expansion demonstrating that higher-order density-dependent corrections remain negligible at the densities and length scales relevant to the droplet and vortex calculations; this assumption is load-bearing for all universality and dynamical claims.

    Authors: The logarithmic density dependence is introduced to capture the leading renormalization effect arising from the 2D quantum anomaly, as obtained from the two-body scattering problem and the running of the coupling constant with density. We agree that the manuscript would benefit from a more explicit discussion of this motivation and an estimate of the regime where higher-order corrections can be neglected. We will add a short subsection in §2 that recalls the renormalization-group origin of the log term and provides a scaling argument showing that, at the low densities characteristic of the quantum droplets and vortices considered here, the leading anomalous contribution dominates over sub-leading density-dependent terms. revision: yes

  2. Referee: [§4] §4 (Quench dynamics and breathing modes): The reported frequencies and stability windows are obtained from the pure logarithmic g(n); if residual beyond-mean-field terms appear at the trap densities used in the numerics, the predicted oscillation periods and post-quench evolution would be quantitatively altered.

    Authors: The numerical results in §4 are obtained within the generalized Gross-Pitaevskii framework that retains only the logarithmic correction. We acknowledge that additional beyond-mean-field contributions, if present at the densities realized in the trapped geometries, could shift the quantitative values of the breathing frequencies and the precise boundaries of the stability windows. In the revised manuscript we will add a paragraph in §4 that (i) reiterates the assumptions underlying the model, (ii) estimates the relative size of neglected terms using the same scaling argument introduced in §2, and (iii) states that the reported frequencies and qualitative dynamical features are expected to be robust within the validity range of the leading-log approximation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation chain.

full rationale

The paper introduces a generalized Gross-Pitaevskii equation featuring a logarithmic density dependence in the coupling constant as an effective framework to capture the quantum anomaly and break scale invariance in 2D attractive Bose systems. Subsequent calculations of universal bound states (quantum droplets), breathing modes, quench dynamics, and excited vortex states are performed directly within this model. No load-bearing step reduces by construction to a self-definition, a fitted parameter renamed as a prediction, or a self-citation chain whose validity depends on the present work; the logarithmic form is positioned as an input encoding known anomaly effects, with the paper's contributions consisting of applications and predictions that remain independently testable against external benchmarks or experiments. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Insufficient information in the provided abstract to enumerate free parameters, axioms, or invented entities; the logarithmic coupling is introduced without derivation details.

pith-pipeline@v0.9.0 · 5489 in / 1020 out tokens · 28828 ms · 2026-05-17T22:35:53.327923+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

60 extracted references · 60 canonical work pages · 2 internal anchors

  1. [1]

    R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett.13, 479–482 (1964)

    work page 1964

  2. [2]

    V. E. Zakharov, Sov. Phys. JETP35, 908 (1972)

    work page 1972

  3. [3]

    V. E. Zakharov and E. A. Kuznetsov, Zh. Eksp. Teor. Fiz.91, 1310 (1986)

    work page 1986

  4. [4]

    Pitaevskii, Phys

    L. Pitaevskii, Phys. Lett. A221, 14–18 (1996)

    work page 1996

  5. [5]

    L. P. Pitaevskii and A. Rosch, Phys. Rev. A55, R853–R856 (1997)

    work page 1997

  6. [6]

    Olshanii, H

    M. Olshanii, H. Perrin, and V. Lorent, Phys. Rev. Lett. 105, 095302 (2010)

    work page 2010

  7. [7]

    Hammer and D

    H.-W. Hammer and D. T. Son, Phys. Rev. Lett.93, 250408 (2004)

    work page 2004

  8. [8]

    D. S. Petrov, SciPost Phys.19, 037 (2025)

    work page 2025

  9. [9]

    Tononi, G

    A. Tononi, G. E. Astrakharchik, and D. S. Petrov, AVS Quantum Sci.6, 023201 (2024)

    work page 2024

  10. [10]

    Brauneis, A

    F. Brauneis, A. G. Volosniev, and H.-W. Hammer, Phys. Rev. A111, 063308 (2025)

    work page 2025

  11. [11]

    D. J. Gross and F. Wilczek, Phys. Rev. Lett.30, 1343 (1973)

    work page 1973

  12. [12]

    H. D. Politzer, Phys. Rev. Lett.30, 1346 (1973)

    work page 1973

  13. [13]

    J. C. Collins and M. J. Perry, Phys. Rev. Lett.34, 1353 (1975)

    work page 1975

  14. [14]

    Kurkela, P

    A. Kurkela, P. Romatschke, and A. Vuorinen, Phys. Rev. D81, 105021 (2010)

    work page 2010

  15. [15]

    Iida and G

    K. Iida and G. Baym, Phys. Rev. D66, 014015 (2002)

    work page 2002

  16. [16]

    M. Eto, M. Nitta, and N. Yamamoto, Phys. Rev. Lett. 104, 161601 (2010)

    work page 2010

  17. [17]

    Schick, Phys

    M. Schick, Phys. Rev. A3, 1067–1073 (1971)

    work page 1971

  18. [18]

    S. I. Shevchenko, Sov. J. Low Temp. Phys.18, 223–230 (1992)

    work page 1992

  19. [19]

    E. H. Lieb, R. Seiringer, and J. Yngvason, Commun. Math. Phys.224, 17 (2001)

    work page 2001

  20. [20]

    E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi, Phys. Rev. Lett.85, 1146–1149 (2000)

    work page 2000

  21. [21]

    2(b), (S5) derivative of the energy presented in Fig

    See Supplemental Material for (S1) description of our choice of the parameterα, (S2) additional information about the flow-equations method, (S3) description of the numerical method used to solve the extended GPE, (S4) additional information about variational data presented Fig. 2(b), (S5) derivative of the energy presented in Fig. 2(b) (S6) detailed deri...

    work page

  22. [22]

    Levinsen and M

    J. Levinsen and M. M. Parish, Strongly interacting two- dimensional Fermi gases, inAnnual Review of Cold Atoms and Molecules(WORLD SCIENTIFIC, 2015) p. 1–75

    work page 2015

  23. [23]

    For simplicity, we omit the paren- theses in the numerator and write it simply as lnB 2/N in what follows

    (lnB 2)/Nis a parameter that we shall use to quantify the interaction strength. For simplicity, we omit the paren- theses in the numerator and write it simply as lnB 2/N in what follows

    work page

  24. [24]

    However, in finite systems and time-dependent scenarios, it is essential for preserving energy conservation

    It may be tempting to disregard theg 2 term due to its typically minor energy contribution, just as it is typi- cally done for repulsive systems [42]. However, in finite systems and time-dependent scenarios, it is essential for preserving energy conservation

    work page

  25. [25]

    Bazak and D

    B. Bazak and D. S. Petrov, New J. Phys.20, 023045 (2018)

    work page 2018

  26. [26]

    Bakkali-Hassani and J

    B. Bakkali-Hassani and J. Dalibard, inProceedings of the International School of Physics ”Enrico Fermi”, Course 211 – Quantum Mixtures with Ultra-cold Atoms, edited by R. Grimm, M. Inguscio, and S. Stringari (IOS Press, Amsterdam, 2024)

    work page 2024

  27. [27]

    See the Supplemen- 6 tal Material for further information about the numerical method [21]

    In practice, calculations are performed in the presence of a very weak external potential, which enhances the stability of our numerical approach. See the Supplemen- 6 tal Material for further information about the numerical method [21]

    work page

  28. [28]

    H. A. Haus, Appl. Phys. Lett.8, 128–129 (1966)

    work page 1966

  29. [29]

    Kruglov, Y

    V. Kruglov, Y. Logvin, and V. Volkov, J. Mod. Opt.39, 2277–2291 (1992)

    work page 1992

  30. [30]

    Bloch, J

    I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885–964 (2008)

    work page 2008

  31. [31]

    Kehrein,The Flow Equation Approach to Many- Particle Systems(Springer Berlin Heidelberg, 2006)

    S. Kehrein,The Flow Equation Approach to Many- Particle Systems(Springer Berlin Heidelberg, 2006)

    work page 2006

  32. [32]

    Tsukiyama, S

    K. Tsukiyama, S. K. Bogner, and A. Schwenk, Phys. Rev. Lett.106, 222502 (2011)

    work page 2011

  33. [33]

    A. G. Volosniev and H.-W. Hammer, New J. Phys.19, 113051 (2017)

    work page 2017

  34. [34]

    Brauneis, H.-W

    F. Brauneis, H.-W. Hammer, M. Lemeshko, and A. Volosniev, SciPost Phys.11, 008 (2021)

    work page 2021

  35. [35]

    In that case, the transition involves spontaneous symmetry breaking, potentially leading to dynamics that differ sig- nificantly from those presented here

    This weak dependence stands in contrast to the behavior observed in a bubble trap [9], which features a transi- tion from a uniform density to a localized droplet. In that case, the transition involves spontaneous symmetry breaking, potentially leading to dynamics that differ sig- nificantly from those presented here. The existence of a vanishing energy B...

    work page

  36. [36]

    Guo and R

    Y. Guo and R. Seiringer, Lett. Math. Phys.104, 141–156 (2013)

    work page 2013

  37. [37]

    In other words, the physics of the high-density regime is governed primarily by kinetic energy, which does not allow the system to collapse

    Collapse cannot occur in our ‘asymptotically free’ theory, because the coupling constant vanishes at high densities. In other words, the physics of the high-density regime is governed primarily by kinetic energy, which does not allow the system to collapse

    work page

  38. [38]

    Efimov, Phys

    V. Efimov, Phys. Lett. B33, 563 (1970)

    work page 1970

  39. [39]

    Braaten and H

    E. Braaten and H. W. Hammer, Phys. Rept.428, 259 (2006)

    work page 2006

  40. [40]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat- tore, Rev. Mod. Phys.83, 863–883 (2011)

    work page 2011

  41. [41]

    M. D. Lee and S. A. Morgan, J. Phys. B: At. Mol. Opt. Phys35, 3009 (2002)

    work page 2002

  42. [42]

    A. Y. Cherny and A. A. Shanenko, Phys. Rev. E64, 027105 (2001)

    work page 2001

  43. [43]

    According to Eq

    Notably, it is already known that the 2D three-body (four-body) system supports a single excited state with zero angular momentum and energy 1.27B 2 [58, 59] (25.1B2 [60]). According to Eq. (4) one could expect that the large-Nlimit for such excited states readsB 2e4πN/|G|, where|G|= 38.6 corresponds to the excited state of the generalized GPE withs= 0. I...

    work page

  44. [44]

    S. G. Rajeev, A condensation of interacting bosons in two dimensional space (1999), arXiv:hep-th/9905120 [hep- th]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  45. [45]

    S. G. Rajeev, Bound states in models of asymptotic free- dom (1999), arXiv:hep-th/9902025 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  46. [46]

    [8],Ris de- noted asR, unlike the notation used in the main text of this Letter

    Note that in many sources, including Ref. [8],Ris de- noted asR, unlike the notation used in the main text of this Letter

    work page

  47. [47]

    A. G. Volosniev and H.-W. Hammer, Phys. Rev. A96, 031601 (2017)

    work page 2017

  48. [48]

    Brauneis, T

    F. Brauneis, T. G. Backert, S. I. Mistakidis, M. Lemeshko, H.-W. Hammer, and A. G. Volosniev, New J. Phys.24, 063036 (2022)

    work page 2022

  49. [49]

    Brauneis, A

    F. Brauneis, A. Ghazaryan, H.-W. Hammer, and A. G. Volosniev, Commun. Phys6, 224 (2023)

    work page 2023

  50. [50]

    Brauneis, H.-W

    F. Brauneis, H.-W. Hammer, S. M. Reimann, and A. G. Volosniev, Phys. Rev. A111, 013303 (2025)

    work page 2025

  51. [51]

    Bao and Q

    W. Bao and Q. Du, SIAM J. Sci. Comput.25, 1674 (2004)

    work page 2004

  52. [52]

    Antoine and R

    X. Antoine and R. Duboscq, Comput. Phys. Commun. 185, 2969 (2014)

    work page 2014

  53. [53]

    Saad,Iterative Methods for Sparse Linear Systems, 2nd ed., Other Titles in Applied Mathematics (SIAM, 2003)

    Y. Saad,Iterative Methods for Sparse Linear Systems, 2nd ed., Other Titles in Applied Mathematics (SIAM, 2003)

    work page 2003

  54. [54]

    Krylovkit.jl,https://jutho.github.io/KrylovKit.jl (2022)

    work page 2022

  55. [55]

    Bao and H

    W. Bao and H. Wang, J. Comput. Phys.217, 612 (2006)

    work page 2006

  56. [56]

    G. M. Koutentakis, S. I. Mistakidis, and P. Schmelcher, Atoms10(1), 3 (2022)

    work page 2022

  57. [57]

    Suchorowski, Julia Generalized GPE Solver for 2D Attractive Bose Gas (2025)

    M. Suchorowski, Julia Generalized GPE Solver for 2D Attractive Bose Gas (2025)

    work page 2025

  58. [58]

    L. W. Bruch and J. A. Tjon, Phys. Rev. A19, 425–432 (1979)

    work page 1979

  59. [59]

    Nielsen, D

    E. Nielsen, D. V. Fedorov, and A. S. Jensen, Few-Body Syst.27, 15–55 (1999)

    work page 1999

  60. [60]

    Platter, H.-W

    L. Platter, H.-W. Hammer, and U.-G. Meissner, Few- Body Syst. , 169–174 (2004). 7 Supplemental Material S1. P ARAMETERα To determineαin Eq. (3) from the main text, we first compute the energies of the system in the limitW→ 0, where according to Ref. [8] the ground state energy forN≫1 approachesB s=0 N →B 2e 4N C +c1 withC≃ 1.862 andc 1 ≃ −1.91. By trial a...

    work page 2004