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arxiv: 2604.23673 · v1 · submitted 2026-04-26 · 🪐 quant-ph · cond-mat.mes-hall

Entanglement (1+2) QED in a double layer of Dirac Materials

Facundo Arreyes , Federico Escudero , Ari\'an Gorza , Sebasti\'an Ardenghi This is my paper

Pith reviewed 2026-05-08 06:09 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hall
keywords entanglement entropyDirac quasiparticlesdouble-layer honeycomb latticeBethe-Salpeter equationvon Neumann entropyself-energy dressingmomentum-space entanglementcavity QED
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The pith

Phenomenological self-energy dressing enhances entanglement entropy between Dirac quasiparticles in a double-layer cavity system when coherence time exceeds photon travel time.

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

The paper studies momentum-space entanglement between two massive Dirac fermions placed in separate layers of a honeycomb lattice and coupled through a planar electromagnetic cavity. It derives the two-body bound-state equation in the ladder approximation, applies a Born expansion around the free quasiparticle state, and extracts the von Neumann entropy from the reduced sublattice density matrix. In the perturbatively controlled regime the entropy stays small, yet introducing phenomenological self-energy dressing produces a marked crossover to large entanglement values. Stationary entanglement persists only when the quasiparticle coherence lifetime is longer than the time required for a cavity photon to travel between the layers. The maximum-entropy regime is presented as a practical route to Bell-like states.

Core claim

Within the perturbatively controlled regime the entropy remains small, while phenomenological self-energy dressing drives a crossover to strong enhancement of the entanglement entropy. Stationary entanglement is obtained only when the quasiparticle coherence time exceeds the photon propagation time between the layers.

What carries the argument

Ladder-approximated Bethe-Salpeter equation for the two-body state of massive Dirac fermions, from which the reduced sublattice density matrix is formed to yield the momentum-resolved von Neumann entropy.

If this is right

  • Within the perturbatively controlled regime the entanglement entropy remains small.
  • Phenomenological self-energy dressing produces a crossover to strong enhancement of the entanglement entropy.
  • Stationary entanglement appears only when quasiparticle coherence time exceeds photon propagation time between the layers.
  • The maximum-entropy regime supplies a viable route to Bell-like states.

Where Pith is reading between the lines

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

  • Engineering longer coherence times in layered Dirac structures could make stationary entanglement experimentally accessible.
  • The same combination of virtual photon exchange and spinor geometry may affect entanglement in other two-dimensional materials with similar cavity coupling.
  • Layer separation could serve as a tunable parameter that controls the crossover between weak and strong entanglement regimes.

Load-bearing premise

The ladder approximation together with a Born-level treatment around a free two-body state remains valid after a phenomenological self-energy dressing is added without a self-consistent recalculation.

What would settle it

A measurement showing that the momentum-resolved von Neumann entropy fails to increase under self-energy dressing, or that stationary entanglement disappears whenever the measured coherence time is shorter than the calculated inter-layer photon travel time.

Figures

Figures reproduced from arXiv: 2604.23673 by Ari\'an Gorza, Facundo Arreyes, Federico Escudero, Sebasti\'an Ardenghi.

Figure 1
Figure 1. Figure 1: Schematic representation of the theoretical setup view at source ↗
Figure 3
Figure 3. Figure 3: Entanglement entropy S1 as a function of the inter-layer distance d1 along the symmetric configuration d1+ d2 = L for fixed (Σ1, Σ2) and fixed kinematic configuration (p1, p2, ϕ1, ϕ2). Here we use Re Σ1 = Re Σ2 = 4.2 × 10−3 eV, chosen from view at source ↗
Figure 2
Figure 2. Figure 2: Top: Entanglement entropy as a function of the inter-layer distances d1 and d2 for different values of the mode cutoff Nmax. Bottom: one-dimensional cuts along the symmetric configuration d1 + d2 = L. All panels use λso = 3.9 meV, corresponding to m ≃ 2.13 eV in the numeri￾cal convention. We begin by validating the proposed method through the reproduction of the results obtained in our previous work. As shown in view at source ↗
Figure 4
Figure 4. Figure 4: Entanglement entropy S1 as a function of the real parts of the self-energy parameters Σ1 and Σ2 (logarithmic axes, in eV), for fixed quasiparticle momenta p1 = 0.13 eV, p2 = 0.12 eV, collinear angles ϕ1 = ϕ2 = 0, and emitter positions d1 = 0.9, d2 = 1.1 eV−1 inside the cavity. The region to the left of it lies outside the perturbative regime. Results in that region are shown for completeness but should not… view at source ↗
Figure 5
Figure 5. Figure 5: Entanglement entropy as a function of coherence view at source ↗
Figure 6
Figure 6. Figure 6: Entanglement entropy S1 as a function of the quasi￾particle momenta p1 and p2 (in eV), for fixed collinear angles ϕ1 = ϕ2 = 0 and emitter positions d1 = 0.9, d2 = 1.1 eV−1 inside the cavity. Top: full 2D map; the red contour marks the Born validity boundary (∥Ψ (1)∥ = ∥Ψ (0)∥), beyond which results lie outside the perturbative regime. Bottom: 1D cuts at fixed p1 = p mid and p2 = p mid, where p mid = 0.10 e… view at source ↗
read the original abstract

We investigate the momentum-space entanglement between two Dirac quasiparticles in a double-layer honeycomb lattice coupled via a planar electromagnetic cavity. We model the low-energy excitations as massive Dirac fermions in $(1+2)$ dimensions and derive the Bethe-Salpeter equation using the ladder approximation. We use a Born-level approximation around a free two-body quasiparticle state, where the interaction is mediated by the cavity photon propagator. From the reduced sublattice density matrix, we compute a momentum-resolved von Neumann entropy. Within the perturbatively controlled regime, the entropy remains small, while phenomenological self-energy dressing drives a crossover to strong enhancement of the entanglement entropy. Stationary entanglement is obtained only when the quasiparticle coherence time exceeds the photon propagation time between the layers. The maximum-entropy regime appears to be a viable method for achieving Bell-like states. These results demonstrate how self-energy renormalization, virtual particle exchange, and spinor geometry combine to reshape the entanglement landscape of Dirac materials.

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

1 major / 2 minor

Summary. The manuscript investigates momentum-space entanglement between two Dirac quasiparticles in a double-layer honeycomb lattice coupled via a planar electromagnetic cavity. Low-energy excitations are modeled as massive Dirac fermions in (1+2) dimensions. The Bethe-Salpeter equation is derived in the ladder approximation, with a Born-level treatment around a free two-body quasiparticle state where the interaction is mediated by the cavity photon propagator. From the reduced sublattice density matrix, a momentum-resolved von Neumann entropy is computed. The central results are that the entropy remains small within the perturbatively controlled regime, while phenomenological self-energy dressing induces a crossover to strong enhancement; stationary entanglement occurs only when the quasiparticle coherence time exceeds the photon propagation time between layers, and the maximum-entropy regime is proposed as a route to Bell-like states.

Significance. If the results hold under controlled approximations, the work illustrates how self-energy renormalization, virtual particle exchange, and Dirac spinor geometry can reshape entanglement in cavity-coupled Dirac materials. This could inform strategies for generating stationary entangled states in 2D material platforms, bridging cavity QED techniques with quantum information measures in condensed-matter systems.

major comments (1)
  1. [Abstract] The abstract states that 'phenomenological self-energy dressing drives a crossover to strong enhancement of the entanglement entropy' while 'within the perturbatively controlled regime, the entropy remains small.' However, the ladder approximation plus Born-level treatment around a free two-body state is perturbative by construction. No derivation is provided showing that the dressing can be introduced self-consistently from the Bethe-Salpeter equation or that it preserves the small-coupling expansion without effectively resumming higher-order terms, which directly affects the reliability of the reported entropy crossover and stationary-entanglement condition.
minor comments (2)
  1. The notation '(1+2) QED' in the title and abstract could be clarified for consistency with standard (2+1)-dimensional QED terminology used in the literature on Dirac materials.
  2. The manuscript would benefit from explicit statements of the parameter regime (e.g., coupling strength, layer separation) where the Born approximation around the free two-body state is expected to hold, including any estimates of neglected diagrams.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments on our manuscript. We appreciate the opportunity to clarify the scope and limitations of our perturbative treatment and the phenomenological aspects of the self-energy dressing. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] The abstract states that 'phenomenological self-energy dressing drives a crossover to strong enhancement of the entanglement entropy' while 'within the perturbatively controlled regime, the entropy remains small.' However, the ladder approximation plus Born-level treatment around a free two-body state is perturbative by construction. No derivation is provided showing that the dressing can be introduced self-consistently from the Bethe-Salpeter equation or that it preserves the small-coupling expansion without effectively resumming higher-order terms, which directly affects the reliability of the reported entropy crossover and stationary-entanglement condition.

    Authors: We agree that the ladder approximation combined with the Born-level treatment around a free two-body state is perturbative by construction, and the manuscript does not provide a self-consistent derivation of the self-energy dressing from the Bethe-Salpeter equation. The dressing is introduced phenomenologically to model quasiparticle renormalization and finite coherence effects arising from interactions outside the strict ladder approximation, as is common in exploratory studies of cavity-coupled Dirac systems. We do not claim that this preserves the small-coupling expansion or avoids resummation; the crossover to enhanced entropy is presented as an illustrative result within this extended model. The stationary-entanglement condition (coherence time exceeding interlayer photon propagation time) follows directly from the photon propagator and reduced density matrix construction and remains valid within the stated regime. To address the concern, we will revise the abstract to more explicitly qualify the dressing as phenomenological and add a paragraph in the discussion section outlining the limitations and the need for future non-perturbative treatments. This clarification will better frame the reliability of the reported crossover without altering the core calculations. revision: partial

Circularity Check

1 steps flagged

Phenomenological self-energy dressing drives the crossover to strong entanglement enhancement, reducing the central result to an inserted input rather than a derivation from the ladder/Bethe-Salpeter setup.

specific steps
  1. fitted input called prediction [Abstract]
    "Within the perturbatively controlled regime, the entropy remains small, while phenomenological self-energy dressing drives a crossover to strong enhancement of the entanglement entropy. Stationary entanglement is obtained only when the quasiparticle coherence time exceeds the photon propagation time between the layers."

    The small-entropy result follows from the ladder + Born setup, but the reported strong enhancement and stationary regime are not outputs of that setup; they are produced by inserting the phenomenological dressing as an external driver. The crossover magnitude is therefore chosen rather than computed from the cavity propagator or free-state assumptions, rendering the headline claim dependent on the input parameter.

full rationale

The paper derives the Bethe-Salpeter equation in the ladder approximation with Born-level treatment around a free two-body state using the cavity photon propagator, yielding small entropy in the perturbatively controlled regime. However, the headline claims of strong enhancement, crossover, and stationary entanglement (when coherence time exceeds propagation time) are explicitly attributed to an additional phenomenological self-energy dressing whose magnitude and form are not derived self-consistently from the same equations or shown to preserve perturbative control. This makes the key prediction equivalent to the choice of that dressing term.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on two standard QED approximations plus one phenomenological addition whose magnitude is not fixed by the model equations.

free parameters (1)
  • phenomenological self-energy dressing strength
    Introduced to produce the crossover from small to strong entanglement entropy; its value is not derived from the cavity photon propagator or the free two-body state.
axioms (2)
  • domain assumption Ladder approximation is sufficient for the Bethe-Salpeter kernel
    Invoked to close the equation for the two-particle amplitude without higher-order diagrams.
  • domain assumption Born-level treatment around the free two-body quasiparticle state
    Used to linearize the interaction mediated by the cavity photon propagator.

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Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    =−ζ 2X n sin nπd1 L sin nπd2 L × Z d3q (2π)3 e−iq·(x1−x′

    work page

  2. [2]

    The cavity con- finement enters through the discrete transverse momen- tumq z =nπ/L, which shifts the photon denominator by (nπ/L)2

    gµν q2 − nπ L 2 + iϵ ,(11) where we integrate over the off-shell three-momentum qµ = (q 0,q) withq 2 ≡(q 0)2 − |q| 2. The cavity con- finement enters through the discrete transverse momen- tumq z =nπ/L, which shifts the photon denominator by (nπ/L)2. III. TWO-BODY QED FRAMEWORK To investigate the momentum-space entanglement properties of the two-body quas...

    work page

  3. [3]

    The eigenvalues are computed and renormalized by their sum to restore the unit-trace condition

    to remove floating-point asymmetry. The eigenvalues are computed and renormalized by their sum to restore the unit-trace condition. This procedure yields the entanglement pro- fileS 1(p1, p2, ϕ1, ϕ2) over the full momentum grid, form- ing the basis of the analysis presented in Sec. IV; the full numerical pipeline is summarized in Appendix B. A momentum-in...

    work page

  4. [4]

    Reznik, Entanglement from the vacuum , Found

    B. Reznik, Entanglement from the vacuum, Found. Phys. 33, 167 (2003), quant-ph/0212044

    work page arXiv 2003

  5. [5]

    C. H. Bennett, G. Brassard, C. Cr´ epeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and einstein-podolsky- rosen channels, Phys. Rev. Lett.70, 1895 (1993)

    work page 1993

  6. [6]

    A. K. Ekert, Quantum cryptography based on bell’s the- orem, Phys. Rev. Lett.67, 661 (1991)

    work page 1991

  7. [7]

    Jozsa and N

    R. Jozsa and N. Linden, On the role of entanglement in quantum-computational speed-up, Proceedings of the Royal Society of London. Series A: Mathematical, Phys- ical and Engineering Sciences459, 2011 (2003)

    work page 2011

  8. [8]

    Violating Bell's inequalities in the vacuum

    B. Reznik, A. Retzker, and J. Silman, Violating bell’s in- equalities in the vacuum, Phys. Rev. A71, 042104 (2005), quant-ph/0310058

    work page Pith review arXiv 2005

  9. [9]

    Valentini, Non-local correlations in quantum electro- dynamics, Phys

    A. Valentini, Non-local correlations in quantum electro- dynamics, Phys. Lett. A153, 321 (1991)

    work page 1991

  10. [10]

    Quantum entanglement

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009), quant-ph/0702225

    work page Pith review arXiv 2009

  11. [11]

    T. R. Perche, C. Lima, and E. Mart´ ın-Mart´ ınez, Har- vesting entanglement from complex scalar and fermionic fields with linearly coupled particle detectors, Phys. Rev. D105, 065016 (2022), 2111.12779

    work page arXiv 2022

  12. [12]

    Pozas-Kerstjens and E

    A. Pozas-Kerstjens and E. Mart´ ın-Mart´ ınez, Harvesting correlations from the quantum vacuum, Phys. Rev. D92, 064042 (2015), 1506.03081. 10

    work page arXiv 2015

  13. [13]

    Dragoman and M

    D. Dragoman and M. Dragoman, Quantum logic gates based on ballistic transport in graphene, J. Appl. Phys. 119, 094902 (2016)

    work page 2016

  14. [14]

    Asjad, M

    M. Asjad, M. Qasymeh, and H. Eleuch, Continuous- variable quantum teleportation using a microwave- enabled plasmonic graphene waveguide, Phys. Rev. Ap- plied16, 034046 (2021)

    work page 2021

  15. [15]

    Asjad, M

    M. Asjad, M. Qasymeh, and H. Eleuch, A local area quantum teleportation communication network based on an array of electrically activated graphene plasmonic waveguides, Opt. Express30, 21016 (2022)

    work page 2022

  16. [16]

    G. Y. Wu and N.-Y. Lue, Graphene-based qubits in quan- tum communications, Phys. Rev. B86, 045456 (2012)

    work page 2012

  17. [17]

    A.-B. A. Mohamed, A.-H. Abdel-Aty, M. Qasymeh, and H. Eleuch, Non-local correlation dynamics in two- dimensional graphene, Sci. Rep.12, 3581 (2022)

    work page 2022

  18. [18]

    Cayssol, Introduction to Dirac materials and topologi- cal insulators, Comptes Rendus

    J. Cayssol, Introduction to Dirac materials and topologi- cal insulators, Comptes Rendus. Physique14, 760 (2013)

    work page 2013

  19. [19]

    Alonso Calafell, J

    I. Alonso Calafell, J. D. Cox, M. Radonji´ c, J. R. M. Saavedra, F. J. Garc´ ıa de Abajo, L. A. Rozema, and P. Walther, Quantum computing with graphene plas- mons, npj Quantum Information5, 37 (2019)

    work page 2019

  20. [20]

    A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys.81, 109 (2009)

    work page 2009

  21. [21]

    M. E. Peskin and D. V. Schroeder,An Introduction to Quantum Field Theory(CRC Press, Boca Raton, FL,

    work page

  22. [22]

    E. C. Marino, L. O. Nascimento, V. S. Alves, and C. M. Smith, Interaction induced quantum valley hall effect in graphene, Phys. Rev. X5, 011040 (2015)

    work page 2015

  23. [23]

    Lamata, J

    L. Lamata, J. Le´ on, and E. Solano, Entanglement be- tween two interacting atoms in a one-dimensional har- monic trap, Phys. Rev. A73, 012335 (2006)

    work page 2006

  24. [24]

    J. K. Pachos and E. Solano, Generation and degree of entanglement in a relativistic formulation, Quantum Inf. Comput.3, 115 (2003), quant-ph/0203065

    work page arXiv 2003

  25. [25]

    C. L. Kane and E. J. Mele, Quantum spin hall effect in graphene, Phys. Rev. Lett.95, 226801 (2005)

    work page 2005

  26. [26]

    T. C. H. Liew, I. A. Shelykh, and G. Malpuech, Polari- tonic devices, Physica E43, 1543 (2011)

    work page 2011

  27. [27]

    Keller, G

    J. Keller, G. Scalari, M. Gasparinetti, D. Hagenm¨ uller, M. Beck, and J. Faist, Few-electron ultrastrong light- matter coupling at 300 ghz with nanogap hybrid lc mi- crocavities, Nano Lett.17, 7410 (2017)

    work page 2017

  28. [28]

    Byrnes, N

    T. Byrnes, N. Y. Kim, and Y. Yamamoto, Exciton- polariton condensates, Nat. Phys.10, 803 (2014)

    work page 2014

  29. [29]

    Y. Sun, Y. Yoon, M. Steger, G. Liu, L. N. Pfeiffer, K. West, D. W. Snoke, and K. A. Nelson, Direct measure- ment of polariton–polariton interaction strength, Nat. Phys.13, 870 (2017)

    work page 2017

  30. [30]

    Arreyes, F

    F. Arreyes, F. Escudero, and J. S. Ardenghi, Correla- tions in twisted double-layer graphene with virtual pho- tons in a microcavity, J. Phys.: Condens. Matter34, 115602 (2022)

    work page 2022

  31. [31]

    Arreyes, F

    F. Arreyes, F. Escudero, J. S. Ardenghi, and A. Juan, Entanglement harvesting in buckled honeycomb lattices by vacuum fluctuations in a microcavity, J. Phys.: Con- dens. Matter36, 505501 (2024)

    work page 2024

  32. [32]

    M. J. Spencer and T. Morishita,Silicene: Structure, Properties and Applications(Springer, Cham, 2016)

    work page 2016

  33. [33]

    Zhang, J

    Y.-Y. Zhang, J. Pan, S. Du, and H.-J. Gao, Two- dimensional crystals: Graphene, silicene, germanene, and stanene, inSpringer Handbook of Surface Science, edited by M. Rocca, T. S. Rahman, and L. Vattuone (Springer International Publishing, Cham, 2020) pp. 243–266

    work page 2020

  34. [34]

    Bianco, S

    E. Bianco, S. Butler, S. Jiang, O. D. Restrepo, W. Windl, and J. E. Goldberger, Stability and exfoliation of ger- manane: A germanium graphane analogue, ACS Nano 7, 4414 (2013)

    work page 2013

  35. [35]

    Rold´ an, L

    R. Rold´ an, L. Chirolli, E. Prada, J. A. Silva-Guill´ en, P. S. Jos´ e, and F. Guinea, Theory of 2d crystals: graphene and beyond, Chem. Soc. Rev.46, 4387 (2017)

    work page 2017

  36. [36]

    J. S. Ardenghi, Entanglement harvesting in double-layer graphene by vacuum fluctuations in a microcavity, Phys. Rev. D98, 045006 (2018), 1808.03990

    work page arXiv 2018

  37. [37]

    J. S. Ardenghi, Entanglement entropy between real and virtual particles inϕ 4 quantum field theory, Phys. Rev. D91, 085006 (2015), 1504.01315

    work page arXiv 2015

  38. [38]

    J. S. Ardenghi, Entanglement entropy between virtual and real excitations in quantum electrodynamics, Int. J. Mod. Phys. A33, 1850081 (2018)

    work page 2018

  39. [39]

    C.-C. Liu, H. Jiang, and Y. Yao, Low-energy effective hamiltonian involving spin–orbit coupling in silicene and two-dimensional germanium and tin, Phys. Rev. B84, 195430 (2011)

    work page 2011

  40. [40]

    Y. Yao, F. Ye, X.-L. Qi, S.-C. Zhang, and Z. Fang, Spin–orbit gap of graphene: First-principles calculations, Phys. Rev. B75, 041401(R) (2007)

    work page 2007

  41. [41]

    Ezawa, Topological insulator and helical zero mode in silicene under inhomogeneous electric field, New J

    M. Ezawa, Topological insulator and helical zero mode in silicene under inhomogeneous electric field, New J. Phys. 14, 033003 (2012)

    work page 2012

  42. [42]

    Greiner and J

    W. Greiner and J. Reinhardt,Quantum Electrodynamics (Springer, Berlin, 2009) chap. 11

    work page 2009

  43. [43]

    von Neumann,Mathematical Foundations of Quantum Mechanics(Princeton University Press, Princeton, NJ,

    J. von Neumann,Mathematical Foundations of Quantum Mechanics(Princeton University Press, Princeton, NJ,

    work page

  44. [44]

    Zhang, W

    Z. Zhang, W. Hu, E. Perfetto, and G. Stefanucci, Non-hermitian bethe-salpeter equation for open systems: Emergence of exceptional points in excitonic spectra from first principles, arXiv , 2510.09386 (2025), 2510.09386

    work page arXiv 2025

  45. [45]

    Kulig, T

    M. Kulig, T. Mas lowski, K. A. Kouzakov, V. K. Dugaev, P. Kurashvili, S. Wolski, M. Inglot, C. Jasiukiewicz, and L. Chotorlishvili, Fermionic entanglement in altermag- nets, Phys. Rev. B111, 184427 (2025)

    work page 2025

  46. [46]

    Kristinsson, O

    K. Kristinsson, O. V. Kibis, S. Morina, and I. A. She- lykh, Control of electronic transport in graphene by elec- tromagnetic dressing, Sci. Rep.6, 20082 (2016)

    work page 2016

  47. [47]

    Parhizgar, H

    F. Parhizgar, H. Rostami, and R. Asgari, Indirect ex- change interaction between magnetic adatoms in a mono- layer mos2, Phys. Rev. B87, 125401 (2013)

    work page 2013

  48. [48]

    Zare, Thermoelectric transport properties of boro- phane, Phys

    M. Zare, Thermoelectric transport properties of boro- phane, Phys. Rev. B99, 235413 (2019)

    work page 2019

  49. [49]

    O. V. Kibis, Metal-insulator transition in graphene in- duced by circularly polarized photons, Phys. Rev. B81, 165433 (2010)

    work page 2010

  50. [50]

    A. V. Kotikov and S. Teber, Two-loop fermion self-energy in reduced quantum electrodynamics and application to the ultrarelativistic limit of graphene, Phys. Rev. D89, 065038 (2014). 11 Appendix A: Bethe–Salpeter Kernel Derivation In Appendix A1, we derive the effective Bethe-Salpeter kernel in momentum space; in Appendices A3 and A4, we perform the frequ...

    work page 2014