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REVIEW 2 major objections 3 minor 96 references

A Two-Gauge Field Model for Magnetoelectric Boundaries

T0 review · 2 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A coupled Maxwell and planar Chern-Simons field reproduces a magnetoelectric material boundary with two tunable parameters, and the exact propagator shows that in the strong-coupling limit the plane becomes a perfect conductor.

desk verdict A promising two-gauge-field construction, but the printed exact propagator has a block determinant in the numerator where it belongs in the denominator, so the central perfect-conductor limit does not follow as written. read the letter →

arxiv 2508.21697 v1 pith:55TKTSBN submitted 2025-08-29 hep-th cond-mat.other

F.A. Barone , H.L. Oliveira , J.P. Ferreira This is my paper
classification hep-thcond-mat.other
keywords magnetoelectricboundaryChern-SimonsfieldMaxwellexactpropagatordelta-functionpotentialperfectconductortopologicalsourcessourceinteractions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a field-theoretic stand-in for a flat material layer with magnetoelectric properties: the ordinary Maxwell field of 3+1-dimensional space is coupled through delta-function Chern-Simons terms to a second gauge field that lives only on the plane. The planar field has a Chern-Simons mass m and couples to the photon via a strength μ, giving two independent parameters to match a real boundary. The authors compute the exact propagator of the coupled system and use it to derive interaction energies, forces, and classical field configurations for charges and topological sources. In the strong-coupling limit μ→∞ the photon propagator reduces to the known Maxwell propagator with a perfectly conducting plate, while the planar sector becomes pure gauge, so the model interpolates between a transparent plane and a perfect conductor.

What carries the argument

The load-bearing object is a 2×2 matrix of differential operators built from the Maxwell block and the planar Maxwell-Chern-Simons block, joined by antisymmetric Chern-Simons mixing terms localized by δ(x3−a). The exact propagator is obtained by inverting this matrix through reduced propagators, i.e. propagators Fourier-transformed only in the directions parallel to the plane. The inversion hinges on the matrix M in Eq. (28), whose inverse enters every interaction computed later. A secondary mechanism is the reading of the delta-localized coupling as a surface polarization and magnetization induced by the planar field, which makes the magnetoelectric character of the model visible.

What would settle it

Compute the determinant or residue of the matrix M in Eq. (28) in a neighborhood of p∥²=m² and p∥=0 under the standard iε prescription: an unregulated pole would show up as a divergence in one of the integral representations of the interaction energies, such as Eq. (44) or Eq. (58), that the closed-form results do not contain. Alternatively, check Eq. (39) directly against the standard perfect-conductor boundary conditions, tangential E = 0 and normal B = 0 on the plane.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is Eq. (38): the exact matrix propagator of the two-gauge-field system. The propagator is obtained by writing the action as a quadratic form with a 2×2 matrix of differential operators and solving the Green-function equation through reduced propagators in the coordinates parallel to the plane. All interaction corrections are packaged through a function χ(p∥) defined in Eq. (37). Taking μ→∞, the off-diagonal sectors decouple: the electromagnetic block becomes the free Maxwell propagator in Feynman gauge corrected by a perfectly conducting plane at x3=a, while the planar block becomes pure gauge and loses physical observables. The paper then reads c

Load-bearing premise

The derivation assumes the matrix M in Eq. (28) is invertible for every parallel momentum, in particular at p∥²=m² and p∥=0, with the Feynman iε prescription silently handling the poles; if that inverse develops extra unregulated poles, the exact propagator and all derived energies would be ill-defined.

Editorial extensions

If this is right

  • In the strong-coupling limit the model's predictions reduce, by construction of the propagator, to image-charge electrostatics: a point charge is attracted to the plane with force −Q²/(16πR⊥²), independent of the Chern-Simons mass m.
  • For finite μ and m the charge–plane force contains a term decaying more slowly than Coulomb plus a short-range term, so the boundary acts as a tunable attractive layer whose range shrinks as μ or m grows.
  • Planar Chern-Simons charges interact like electromagnetism with a Yukawa-like kernel when μ=0; increasing μ suppresses planar propagation until, at μ→∞, all planar observables vanish.
  • A stationary electric charge generates a magnetic field in the bulk and electric and magnetic fields in the plane, a concrete magnetoelectric signature that disappears when the Chern-Simons mass m goes to zero.
  • Topological planar sources behave as Dirac points (time-like) or electric dipoles (space-like), and their anisotropic interaction produces a torque on the sources.

Reading between the lines

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

  • Because μ continuously interpolates between a decoupled plane (μ=0) and a perfect conductor (μ→∞) while m tunes the magnetoelectric response, the model could be used to extract an effective surface conductivity or an axion-like θ for a boundary from the two parameters—a step the paper does not explicitly take.
  • The exact propagator opens a direct route to Casimir-energy calculations for one or two such layers by summing zero-point fluctuations; in the strong-coupling limit that calculation should reproduce the standard perfect-conductor Casimir result, providing a clean cross-check.
  • The same inversion technique should carry over to multiple parallel planes or to a smooth profile instead of a delta-layer: the matrix would grow, but the structure of the reduced-propagator equations would be unchanged.
  • The field solutions suggest a direct experimental test: measure the magnetic field induced by a static charge near a candidate magnetoelectric material and compare its m-dependent amplitude to Eq. (74).
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper proposes a field-theoretic model of a magnetoelectric boundary: a 3+1-dimensional Maxwell field coupled through Chern-Simons-like terms, localized on a plane x3=a, to a planar Maxwell-Chern-Simons field. The central result is an exact propagator, Eq. (38), obtained by inverting a 2×2 block matrix of differential operators in Eq. (28). The authors claim that in the strong-coupling limit μ→∞ the electromagnetic sector reduces to the Maxwell propagator with a perfectly conducting plate, Eq. (39), while the planar sector becomes pure gauge, Eq. (40). The propagator is then used to compute charge-plane interactions, planar source-source interactions, topological-source interactions, and field solutions for a stationary charge. Several decoupling and massless limits are compared with known results.

Significance. The two-parameter structure of the model is conceptually appealing, and the paper provides explicit formulas for a wide class of observables. The μ=0 limit correctly reduces to the decoupled Maxwell plus planar Maxwell-Chern-Simons theory, and several m→∞ limits are consistent with the expected suppression of planar propagation. However, the central algebraic step—the inversion of M in Eq. (28)—is not demonstrated and, as printed, the resulting propagator is not the inverse of M. The claimed perfect-conductor limit therefore does not follow, and all subsequent interaction energies and field solutions inherit the issue. The paper ships no machine-checkable proof or numerical code, so the correctness of this algebra is essential and must be established explicitly.

major comments (2)
  1. [Section II, Eqs. (33)–(36) and (39)] The factor {1+p∥²[2χ(p∥)+χ(p∥)²(p∥²−m²)]} appears in the numerator of every element of ∆G. For the block matrix M=[[1,A],[B,1]] in Eq. (28), Schur-complement inversion gives (M^{-1})_{11}=(1−AB)^{-1}, etc. On the transverse subspace det(1−AB)=1+p∥²[2χ+χ²(p∥²−m²)], so this determinant must appear in the denominator, not the numerator. As printed, ∆G(11) at fixed p∥ is O(χ⁴)=O(μ⁸) as μ→∞, so Eq. (33) does not approach the image propagator and Eq. (39) is not a consequence. Equations (34)–(36) have the same structural error, so the claimed limits (39), (40) and every downstream result using this propagator—Eqs. (44), (45), (54), (58), (62), and (73)–(76)—are unsupported as written. If the determinant was intended in the denominator, the displayed expressions are misprinted; if it was intended in the numerator, they are not the inverse of Eq. (28). Either way, the central derivation needs to
  2. [Section II, Eqs. (27)–(32)] The inversion of M is not shown; the text says only 'performing some algebraic manipulations.' This is not a presentation quibble: the placement of the determinant is exactly the error identified above. Moreover, the pole structure is not discussed. The matrix M contains 1/(p∥²−m²) and 1/√(−p∥²), and the determinant itself develops a pole at p∥²=m² through χ. Since Secs. III–V integrate the propagator over momenta, the iε prescription and the domain of M^{-1} must be specified. Without this, the 'exact' character of Eq. (38) cannot be verified even after the algebraic sign issue is fixed.
minor comments (3)
  1. [Eq. (2) and Conclusions] The dimensional analysis in Eq. (2) gives [μ]=[ℓ]^{-1/2}, which is consistent with the use of μ² in χ. The Conclusions, however, state that μ has 'dimensions of mass squared.' Please correct the latter.
  2. [Throughout] There are several typos and leftover placeholders: 'collunm' before Eq. (14), 'refrence' in Sec. IV.C, and the word 'AQUI' appearing after Figs. 5 and 9. These should be cleaned up.
  3. [Sec. IV.A, after Eq. (54)] The text says 'opposite-sign sources q1q2 > 0 experience an attractive one'; the condition for attraction should be q1q2 < 0.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the propagator is obtained by inverting the model's kinetic operator, and later limits are consistency checks rather than fitted or assumed inputs.

full rationale

The paper's derivation chain starts from the Lagrangian (1) with free parameters m and mu; the propagator is defined as the inverse of the quadratic operator O in Eq. (17). Equations (26)-(30) are a standard reduced-propagator solution for the delta-localized interaction, and the displayed inverse (33)-(36) is the algebraic output of that inversion. None of the later results -- charge-plane energy (45), source-source energies (54), (58), (62), or field solutions (73)-(76) -- is used to fit m or mu; they are computed as integrals of the derived propagator with prescribed sources. The mu->infinity and mu=0 limits are consistency checks against the known perfect-conductor propagator [64] and Maxwell-Chern-Simons results [30]. Even where those references share authors with the present paper, they are external, previously established benchmarks rather than inputs to the derivation, and the present propagator is derived independently of those target limits. No parameter is fitted to the quantities being predicted, and no uniqueness theorem from prior same-author work is invoked to exclude alternatives. The 'AQUI' placeholders after Fig. 5 and Fig. 9 are editorial artifacts and do not alter this assessment. Whether the block-matrix inverse (33)-(36) is algebraically correct as printed -- e.g., whether the chi-bracket belongs in the numerator or denominator -- is a correctness and invertibility concern, not a circularity concern; an erroneous inverse would leave the strong-coupling claim unsupported, but that is the opposite of circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The model rests on two free parameters (m, mu) and a newly invented planar gauge field. The axioms are standard for this class of effective field theory, but the specific coupling form is ad hoc. No independent experimental handle is provided.

free parameters (2)
  • m (Chern-Simons mass)
    Introduced as a free parameter of the planar Maxwell-Chern-Simons sector. Not fitted to data in this paper; it can be adjusted to material properties.
  • mu (interfield coupling constant)
    Introduced ad hoc as the strength of the Chern-Simons-like coupling between the planar field and the photon field. Not fitted here.
assumptions (5)
  • domain assumption The planar Chern-Simons field A_bar obeys Maxwell-Chern-Simons dynamics with mass m.
    The model postulates this form, motivated by planar condensed matter systems; it is the core descriptive assumption.
  • domain assumption A Dirac delta function localizes the planar field and its couplings to the plane x3 = a.
    Used throughout Sec. II to model a thin material layer; standard idealization, but the paper does not justify its validity at the quantum level.
  • standard math Feynman gauge (alpha = beta = 1) is a legitimate gauge choice for both fields.
    Gauge-fixing parameters are set to 1 in Sec. II; standard practice.
  • standard math The standard Feynman i-epsilon prescription regularizes propagator poles.
    Invoked implicitly when inverting operators with poles at p_parallel^2 = m^2; not discussed in the paper.
  • ad hoc to paper The specific form of the Chern-Simons-like coupling with parameter mu is the correct minimal emulation of magnetoelectric boundaries.
    The coupling terms in Eq. (1) are chosen by the authors as the central new ingredient, without derivation from a microscopic model.
invented entities (1)
  • Planar Chern-Simons gauge field A_bar_mu
    purpose: Confined to the material layer to emulate the crystalline lattice structure and magnetoelectric properties of the boundary.
    The field is introduced as a new dynamical degree of freedom. It is not observed directly and produces no falsifiable prediction outside the model's own derived effects.

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Cite this review

Pith. "Pith review of A Two-Gauge Field Model for Magnetoelectric Boundaries." pith.science (2026). https://pith.science/paper/55TKTSBN

@misc{pith2026250821697,
  author       = {Pith},
  title        = {Pith review of: A Two-Gauge Field Model for Magnetoelectric Boundaries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/55TKTSBN}},
  note         = {Machine review of arXiv:2508.21697}
}
read the original abstract

This work introduces a field-theoretical model designed to simulate the presence of material layers with magnetoelectric properties. The model comprises the standard Maxwell field coupled to a Chern-Simons field confined to a planar layer. The electromagnetic behavior of the boundary is emulated through the interaction between the Chern-Simons and Maxwell fields, governed by two parameters: the Chern-Simons mass and the coupling constant between the fields. Both parameters can be adjusted to reflect the specific properties of different materials. We compute the exact propagator of the theory and employ it to investigate several physical properties. Our analysis focuses on phenomena that arise from the presence of external sources coupled to both the Maxwell and Chern-Simons fields, considering various scenarios. In the Chern-Simons sector, the sources emulate defects in the crystal lattice of the material layer. The main objective of this paper is to present the proposed model and to explore its behavior in the simple context of a single planar material interface. We also suggest possible extensions of the model to more general configurations.

Figures

Figures reproduced from arXiv: 2508.21697 by the authors.

Figure 1
Figure 1. FIG. 1: Energy (45), normalized by [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Force (47), normalized by [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Energy (50), normalized by [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Force (51), normalized by [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: displays the interaction energy given by Eq. (54), normalized by q1q2, as a function of R∥. FIG. 5: Energy (54), normalized by q1q2, as a function of R∥ with m = 1 and µ = 1 (solid line), µ = 10 (dashed line) and µ = 15 (dotted line). AQUI B. Point-Like Sources in Dist…
Figure 6
Figure 6. Figure 6: FIG. 6: Energy (59), normalized by [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Energy (59), normalized by [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Plot of (58), normalized by [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Plot of (58), normalized by [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Plot of (63), normalized by [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Plot of (63), normalized by [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: , we plot Eq. (68), normalized by ξV (1)V (2), as a function of R∥ for several values of the coupling parameter. FIG. 12: Plot of (68), normalized by ξV (1)V (2), as a function of R∥ with µ = 1 (solid line), µ = 3 (dashed line) and µ = 10 (dotted line). The interactio…
Figure 13
Figure 13. Figure 13: FIG. 13: Plot of the component of (69), normalized by [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]

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Works this paper leans on

96 extracted references · 78 canonical work pages

  1. [1]

    (63) We now present some remarks regarding the result above

    Time-like case For the time-like case, the interaction energy (62) becomes EV (1)V (2) (m, µ, R∥) = V (1)0V (2)0 4 Re n 2 R∥ (µ2/8 + im)H−1((µ2/8 + im)R∥) + (µ2/8 + im)2 h Y0((µ2/8 + im)R∥) + H−2((µ2/8 + im)R∥) io . (63) We now present some remarks regarding the result above. In the temporal case, the topological source given by (61) is equivalent to a Di...

  2. [2]

    ˆR∥ V(2)

    Space-like case In the spatial case, where V (1)0 = V (2)0 = 0, the interaction energy given by (62) becomes EV (1)V (2) (m, µ, R∥) = V(1).V(2) 4R∥ Re n (µ2/8 + im) h 2 H−1((µ2/8 + im)R∥) + 2Y1((µ2/8 + im)R∥) − (µ2/8 + im)R∥ H−2((µ2/8 + im)R∥) + Y0((µ2/8 + im)R∥) io + V(1). ˆR∥ V(2). ˆR∥ Re n (µ2/8 + im)2 × h H−2((µ2/8 + im)R∥) − Y2((µ2/8 + im)R∥) io , (6...

    2024

  3. [3]

    Zhang, Int

    S. Zhang, Int. J. Mod. Phys. B 06, 25 (1992)

    1992

  4. [4]

    Banerjee, P

    R. Banerjee, P. Mukherjee, Nucl. Phys. B 478, 235 (1996)

    1996

  5. [5]

    Banerjee, P

    R. Banerjee, P. Mukherjee, Prog. Theor. Phys. 101, 1189 (1999)

    1999

  6. [6]

    Frohlich and A

    J. Frohlich and A. Zee, Nucl. Phys. B 364, 517 (1991)

    1991

  7. [7]

    de Gracia and B.M

    G.B. de Gracia and B.M. Pimentel, Rev. Bras. Ens. Fis. 45, e20230042 (2023). 20

    2023

  8. [8]

    Hasan and C.L

    M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)

    2010

Show all 96 references
  1. [9]

    Kaufmann and D

    R.M. Kaufmann and D. Li, Rev. Math. Phys. 28, 1630003 (2016)

    2016

  2. [10]

    Shen, Topological Insulators; Dirac equation in condensed matter , Springer-Verlag, Berlim, (2012)

    S.Q. Shen, Topological Insulators; Dirac equation in condensed matter , Springer-Verlag, Berlim, (2012)

    2012

  3. [11]

    Marino, Quantum Field Theory Approach to Condensed Matter Physics , Cambridge University Press, (2017)

    E.C. Marino, Quantum Field Theory Approach to Condensed Matter Physics , Cambridge University Press, (2017)

    2017

  4. [12]

    Metlitski and Ashvin Vishwanath, Phys

    M.A. Metlitski and Ashvin Vishwanath, Phys. Rev. B 93, 245151 (2016)

    2016

  5. [13]

    Wang and T

    C. Wang and T. Senthil, Phys. Rev. X 5, 041031 (2015)

    2015

  6. [14]

    Santos, C.W

    R.A. Santos, C.W. Huang, Y. Gefen and D.B. Gutman, Phys. Rev. B 91, 205141 (2015)

    2015

  7. [15]

    Mechelen and Z

    T.V. Mechelen and Z. Jacob, Phys. Rev. B 102, 155425 (2020)

    2020

  8. [16]

    Chern and J

    S.S. Chern and J. Simons, Annals Math. 99, 48 (1974)

    1974

  9. [17]

    G.V. Dunne, Aspects of Chern-Simons theory, in: Les Houches Summer School in Theoretical Physics, Session 69: Topo- logical Aspects of Low-Dimensional Systems, Les Houches, France, (1998) [arXiv:hep-th/9902115]

    1998 arXiv

  10. [18]

    Oliveira, L.H.C

    H.L. Oliveira, L.H.C. Borges, F.E. Barone and F.A. Barone, Eur. Phys. J. C 81, 558 (2021)

    2021

  11. [19]

    Meier, M

    Q.N. Meier, M. Fechner, T. Nozaki, M. Sahashi, Z. Salman, T. Prokscha, A. Suter, P. Schoenherr, M. Lilienblum, P. Borisov, I. E. Dzyaloshinskii, M. Fiebig, H. Luetkens and N.A. Spaldin, Phys. Rev. X 9, 011011 (2019)

    2019

  12. [20]

    Milton and Y

    K.A. Milton and Y. Jack Ng, Phys. Rev. D 42, 2875 (1990)

    1990

  13. [21]

    Milton and Y

    Kimball A. Milton and Y. Jack Ng, Phys. Rev. D 46, 842 (1992)

    1992

  14. [22]

    Alves, E

    D.T. Alves, E. R. Granhen, J.F. Medeiros Neto and S. Perez, Phys. Lett. A 374, 2113 (2010)

    2010

  15. [23]

    T. -Y. Zheng, L. Zheng and C. -S. Li, Commun. Theor. Phys. 36 , 431 (2001)

    2001

  16. [24]

    Zheng, Phys

    T.-Y. Zheng, Phys. Lett. A 305, 337 (2002)

    2002

  17. [25]

    de Medeiros Neto, Rudnei O

    J.F. de Medeiros Neto, Rudnei O. Ramos and Carlos Rafael M. Santos, Phys. Rev. D 86, 125034 (2012)

    2012

  18. [26]

    Bordag and D.V

    M. Bordag and D.V. Vassilevich, Phys. Lett. A 268, 75 (2000)

    2000

  19. [27]

    Marachevsky, Phys

    V.N. Marachevsky, Phys. Rev. B 99, 075420 (2019)

    2019

  20. [28]

    Bashir, A

    A. Bashir, A. Raya, and S. S´ anchez Madrigal, J. Phys. A 41, 505401 (2008)

    2008

  21. [29]

    Concha-S´ anchez, A

    Y. Concha-S´ anchez, A. Raya, and M. E. Tejeda-Yeomans, Phys. Rev. D87, 035001 (2013)

    2013

  22. [30]

    2, 023B03 (2015)

    Yuichi Hoshino, Tomohiro Inagaki, Yuichi Mizutani, PTEP 2015, no. 2, 023B03 (2015)

    2015

  23. [31]

    V. N. Marachevsky, Y.M. Pis’mak, Phys. Rev. D 81, 065005 (2010)

    2010

  24. [32]

    Borges, F.E

    L.H.C. Borges, F.E. Barone, C.C.H. Ribeiro, H.L. Oliveira, R.L. Fernandez and F.A. Barone, Eur. Phys. J. C 80, 238 (2020)

    2020

  25. [33]

    Pis’mak, Yu.M

    D.Yu. Pis’mak, Yu.M. Pis’mak and F.J. Wegner, Phys. Rev. E 92, 013204 (2015)

    2015

  26. [34]

    Carroll, G.B

    S.M. Carroll, G.B. Field and R. Jackiw, Phys. Rev. D 41, 1231 (1990)

    1990

  27. [35]

    Ribeiro, P.D.S

    F.S. Ribeiro, P.D.S. Silva and M.M. Ferreira Jr., Physical Review D 107, 096018 (2023)

    2023

  28. [36]

    Belich, M.M

    H. Belich, M.M. Ferreira Jr., J.A. Helay¨ el-Neto, Eur. Phys. J. C 38, 511 (2005)

    2005

  29. [37]

    Soares, L.H

    D.M. Soares, L.H. Borges, G. Dallabona and L.C.T. Brito, Eur. Phys. J. Plus 139, 152 (2024)

    2024

  30. [38]

    Cremmer and J

    E. Cremmer and J. Scherk, Nucl. Phys. B 72, 117 (1974)

    1974

  31. [39]

    Allen, M.J

    T.J. Allen, M.J. Bowick and A. Lahiri, Mod. Phys. Lett. A 6, 559 (1991)

    1991

  32. [40]

    Belich, L.M

    H. Belich, L.M. Silva, J.A. Helay¨ el-Neto and A.E. Santana, Phys. Rev. D 84, 045007 (2011)

    2011

  33. [41]

    Barone, F.E

    F.A. Barone, F.E. Barone and J. A. Helay¨ el-Neto, Phys. Rev. D 84, 065026 (2011)

    2011

  34. [42]

    Barone, L.M

    F.A. Barone, L.M. de Moraes and J. A. Helay¨ el-Neto, Phys. Rev. D72, 105012 (2005); Erratum-ibid. D 73, 089901 (2006)

    2005

  35. [43]

    Deser and R

    S. Deser and R. Jackiw, Phys. Lett. B 451, 73 (1999)

    1999

  36. [44]

    Nascimento, Albert Yu

    Ricardo Avila, Jose R. Nascimento, Albert Yu. Petrov, Carlos M. Reyes and Marco Schreck, Phys. Rev. D 101, 055011 (2020)

    2020

  37. [45]

    Anacleto, F.A

    M.A. Anacleto, F.A. Brito, O. Holanda, E. Passos and A. Yu. Petrov, International Journal of Modern Physics A 31, 1650140 (2016)

    2016

  38. [46]

    Salgado-Rebolledo, G

    P. Salgado-Rebolledo, G. Palumbo and J.K. Pachos, Sci. Rep. 10, 21998 (2020)

    2020

  39. [47]

    Van Mechelen and Z

    T. Van Mechelen and Z. Jacob, Phys. Rev. B 102, 155425 (2020)

    2020

  40. [48]

    Borges, F.A

    L.H.C. Borges, F.A. Barone and H.L. Oliveira, Phys. Rev. D 105, 025008 (2022)

    2022

  41. [49]

    Milton, The Casimir Effect, Physical Manifestations of Zero-Point Energy, World Scientific, Singapore (2001)

    K.A. Milton, The Casimir Effect, Physical Manifestations of Zero-Point Energy, World Scientific, Singapore (2001)

    2001

  42. [50]

    Bordag, U

    M. Bordag, U. Mohideen, and V. M. Mostepanenko, Phys. Rep. 353, 1 (2001)

    2001

  43. [51]

    Bordag, D

    M. Bordag, D. Hennig, and D. Robaschik, J. Phys. A 25, 4483 (1992)

    1992

  44. [52]

    Milton, J

    Kimball A. Milton, J. Phys. A: Math. Gen. 37, 6391 (2004)

    2004

  45. [53]

    Bordag, K

    M. Bordag, K. Kirsten and D. Vassilevich, Phys. Rev. D 59, 085011 (1999)

    1999

  46. [54]

    Graham, R.L

    N. Graham, R.L. Jaffe, V. Khemani, M. Quandt, M. Scandurra and H. Weigel, Nucl. Phys. B 645, 49 (2002)

    2002

  47. [55]

    Graham, R.L

    N. Graham, R.L. Jaffe, V. Khemani, M. Quandt, M. Scandurra and H. Weigel, Phys. Lett. B 572, 196 (2003)

    2003

  48. [56]

    Sundberg and R.L

    P. Sundberg and R.L. Jaffe, Annals Phys. 309, 442 (2004)

    2004

  49. [57]

    Cavalcanti, [arXiv:hep-th/0201150] (2002)

    R.M. Cavalcanti, [arXiv:hep-th/0201150] (2002)

    2002 arXiv

  50. [58]

    Fosco, E.L

    C.D. Fosco, E.L. Losada, Phys. Rev. D 78, 025017 (2008)

    2008

  51. [59]

    Ccapa Ttira, C.D

    C. Ccapa Ttira, C.D. Fosco, E. Losada, Phys. Rev. D 82, 085008 (2010)

    2010

  52. [60]

    C. D. Fosco, F. C. Lombardo and F. D. Mazzitelli, Phys. Rev. D 85, 125037 (2012)

    2012

  53. [61]

    G. T. Camilo, F. A. Barone and F. E. Barone, Phys. Rev. D 87, 025011 (2013)

    2013

  54. [62]

    C. D. Fosco and , M. L. Remaggi, Eur. Phys. J. C 77, 155 (2017)

    2017

  55. [63]

    Parashar, K

    P. Parashar, K. Milton, K. V. Shajesh and M. Shaden, Phys. Rev. D 86, 085021 (2012)

    2012

  56. [64]

    Silva, A.N

    J.D.L. Silva, A.N. Braga and D.T. Alves, Phys. Rev. D 94, 105009 (2016)

    2016

  57. [65]

    Barone, L.H.C

    F.A. Barone, L.H.C. Borges, G. Flores-Hidalgo, H.L. Oliveira and W.Y.A. da Silva, Eur. Phys. J. C 84, 322 (2024)

    2024

  58. [66]

    Barone and F.E

    F.A. Barone and F.E. Barone, Phys. Rev. D 89, 065020 (2014). 21

    2014

  59. [67]

    Barone and F

    F.A. Barone and F. E. Barone, Eur. Phys. J. C 74, 3113 (2014)

    2014

  60. [68]

    Blasi, N

    A. Blasi, N. Maggiore, N Magnoli and S. Storace, Class. Quantum Grav. 27, 165018 (2010)

    2010

  61. [69]

    Mart ´ ın-Ruiz,M

    A. Mart ´ ın-Ruiz,M. Cambiaso and L.F. Urrutia, Phys. Rev. D92, 125015 (2015)

    2015

  62. [70]

    Mart ´ ın-Ruiz,M

    A. Mart ´ ın-Ruiz,M. Cambiaso and L.F. Urrutia, Phys. Rev. D93, 045022 (2015)

    2015

  63. [71]

    Mart ´ ın-Ruiz,M

    A. Mart ´ ın-Ruiz,M. Cambiaso and L.F. Urrutia, Phys. Rev. D94, 085019 (2016)

    2016

  64. [72]

    Mart ´ ın-Ruiz, M

    A. Mart ´ ın-Ruiz, M. Cambiaso and L.F. Urrutia, EPL113, 60005 (2016)

    2016

  65. [73]

    Palumbo and J.K

    G. Palumbo and J.K. Pachos, Phys. Rev. Lett. 110, 211603 (2013)

    2013

  66. [74]

    Hughes and Shou-Cheng Zhang, Phys

    Xiao-Liang Qi, T.L. Hughes and Shou-Cheng Zhang, Phys. Rev. B 78, 195424 (2008). Erratum Phys. Rev. B 81, 159901 (2010)

    2008

  67. [75]

    Plummer, E

    D.Z. Plummer, E. D’Alessandro, A. Burrowes, J. Fleischer, A.M. Heard and Y. Wu, J. Low Power Electron. Appl. 15(2) (2025)

    2025

  68. [76]

    Chen, H.T

    X. Chen, H.T. Lam and X. Ma, [arXiv:2211.10458] (2022)

    2022 arXiv

  69. [77]

    X. Ma, W. Shirley, M. Cheng, M. Levin, J. McGreevy and X. Chen, Phys. Rev. B 105, 195124 (2022)

    2022

  70. [78]

    Son, Prog

    D.T. Son, Prog. Theor. Exp. Phys. 2016, 12C103 (2016)

    2016

  71. [79]

    Moroz, A

    S. Moroz, A. Prem, V. Gurarie and L. Radzihovsky, Phys. Rev. B 95, 014508 (2017)

    2017

  72. [80]

    Sohal, L.H

    R. Sohal, L.H. Santos and E. Fradkin, Phys. Rev. B 97, 125131 (2018)

    2018

  73. [81]

    Palumbo, Ann

    G. Palumbo, Ann. Phys. 386, 15 (2017)

    2017

  74. [82]

    Nayak, S.H

    C. Nayak, S.H. Simon, A. Stern, M. Freedman and S.D. Sarma, Rev. Mod. Phys. 80, 1083 (2008)

    2008

  75. [83]

    Hansson, V

    T.H. Hansson, V. Oganesyan and S.L. Sondhi, 313, 497 (2004)

    2004

  76. [84]

    L. H. C. Borges, A.F. Ferrari and F.A. Barone, Nucl. Phys. B 954, 114974 (2020)

    2020

  77. [85]

    Borges and F.A

    L.H.C. Borges and F.A. Barone, Eur. Phys. J. C 77, 693 (2017)

    2017

  78. [86]

    Borges and F.A

    L.H.C. Borges and F.A. Barone, Braz. J. Phys. 50, 647 (2020)

    2020

  79. [87]

    Barone and A.A

    F.A. Barone and A.A. Nogueira, Eur. Phys. J. C 75, 339 (2015)

    2015

  80. [88]

    Barone, G

    F.A. Barone, G. Flores-Hidalgo and A.A. Nogueira, Phys. Rev. D 88, 105031 (2013)

    2013

  81. [89]

    Borges, F.A

    L.H.C. Borges, F.A. Barone and J. A. Helayel-Neto, Eur. Phys. J. C 74, 2937 (2014)

    2014

  82. [90]

    Borges, A.F

    L.H.C. Borges, A.F. Ferrari and F.A. Barone, Eur. Phys. J. C 76, 599 (2016)

    2016

  83. [91]

    Borges and F.A

    L.H.C. Borges and F.A. Barone, Braz. J. Phys. 49, 571-582 (2019)

    2019

  84. [92]

    Barone and G

    F.A. Barone and G. Flores-Hidalgo, Phys. Rev. D 78, 125003 (2008)

    2008

  85. [93]

    Arfken and H.J

    G.B. Arfken and H.J. Weber, Mathematical Methods for Physicists , Academic Press (1995)

    1995

  86. [94]

    Gradshteyn and I.M

    I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products , Academic Press (2000)

    2000

  87. [95]

    Zee, Quantum Field Theory in a Nutshell, Princeton University Press, (2003)

    A. Zee, Quantum Field Theory in a Nutshell, Princeton University Press, (2003)

    2003

  88. [96]

    Medeiros, F.E

    M.F.X.P. Medeiros, F.E. Barone, F.A. Barone, Eur. Phys. J. C 78, 12 (2018)

    2018

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