Pith. sign in

REVIEW 4 major objections 5 minor 70 references

Fractional vorticity, Bogomol'nyi-Prasad-Sommerfield systems and complex structures for the (generalized) spinor Gross-Pitaevskii equations

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read This paper claims that generalized two-dimensional Gross–Pitaevskii equations admit first-order BPS systems whose simplest vortices carry fractional charge set by the nonlinearity.

desk verdict The BPS construction is mostly right, but the fractional-vorticity solutions are multivalued on the plane, and the sextic section has coefficient errors; worth refereeing, not publishable as is. read the letter →

arxiv 2502.00578 v2 pith:U4AQU63D submitted 2025-02-01 cond-mat.quant-gas hep-phhep-thnucl-th

Fabrizio Canfora , Pablo Pais This is my paper · ORCID
classification cond-mat.quant-gashep-phhep-thnucl-th MSC 35Q5535J61
keywords fractionalvorticityBPSboundGross-PitaevskiiequationsuperpotentialspinorBose-Einsteincondensatesquantummixturescomplexstructure
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

Using holomorphic superpotentials in the field space of a two-dimensional complex scalar, the paper proves that an infinite family of self-interaction potentials, including the quartic $|\Psi|^4$, the quintic $|\Psi|^5$, and the sextic $|\Psi|^6$, give rise to first-order BPS systems whose solutions automatically solve the full second-order Gross-Pitaevskii equation (GPE). The simplest topologically nontrivial solutions are annular configurations with quantized fractional vorticity, the fraction being tied to the interaction power. This matters because BPS systems provide exact energy bounds and first-order reduction, tools previously unavailable for superfluids described by the GPE. The same complex-structure construction extends to coupled two-component GPEs, producing BPS systems even for mixtures that do not have a Hamiltonian and therefore are not supersymmetric in the usual sense.

What carries the argument

The key mechanism is the field-space complex structure: identifying the two real fields as $Z = \Phi_1 + i\Phi_2$ and writing the first-order system through a holomorphic superpotential $W(Z)$ via $A = \mathrm{Re}\,W$, $B = \mathrm{Im}\,W$. Imposing the field-space Cauchy–Riemann equations ensures the derived second-order equations contain no first derivatives, so they match the GPE with potential $\frac12(A^2+B^2)$. This structure does the dual work of producing the BPS bound (the energy becomes squares of the first-order equations plus boundary terms) and guaranteeing that BPS solutions satisfy the second-order equations. For multicomponent systems, two superpotentials analytic in both complex components play the analogous role.

What would settle it

Recompute the coefficient matching in the general derivation for $n=3$; if the second-order equation acquires a factor $n$ in the coupling (so $g_{\mathrm{eff}}=n\kappa$ instead of $\kappa$), then the printed radial profile $\rho(r)=(A r^{1/2}-4\sqrt{3}\kappa r)^{-1/2}$ with $S=\theta/4$ is not a solution of the $|\Psi|^6$ GPE as written, which would settle the automatic-solution claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that for every real $n$, the superpotential $W = \kappa\sqrt{n}\,Z^n$, with $Z = \Phi_1 + i\Phi_2$, generates a first-order system, $\partial_x\Phi_1 + \partial_y\Phi_2 = \mathrm{Re}\,W$ and $\partial_y\Phi_1 - \partial_x\Phi_2 = \mathrm{Im}\,W$, whose solutions solve the generalized GPE $\Delta\Phi_j = \kappa^2(\Phi_1^2+\Phi_2^2)^{n-1}\Phi_j$ for $j=1,2$. The energy admits a BPS bound written as a sum of squares plus two boundary terms. For $n=2,5/2,3$, the simplest solutions have phase $S(\theta)=\theta/(n+1)$ and radial profile $\rho(r) = (A r^{(n-1)/(n+1)} + \frac{(1-n^2)\kappa}{2\sqrt{n}} r)^{1/(1-n)}$, giving fractional vorticity $1/3$, $2/7$, and $1/4$, respectively. For two condensates, choosing two superpotentials $W_1(Z_1,Z_2)$ and $W_2(Z_1,Z_2)$ yields first-order systems for coupled GPEs whose couplings satisfy specific factorization conditions, and these systems are generally not supersymmetric.

Load-bearing premise

The construction stands on the assumption that field configurations with the multivalued phase $S(\theta)=\theta/(n+1)$ are acceptable physical solutions on an annulus, together with the coefficient identities that identify the BPS system's derived second-order equations with the displayed GPE.

Editorial extensions

If this is right

  • For each superpotential $W=\kappa\sqrt{n}\,Z^n$, every solution of the BPS system is an exact static solution of the corresponding generalized GPE, and the energy is fixed by the boundary terms $Q_1$ and $Q_2$.
  • The fractional vorticity is determined by the nonlinearity: $1/3$ for $|\Psi|^4$, $2/7$ for $|\Psi|^5$, and $1/4$ for $|\Psi|^6$, without any extra gauge field or external ingredient.
  • The BPS equations link the amplitude to the phase through an algebraic relation, reducing the GPE system to a single master equation for the phase, which can be used to construct multi-vortex configurations numerically.
  • For two-component mixtures, a BPS system exists whenever the four coupling constants satisfy $G_1 = G_3 = g_1 g_2$, $G_2 = g_1^2$, and $G_4 = g_2^2$, bringing moduli-space and hydrodynamic methods to spinor condensates.
  • BPS first-order systems can exist for systems without a Hamiltonian, so the BPS toolbox is not restricted to supersymmetric field theories.

Reading between the lines

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

  • The same superpotential construction with fractional powers suggests a continuous family of fractional vorticities, and non-polynomial potentials could be treated by the same one-function reduction.
  • The branch-cut structure of the fractional vortex implies the order parameter lives on a Riemann surface, so the solutions may be materially realizable in ring-shaped traps or in systems with engineered three-body losses.
  • For three or more components, recursive superpotential choices should produce analogous BPS systems, provided the coupling matrix satisfies the corresponding factorized identities.
  • The amplitude-phase master equation could serve as a numerical tool to search for fractional-vortex lattices and to test whether their energy ordering matches the boundary-term formula.
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

4 major / 5 minor

Summary. The paper develops a first-order BPS-type formalism for two-dimensional generalized Gross-Pitaevskii equations. Starting from a first-order elliptic system whose right-hand sides are the real and imaginary parts of an analytic superpotential W(Z), it shows that solutions of the first-order system solve the second-order semilinear equation with potential (1/2)(A^2+B^2). It then applies this to quartic, quintic, and sextic self-interactions, claiming that the sextic and quintic cases admit BPS bounds and explicit fractional-vorticity solutions with vorticity 1/(n+1), and extends the construction to two-component mixtures. The paper closes with remarks distinguishing the construction from standard supersymmetry.

Significance. The general mechanism in Section 2 is elegant and potentially useful: any analytic superpotential gives a first-order system whose solutions solve a second-order generalized GPE, and the Appendix derivations are self-contained. If the fractional-vorticity solutions were legitimate, the paper would provide a new toolbox for non-perturbative vortex physics in BECs and (2+1)-dimensional field theories. However, the advertised results for the higher-order GPEs rely on incorrect coefficient identifications, and the explicit fractional-vorticity solutions are not single-valued order parameters on the physical annulus. These issues affect the central claims of the paper, not just its presentation.

major comments (4)
  1. [§3.2, Eqs. (25)–(30)] The sextic case is internally inconsistent as printed. For W = (κ/√3) Z³, the BPS system (26)–(27) gives A²+B² = (κ²/3) ρ⁶, and a direct computation gives ΔΦ = κ² ρ⁴ Φ. Therefore the GPE (28) should read g_eff = κ², not g_eff = κ. Moreover, the potential term in the energy (29) is written as (g_eff/2)ρ⁶; for a |Ψ|⁶ GPE the potential coefficient should be g_eff/3 (equivalently, g_eff/n for |Ψ|^{2n}), so even after correcting g_eff to κ², the coefficient in (29) should be κ²/3, not κ²/2. As written, expanding the squares in (30) with A,B from (31) gives a ρ⁶ coefficient κ²/3, which disagrees with (29) both as printed (κ/2) and after the correction g_eff=κ² (κ²/2). Thus the claimed BPS bound is not a bound for the stated energy functional, and the claim of an infinite family of BPS-bounded generalized GPEs is not supported by the equations as written.
  2. [Appendix A] The general derivation in Appendix A does not match the normalizations used in the main text. The appendix starts from BPS equations with right-hand side α√n ρⁿ cos(nS), but then substitutes the equations as though the coefficient were κ√n, implicitly setting α=κ. The main text, however, uses superpotentials W = (κ/√n) Zⁿ, whose BPS right-hand side is (κ/√n) ρⁿ cos(nS), i.e., α = κ/n in the appendix notation. Carrying out the appendix calculation with the main-text normalization gives a different coupling constant in the derived GPE. This is the source of the erroneous g_eff=κ in Eq. (28); the derivation needs to be harmonized with the definitions of W used in Sections 2 and 3.
  3. [§3.2.1, Eq. (36) and Appendix B] The explicit sextic solution (36) does not solve the BPS system with the coefficient stated in the main text. Appendix B, Eq. (96), for n=3 gives ρ(r) = (A r^{1/2} − (4κ/√3) r)^{-1/2}, whereas Eq. (36) contains the coefficient 4√3 κ. The two differ by a factor of 3. Since this solution is used to compute the topological charges Q1 and Q2 in Eqs. (37)–(38), the printed solution and its derived charges are not correct.
  4. [§3.2.1, Fig. 1 and Eq. (35)] The fractional-vorticity solutions are multivalued on the annulus. For S(θ)=θ/4, Ψ(r,2π)=e^{iπ/2}Ψ(r,0), so the order parameter jumps by a factor i across the branch cut; similarly S=θ/3 and S=2θ/7 for the quartic and quintic examples. Such a field is not a single-valued solution of the original two-dimensional GPE on the annulus. The distributional derivative ∂_θ S contains a delta-function contribution along the cut, producing an infinite gradient-energy term that the atomic-size cutoff at the inner radius does not cure. Moving to a Riemann surface, as suggested in the text, changes the domain and amounts to studying a different problem from the 2D GPE. This undermines the central physical claim of quantized fractional vorticity in the generalized GPE.
minor comments (5)
  1. [Eq. (34) and Eq. (38)] The expression for J^y in Eq. (38) is printed as κ/(2√3) ρ⁴ cos(4S), but the correct expression from Eq. (34) is κ/(2√3) ρ⁴ sin(4S); the subsequent computation of n·J uses the sine form, so the printed line is a typographical error.
  2. [Eq. (37)] In the displayed formula for Q1, the same letter r is used in both boundary terms; the first term should involve the outer radius R and the second the inner radius a. The coefficient should also be updated to match the corrected solution from Appendix B.
  3. [Footnote 3] The sentence 'Thus, only the outer boundary of the annulus contributes to the vorticity' is repeated twice in the footnote.
  4. [Text before Eq. (37)] The phrase 'as costume calculated' appears to be a typo for 'as calculated'.
  5. [General notation] The paper writes the generalized GPE energy with a potential coefficient g_eff/2 in Eq. (14) and again in Eq. (29); for higher-order interactions the coefficient depends on the degree of nonlinearity (g_eff/n for |Ψ|^{2n}). A consistent general formula would prevent the kind of mismatch identified in the sextic case.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the BPS systems are derived by direct first-order completion from an analytic superpotential, and the fractional vorticity follows from the constructed ansatz rather than from any fitted parameter.

full rationale

The paper's central derivation is self-contained algebra. Equations (1)-(9) define A and B as the real and imaginary parts of an analytic superpotential W(Z), which forces C1=C2=0 and yields the second-order system (6)-(7); for W proportional to Z^n this reproduces the stated generalized GPE (Appendix A). The BPS energy bounds in Eqs. (16) and (30) are exact square completions, with the topological charges Q1 and Q2 appearing as boundary terms. No parameter is fitted to a target vorticity: the angular ansatz S = theta/(n+1) is imposed in Appendix B, and the resulting fraction 1/(n+1) is a consequence of the BPS equations, not an input tuned to reproduce a desired value. The citations to [35]-[37] describe earlier BPS constructions, but the present derivation re-derives the necessary identities and does not use those papers as the load-bearing evidence, so the self-citation is not circular in a damaging sense. The reader-identified coefficient mismatches (e.g., g_eff = kappa in Eq. (28) versus Appendix A's kappa^2) and the multivaluedness of the order parameter across the branch cut are correctness or physical-validity concerns rather than circularity: they do not make the claimed derivation equivalent to its own inputs. Thus no concrete circular step is exhibited, and the paper receives a low score reflecting only a minor non-load-bearing self-citation.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

No data fitting is involved; the constants A and S0 are integration constants. The main additional input is the short-distance cutoff a and the multivalued-phase ansatz.

free parameters (1)
  • atomic short-distance cutoff a = not specified
    The amplitude rho(r) diverges at r=0 for all n; the annulus inner radius a (interpreted as atomic size) is introduced to regularize the solutions and to make the fractional vorticity well-defined. Its value is not determined by the theory.
assumptions (3)
  • domain assumption The physical order parameter is a complex scalar field whose energy is the GPE functional (14)/(29), and the equivalence between the first-order BPS system and the second-order GPE holds when the right-hand sides A,B satisfy the Cauchy-Riemann equations.
    Invoked in Section 2, Eqs. (5)-(9); this restricts to potential terms with no first derivatives, which is the standard GPE form.
  • ad hoc to paper The ansatz S(theta)=theta/(n+1), partial_theta rho=0, partial_r S=0, used to solve the BPS system, yields admissible single-valued fields on the annulus despite the branch cut.
    Appendix B, 'Let us suppose partial_theta rho=0 and partial_r S=0'; the resulting phase is multivalued around the inner boundary, which is not a standard single-valued condensate order parameter.
  • standard math Analyticity of the superpotential W(Z) implies the compatibility conditions (63)-(66), so that first-order solutions solve the second-order system.
    This is the standard Bogomolnyi completion trick; the paper derives it in Section 2 and Appendix A.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Fractional vorticity, Bogomol'nyi-Prasad-Sommerfield systems and complex structures for the (generalized) spinor Gross-Pitaevskii equations." pith.science (2026). https://pith.science/paper/U4AQU63D

@misc{pith2026250200578,
  author       = {Pith},
  title        = {Pith review of: Fractional vorticity, Bogomol'nyi-Prasad-Sommerfield systems and complex structures for the (generalized) spinor Gross-Pitaevskii equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U4AQU63D}},
  note         = {Machine review of arXiv:2502.00578}
}
abstract

The (generalized) Gross-Pitaevskii equation (GPE) for a complex scalar field in two spatial dimensions is analyzed. It is shown that there is an infinite family of self-interaction potentials which admit Bogomol'nyi-Prasad-Sommerfield (BPS) bounds together with the corresponding first-order BPS systems. For each member of this family, the solutions of the first-order BPS systems are automatically solutions of the corresponding second-order generalized GPE. The simplest topologically non-trivial solutions of these first-order BPS systems describe configurations with quantized fractional vorticity. The corresponding fraction is related to the degree of non-linearity. The case in which the self-interaction potential is of order six (namely $|\Psi |^{6}$, which is a relevant theory both in relativistic quantum field theories in $(2+1)$ dimensions in connection with the quantum Hall effect as well as in the theory of the supersolids) is analyzed in detail. Such formalism can also be extended to the case of quantum mixtures with multi-component GPEs. The relationship between these techniques and supersymmetry will be discussed. In particular, despite several common features, we will show that there are multi-component GPEs that are not supersymmetric (at least, not in the standard sense) and possess a BPS system of the above type.

Figures

Figures reproduced from arXiv: 2502.00578 by the authors.

Figure 1
Figure 1. Level curves for solution (35). We took κ = 1 and A = 1, and internal radius a = 0.3 while the external one is R ≈ 0.91 in arbitrary units. This solution can be embedded on a Riemann surface, as the fractional vorticity feature implies there should be a branch cut. In this case, such a branch cut lies on the positive x-axis, as is evident in the colour discontinuity in both real and imaginary parts of the solution. … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

70 extracted references · 57 canonical work pages

  1. [37]

    A novel BPS bound with a first order BPS system in the 2 D Gross-Pitaevskii equation,

    F. Canfora and P. Pais, “A novel BPS bound with a first order BPS system in the 2 D Gross-Pitaevskii equation,” arXiv:2501.04092 [cond-mat.quant-gas]

    arXiv

  2. [1]

    Pitaevskii and S

    L. Pitaevskii and S. Stringari, Bose-Einstein Condensation and Superfluidity . International Series of Monographs on Physics. OUP Oxford, 2016. https://books.google.cl/books?id=yHByCwAAQBAJ

    work page 2016

  3. [2]

    Theory of Bose-Einstein condensation in trapped gases,

    F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71 (1999) 463–512, arXiv:cond-mat/9806038

    arXiv 1999

  4. [3]

    Nobel Lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments,

    E. A. Cornell and C. E. Wieman, “Nobel Lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments,” Rev. Mod. Phys. 74 (2002) 875–893

    work page 2002

  5. [4]

    Three Lectures on QCD Phase Transitions,

    R. Pisarski, “Three Lectures on QCD Phase Transitions,” Lect. Notes Phys. 999 (2022) 89–145. 20

    work page 2022

  6. [5]

    N. S. Manton and P. Sutcliffe, Topological solitons. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004

    work page 2004

  7. [6]

    Shuryak, Nonperturbative Topological Phenomena in QCD and Related Theories

    E. Shuryak, Nonperturbative Topological Phenomena in QCD and Related Theories . Lecture Notes in Physics. Springer International Publishing, 2021. https://books.google.cl/books?id=Ee4lEAAAQBAJ

    work page 2021

  8. [7]

    Shifman, Advanced Topics in Quantum Field Theory: A Lecture Course

    M. Shifman, Advanced Topics in Quantum Field Theory: A Lecture Course . Cambridge University Press, 2022. https://books.google.cl/books?id=CS9lEAAAQBAJ

    work page 2022

Show all 70 references
  1. [8]

    E. J. Weinberg, Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics . Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2012

    2012

  2. [11]

    Volume of vortex moduli spaces,

    N. S. Manton and S. M. Nasir, “Volume of vortex moduli spaces,” Communications in Mathematical Physics 199 no. 3, (Jan, 1999) 591–604. https://doi.org/10.1007/s002200050513

    1999 doi

  3. [12]

    Spinor bose gases: Symmetries, magnetism, and quantum dynamics,

    D. M. Stamper-Kurn and M. Ueda, “Spinor bose gases: Symmetries, magnetism, and quantum dynamics,” Rev. Mod. Phys. 85 (Jul, 2013) 1191–1244. https://link.aps.org/doi/10.1103/RevModPhys.85.1191

    2013 doi

  4. [13]

    Vortices in multicomponent Bose–Einstein condensates,

    K. Kasamatsu, M. Tsubota, and M. Ueda, “Vortices in multicomponent Bose–Einstein condensates,” International Journal of Modern Physics B 19 no. 11, (Apr., 2005) 1835–1904. http://dx.doi.org/10.1142/S0217979205029602

    2005 doi

  5. [14]

    Classification of the ground states and topological defects in a rotating two-component Bose-Einstein condensate,

    P. Mason and A. Aftalion, “Classification of the ground states and topological defects in a rotating two-component Bose-Einstein condensate,” Phys. Rev. A 84 (Sep, 2011) 033611. https://link.aps.org/doi/10.1103/PhysRevA.84.033611

    2011 doi

  6. [15]

    Leggett, Quantum Liquids: Bose condensation and Cooper pairing in condensed-matter systems

    A. Leggett, Quantum Liquids: Bose condensation and Cooper pairing in condensed-matter systems. Oxford Graduate Texts. OUP Oxford, 2006. https://books.google.cl/books?id=HnlPAwAAQBAJ

    2006

  7. [16]

    Schmitt, Introduction to Superfluidity: Field-theoretical Approach and Applications

    A. Schmitt, Introduction to Superfluidity: Field-theoretical Approach and Applications . Lecture Notes in Physics. Springer International Publishing, 2014. https://books.google.cl/books?id=vtQlBAAAQBAJ

    2014

  8. [17]

    Pethick and H

    C. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases . Cambridge University Press, 2008. https://books.google.cl/books?id=G8kgAwAAQBAJ

    2008

  9. [18]

    Barenghi, R

    C. Barenghi, R. Donnelly, and W. Vinen, Quantized Vortex Dynamics and Superfluid Turbulence. Lecture Notes in Physics. Springer Berlin Heidelberg, 2001. https://books.google.cl/books?id=GBIgF9X56_oC

    2001

  10. [19]

    Colloquium: Supersolids: What and where are they?

    M. Boninsegni and N. V. Prokof’ev, “Colloquium: Supersolids: What and where are they?” Rev. Mod. Phys. 84 (May, 2012) 759–776. https://link.aps.org/doi/10.1103/RevModPhys.84.759

    2012 doi

  11. [20]

    The enigma of supersolidity,

    S. Balibar, “The enigma of supersolidity,” Nature 464 no. 7286, (Mar, 2010) 176–182. https://doi.org/10.1038/nature08913

    2010 doi

  12. [21]

    What makes a crystal supersolid?

    N. Prokof’ev, “What makes a crystal supersolid?” Advances in Physics 56 no. 2, (2007) 381–402, https://doi.org/10.1080/00018730601183025. https://doi.org/10.1080/00018730601183025

    2007 doi

  13. [22]

    Saga of superfluid solids,

    V. I. Yukalov, “Saga of superfluid solids,” Physics 2 no. 1, (2020) 49–66. https://www.mdpi.com/2624-8174/2/1/6. 21

    2020

  14. [23]

    Observation of a dipolar quantum gas with metastable supersolid properties,

    L. Tanzi, E. Lucioni, F. Fam` a, J. Catani, A. Fioretti, C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno, “Observation of a dipolar quantum gas with metastable supersolid properties,” Phys. Rev. Lett. 122 (Apr, 2019) 130405. https://link.aps.org/doi/10.1103/PhysRevLett.1...

    2019 doi

  15. [24]

    Transient supersolid properties in an array of dipolar quantum droplets,

    F. B¨ ottcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo, T. Langen, and T. Pfau, “Transient supersolid properties in an array of dipolar quantum droplets,” Phys. Rev. X 9 (Mar, 2019) 011051. https://link.aps.org/doi/10.1103/PhysRevX.9.011051

    2019 doi

  16. [25]

    Long-lived and transient supersolid behaviors in dipolar quantum gases,

    L. Chomaz, D. Petter, et al. , “Long-lived and transient supersolid behaviors in dipolar quantum gases,” Phys. Rev. X 9 (Apr, 2019) 021012. https://link.aps.org/doi/10.1103/PhysRevX.9.021012

    2019 doi

  17. [26]

    Two-dimensional supersolid formation in dipolar condensates,

    T. Bland, E. Poli, C. Politi, L. Klaus, M. A. Norcia, F. Ferlaino, L. Santos, and R. N. Bisset, “Two-dimensional supersolid formation in dipolar condensates,” Phys. Rev. Lett. 128 (May,

  18. [27]

    Birth, life, and death of a dipolar supersolid,

    M. Sohmen, C. Politi, L. Klaus, L. Chomaz, M. J. Mark, M. A. Norcia, and F. Ferlaino, “Birth, life, and death of a dipolar supersolid,” Phys. Rev. Lett. 126 (Jun, 2021) 233401. https://link.aps.org/doi/10.1103/PhysRevLett.126.233401

    2021 doi

  19. [28]

    Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas,

    L. Tanzi, S. M. Roccuzzo, E. Lucioni, F. Fam` a, A. Fioretti, C. Gabbanini, G. Modugno, A. Recati, and S. Stringari, “Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas,” Nature 574 no. 7778, (Oct, 2019) 382–385. https://doi.org/10.1038/s4158...

    2019 doi

  20. [29]

    A new form of liquid matter: Quantum droplets,

    Z.-H. Luo, W. Pang, B. Liu, Y.-Y. Li, and B. A. Malomed, “A new form of liquid matter: Quantum droplets,” Frontiers of Physics 16 no. 3, (Dec, 2020) 32201. https://doi.org/10.1007/s11467-020-1020-2

    2020 doi

  21. [30]

    A new state of matter of quantum droplets,

    M. Guo and T. Pfau, “A new state of matter of quantum droplets,” Frontiers of Physics 16 no. 3, (Dec, 2020) 32202. https://doi.org/10.1007/s11467-020-1035-8

    2020 doi

  22. [31]

    New states of matter with fine-tuned interactions: quantum droplets and dipolar supersolids,

    F. B¨ ottcher, J.-N. Schmidt, J. Hertkorn, K. S. H. Ng, S. D. Graham, M. Guo, T. Langen, and T. Pfau, “New states of matter with fine-tuned interactions: quantum droplets and dipolar supersolids,” Reports on Progress in Physics 84 no. 1, (Dec, 2020) 012403. https://dx.doi.org/...

    2020 doi

  23. [32]

    Eigenvalues and eigenfunctions of a bose system of hard spheres and its low-temperature properties,

    T. D. Lee, K. Huang, and C. N. Yang, “Eigenvalues and eigenfunctions of a bose system of hard spheres and its low-temperature properties,” Phys. Rev. 106 (Jun, 1957) 1135–1145. https://link.aps.org/doi/10.1103/PhysRev.106.1135

    1957 doi

  24. [33]

    Quantum Mechanical Stabilization of a Collapsing Bose-Bose Mixture,

    D. S. Petrov, “Quantum Mechanical Stabilization of a Collapsing Bose-Bose Mixture,” Phys. Rev. Lett. 115 (Oct, 2015) 155302. https://link.aps.org/doi/10.1103/PhysRevLett.115.155302

    2015 doi

  25. [34]

    Self-trapped quantum balls in binary Bose–Einstein condensates,

    S. Gautam and S. K. Adhikari, “Self-trapped quantum balls in binary Bose–Einstein condensates,” Journal of Physics B: Atomic, Molecular and Optical Physics 52 no. 5, (Feb,

  26. [35]

    Magnetized baryonic layer and a novel bps bound in the gauged-non-linear-sigma-model-maxwell theory in (3+1)-dimensions through hamilton-jacobi equation,

    F. Canfora, “Magnetized baryonic layer and a novel bps bound in the gauged-non-linear-sigma-model-maxwell theory in (3+1)-dimensions through hamilton-jacobi equation,” Journal of High Energy Physics 2023 no. 11, (Nov, 2023) 7. https://doi.org/10.1007/JHEP11(2023)007

    2023 doi

  27. [36]

    Superconducting multi-vortices and a novel bps bound in chiral perturbation theory,

    F. Canfora, M. Lagos, and A. Vera, “Superconducting multi-vortices and a novel bps bound in chiral perturbation theory,” Journal of High Energy Physics 2024 no. 10, (Oct, 2024) 224. https://doi.org/10.1007/JHEP10(2024)224

    2024 doi

  28. [38]

    Vortex Motion Quantifies Strong Dissipation in a Holographic Superfluid,

    P. Wittmer, C.-M. Schmied, T. Gasenzer, and C. Ewerz, “Vortex Motion Quantifies Strong Dissipation in a Holographic Superfluid,” Phys. Rev. Lett. 127 no. 10, (2021) 101601, arXiv:2011.12968 [hep-th]. 22

    2021 arXiv

  29. [39]

    Energy-dependent scattering and the Gross-Pitaevskii equation in two-dimensional Bose-Einstein condensates,

    M. D. Lee, S. A. Morgan, M. J. Davis, and K. Burnett, “Energy-dependent scattering and the Gross-Pitaevskii equation in two-dimensional Bose-Einstein condensates,” Phys. Rev. A 65 (Apr, 2002) 043617. https://link.aps.org/doi/10.1103/PhysRevA.65.043617

    2002 doi

  30. [40]

    Peskin and D

    M. Peskin and D. Schroeder, An Introduction to Quantum Field Theory . Advanced book classics. Addison-Wesley Publishing Company, 1995

    1995

  31. [41]

    Wilczek, Fractional Statistics and Anyon Superconductivity

    F. Wilczek, Fractional Statistics and Anyon Superconductivity . World Scientific, 1990. https://www.worldscientific.com/doi/abs/10.1142/0961

    1990 doi

  32. [42]

    Ezawa, Quantum Hall Effects: Recent Theoretical And Experimental Developments (3rd Edition)

    Z. Ezawa, Quantum Hall Effects: Recent Theoretical And Experimental Developments (3rd Edition). World Scientific Publishing Company, 2013. https://books.google.cl/books?id=6io8DQAAQBAJ

    2013

  33. [43]

    Effective-field-theory model for the fractional quantum hall effect,

    S. C. Zhang, T. H. Hansson, and S. Kivelson, “Effective-field-theory model for the fractional quantum hall effect,” Phys. Rev. Lett. 62 (Jan, 1989) 82–85. https://link.aps.org/doi/10.1103/PhysRevLett.62.82

    1989 doi

  34. [44]

    Self-dual chern-simons vortices,

    R. Jackiw and E. J. Weinberg, “Self-dual chern-simons vortices,” Phys. Rev. Lett. 64 (May,

  35. [45]

    Classical vortex solution of the abelian higgs model,

    H. J. de Vega and F. A. Schaposnik, “Classical vortex solution of the abelian higgs model,” Phys. Rev. D 14 (Aug, 1976) 1100–1106. https://link.aps.org/doi/10.1103/PhysRevD.14.1100

    1976 doi

  36. [46]

    Abelian Chern-Simons vortices at finite chemical potential,

    S. P. Kumar and S. Stratiev, “Abelian Chern-Simons vortices at finite chemical potential,” JHEP 04 (2020) 041, arXiv:1912.11321 [hep-th]

    2020 arXiv

  37. [47]

    Multivortex solutions of the abelian chern-simons-higgs theory,

    J. Hong, Y. Kim, and P. Y. Pac, “Multivortex solutions of the abelian chern-simons-higgs theory,” Phys. Rev. Lett. 64 (May, 1990) 2230–2233. https://link.aps.org/doi/10.1103/PhysRevLett.64.2230

    1990 doi

  38. [49]

    Vortex dynamics in self-dual chern-simons-higgs systems,

    Y. Kim and K. Lee, “Vortex dynamics in self-dual chern-simons-higgs systems,” Phys. Rev. D 49 (Feb, 1994) 2041–2054. https://link.aps.org/doi/10.1103/PhysRevD.49.2041

    1994 doi

  39. [50]

    Wess and J

    J. Wess and J. Bagger, Supersymmetry and supergravity. Princeton University Press, Princeton, NJ, USA, 1992

    1992

  40. [51]

    Kleinert, Gauge fields in condensed matter

    H. Kleinert, Gauge fields in condensed matter. Vol. 2: Stresses and defects. Differential geometry, crystal melting . 1989

    1989

  41. [52]

    Quantum field theory in curved graphene spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking effect, and all that,

    A. Iorio and G. Lambiase, “Quantum field theory in curved graphene spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking effect, and all that,” Phys. Rev. D90 no. 2, (2014) 025006, arXiv:1308.0265 [hep-th]

    2014 arXiv

  42. [53]

    Curved Spacetimes and Curved Graphene: a status report of the Weyl-symmetry approach,

    A. Iorio, “Curved Spacetimes and Curved Graphene: a status report of the Weyl-symmetry approach,” Int. J. Mod. Phys. D24 no. 05, (2015) 1530013, arXiv:1412.4554 [hep-th]

    2015 arXiv

  43. [54]

    (anti-)de sitter, poincar´ e, super symmetries, and the two dirac points of graphene,

    A. Iorio and P. Pais, “(anti-)de sitter, poincar´ e, super symmetries, and the two dirac points of graphene,” Annals of Physics 398 (2018) 265 – 286. http://www.sciencedirect.com/science/article/pii/S0003491618302495

    2018

  44. [55]

    Hitchin, G

    N. Hitchin, G. Segal, N. Woodhouse, and R. Ward, Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces . Oxford Graduate Texts in Mathematics. Clarendon Press,

  45. [56]

    Babelon, D

    O. Babelon, D. Bernard, and M. Talon, Introduction to Classical Integrable Systems . Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2003

    2003

  46. [57]

    Jurdjevic, Optimal Control and Geometry: Integrable Systems

    V. Jurdjevic, Optimal Control and Geometry: Integrable Systems . Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2016. 23

    2016

  47. [58]

    Vortices with fractional flux in two-gap superconductors and in extended faddeev model,

    E. Babaev, “Vortices with fractional flux in two-gap superconductors and in extended faddeev model,” Phys. Rev. Lett. 89 (Jul, 2002) 067001. https://link.aps.org/doi/10.1103/PhysRevLett.89.067001

    2002 doi

  48. [59]

    Ground states of one and two fractional vortices in long josephson 0 −κ junctions,

    E. Goldobin, D. Koelle, and R. Kleiner, “Ground states of one and two fractional vortices in long josephson 0 −κ junctions,” Phys. Rev. B 70 (Nov, 2004) 174519. https://link.aps.org/doi/10.1103/PhysRevB.70.174519

    2004 doi

  49. [60]

    Superconducting vortices carrying a temperature-dependent fraction of the flux quantum,

    Y. Iguchi, R. A. Shi, K. Kihou, C.-H. Lee, M. Barkman, A. L. Benfenati, V. Grinenko, E. Babaev, and K. A. Moler, “Superconducting vortices carrying a temperature-dependent fraction of the flux quantum,” Science 380 no. 6651, (2023) 1244–1247, https://www.science.org/doi/pdf/10...

    2023 doi

  50. [61]

    Quantum liquid crystals in an imbalanced fermi gas: Fluctuations and fractional vortices in larkin-ovchinnikov states,

    L. Radzihovsky and A. Vishwanath, “Quantum liquid crystals in an imbalanced fermi gas: Fluctuations and fractional vortices in larkin-ovchinnikov states,” Phys. Rev. Lett. 103 (Jul,

  51. [62]

    Quantum Fluctuations of Instantons in the Nonlinear Sigma Model,

    V. A. Fateev, I. V. Frolov, and A. S. Shvarts, “Quantum Fluctuations of Instantons in the Nonlinear Sigma Model,” Nucl. Phys. B 154 (1979) 1–20

    1979

  52. [63]

    Some twisted self-dual solutions for the yang-mills equations on a hypertorus,

    G. Hooft, “Some twisted self-dual solutions for the yang-mills equations on a hypertorus,” Communications in Mathematical Physics 81 no. 2, (Jun, 1981) 267–275. https://doi.org/10.1007/BF01208900

    1981 doi

  53. [64]

    Three-dimensional gauge configurations and their properties in qcd,

    J. M. Cornwall and G. Tiktopoulos, “Three-dimensional gauge configurations and their properties in qcd,” Physics Letters B 181 no. 3, (1986) 353–358. https://www.sciencedirect.com/science/article/pii/0370269386900626

    1986

  54. [65]

    Fractional topological charge in su(n) gauge theories without dynamical quarks,

    V. P. Nair and R. D. Pisarski, “Fractional topological charge in su(n) gauge theories without dynamical quarks,” Phys. Rev. D 108 (Oct, 2023) 074007. https://link.aps.org/doi/10.1103/PhysRevD.108.074007

    2023 doi

  55. [66]

    Review on fractional vortex beam,

    H. Zhang, J. Zeng, X. Lu, Z. Wang, C. Zhao, and Y. Cai, “Review on fractional vortex beam,” Nanophotonics 11 no. 2, (2022) 241–273. https://doi.org/10.1515/nanoph-2021-0616

    2022 doi

  56. [67]

    Binary Mixtures of Bose Condensates of Alkali Atoms,

    T.-L. Ho and V. B. Shenoy, “Binary Mixtures of Bose Condensates of Alkali Atoms,” Phys. Rev. Lett. 77 (1996) 3276–3279

    1996

  57. [68]

    The Moduli space metric for well separated BPS monopoles,

    G. W. Gibbons and N. S. Manton, “The Moduli space metric for well separated BPS monopoles,” Phys. Lett. B 356 (1995) 32–38, arXiv:hep-th/9506052. 24

    1995 arXiv

  58. [1990]

    https://link.aps.org/doi/10.1103/PhysRevLett.64.2234

    2234–2237. https://link.aps.org/doi/10.1103/PhysRevLett.64.2234

    doi

  59. [1999]

    https://books.google.cl/books?id=B8gNPwEssmgC

  60. [2009]

    https://link.aps.org/doi/10.1103/PhysRevLett.103.010404

    010404. https://link.aps.org/doi/10.1103/PhysRevLett.103.010404

    doi

  61. [2019]

    https://dx.doi.org/10.1088/1361-6455/aafb92

    055302. https://dx.doi.org/10.1088/1361-6455/aafb92

    doi

  62. [2022]

    https://link.aps.org/doi/10.1103/PhysRevLett.128.195302

    195302. https://link.aps.org/doi/10.1103/PhysRevLett.128.195302

    doi

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.