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arxiv: 2512.00845 · v2 · pith:XJTJTJ5Unew · submitted 2025-11-30 · ❄️ cond-mat.mes-hall

Screened topological plasmons in graphene plasmonic crystals

Andr\'e Oct\'avio Soares , Christos Tserkezis , N. M. R. Peres This is my paper

Pith reviewed 2026-05-21 18:27 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords topological plasmonsgraphene plasmonicsplasmonic crystalsscreened plasmonstopological bandsedge statesgeometric phaseband structure
0
0 comments X

The pith

Periodically modulated graphene on a metal substrate forms a plasmonic crystal with nontrivial topological bands and protected edge states.

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

The paper develops a theory for quantizing screened plasmons in lossless graphene using a Drude conductivity model. This allows analysis of the band structure in a one-dimensional crystal created by periodic modulation of the graphene sheet placed on a metallic substrate. The resulting bands are shown to be topologically nontrivial, featuring a quantized geometric phase. In finite open systems, edge states appear in the band gap, and these states merge with bulk states through a topological phase transition as the modulation strength increases. This provides a framework for engineering topological properties in two-dimensional materials through external modulation.

Core claim

By developing the quantization theory for screened plasmons in a periodically modulated graphene sheet on a metallic substrate, the work shows that the plasmonic crystal sustains nontrivial topological bands with quantized geometric phase, and that finite systems exhibit edge states in the band gap which undergo a topological phase transition and merge with bulk states as modulation increases.

What carries the argument

The quantization of screened plasmons based on the Drude conductivity model for lossless graphene, which determines the band structure and enables computation of the geometric phase.

If this is right

  • The crystal supports nontrivial topological bands characterized by a quantized geometric phase.
  • Edge states emerge within the band gap in finite, open systems.
  • These edge states merge with bulk states as the modulation strength increases, marking a topological phase transition.
  • This approach extends possibilities for engineering two-dimensional materials with external modulation.

Where Pith is reading between the lines

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

  • This framework could apply to other modulated 2D materials to induce similar topological plasmon effects.
  • Experimental verification might focus on optical measurements of the band gap and edge state signatures in fabricated structures.
  • Connections to topological photonics suggest potential for robust plasmonic waveguides immune to certain defects.

Load-bearing premise

The graphene is assumed to be lossless and described by a simple Drude conductivity without considering losses or more complex conductivity models.

What would settle it

Fabricating a modulated graphene structure on a metal substrate and observing the predicted edge states via near-field optical microscopy or similar technique, or measuring a quantized geometric phase through interference experiments.

Figures

Figures reproduced from arXiv: 2512.00845 by Andr\'e Oct\'avio Soares, Christos Tserkezis, N. M. R. Peres.

Figure 1
Figure 1. Figure 1: FIG. 1. Modulation of the Fermi energy level in space, across [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Spectrum of the Hamiltonian given in Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Profile of the particle-component (first entry in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Modulation profile of a single supercell. It has [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Evolution of the spectrum in a finite plasmonic [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

We study topological effects in an one-dimensional plasmonic crystal formed by the screened plasmons emerging in a periodically modulated graphene sheet, placed on top of a metallic substrate. To this end, we develop the theory of quantization of screened plasmons, as appropriate for lossless graphene described by a Drude conductivity. By analyzing the resulting band structure, we show that the crystal sustains nontrivial topological bands, with quantized geometric phase. We further show that in a finite, open system, edge states appear within the band gap, which undergo a topological phase transition and merge with bulk states as the modulation increases. Our work provides a robust theoretical framework for the study of band structure and topology of layered media, and extends the possibilities for engineering two-dimensional materials with external modulation.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript develops a quantization theory for screened plasmons in a one-dimensional plasmonic crystal formed by a periodically modulated graphene sheet on a metallic substrate, assuming lossless graphene described by a Drude conductivity. Analysis of the resulting band structure shows nontrivial topological bands with quantized geometric phase. In finite open systems, edge states appear within the band gap and undergo a topological phase transition, merging with bulk states as the modulation strength increases. The work claims to provide a robust theoretical framework for band structure and topology in layered media.

Significance. If the central claims hold under the stated assumptions, the paper extends topological band theory to screened plasmons in modulated graphene, offering a framework for engineering topological features in 2D plasmonic systems via external modulation. This could inform design of devices exploiting edge states in plasmonic crystals.

major comments (2)
  1. [Theoretical framework] Theoretical framework section: The quantization of screened plasmons and the subsequent Hermitian eigenvalue problem rest on the assumption of lossless graphene with purely real Drude conductivity. When realistic damping is included, the conductivity acquires an imaginary component, the operator becomes non-Hermitian, and neither the quantization of the geometric phase nor the standard bulk-boundary correspondence for the reported edge states is guaranteed to survive. The manuscript provides no perturbation analysis or demonstration that the quantized phase and edge-state merging persist under small losses.
  2. [Band structure analysis] Band structure analysis and finite-system results: The claims of quantized geometric phase in nontrivial bands and the topological phase transition of edge states as modulation increases are load-bearing for the central topological conclusions. These features are derived under the Hermitian lossless model; without explicit checks against non-Hermitian perturbations, the robustness of the reported quantization and merging behavior remains unverified.
minor comments (2)
  1. [Abstract] The abstract and introduction could more explicitly state the topological invariant (e.g., Zak phase) used to quantify the geometric phase.
  2. [Figures] Figure captions for band-structure plots should indicate the specific modulation values at which the edge-state merging occurs.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address each of the major comments below, clarifying the assumptions and scope of our theoretical study.

read point-by-point responses

  1. Referee: [Theoretical framework] Theoretical framework section: The quantization of screened plasmons and the subsequent Hermitian eigenvalue problem rest on the assumption of lossless graphene with purely real Drude conductivity. When realistic damping is included, the conductivity acquires an imaginary component, the operator becomes non-Hermitian, and neither the quantization of the geometric phase nor the standard bulk-boundary correspondence for the reported edge states is guaranteed to survive. The manuscript provides no perturbation analysis or demonstration that the quantized phase and edge-state merging persist under small losses.

    Authors: We appreciate the referee pointing out the implications of including realistic damping. As stated in the abstract and throughout the manuscript, our work develops the quantization theory specifically for lossless graphene described by a real Drude conductivity, resulting in a Hermitian problem. This framework allows us to demonstrate the existence of nontrivial topological bands with quantized geometric phase and the appearance of edge states in finite systems. We acknowledge that with damping, the system becomes non-Hermitian, and the precise quantization and bulk-boundary correspondence may require a different treatment, such as in non-Hermitian topological physics. Since our study focuses on the ideal case to reveal these effects clearly, we did not include a perturbation analysis. We will revise the manuscript to add a short discussion in the conclusion section acknowledging this limitation and noting that the lossless case provides a baseline for understanding potential topological plasmonic phenomena. revision: partial

  2. Referee: [Band structure analysis] Band structure analysis and finite-system results: The claims of quantized geometric phase in nontrivial bands and the topological phase transition of edge states as modulation increases are load-bearing for the central topological conclusions. These features are derived under the Hermitian lossless model; without explicit checks against non-Hermitian perturbations, the robustness of the reported quantization and merging behavior remains unverified.

    Authors: The band structure and finite-system calculations are performed within the Hermitian model as described. The quantization of the geometric phase is shown through the standard Berry phase calculation for the bands, and the edge states are identified in the gap with the merging behavior as modulation strength increases. We agree that without explicit non-Hermitian checks, the robustness is not verified in the manuscript. To strengthen the presentation, we will include a brief remark on the expected persistence for weak damping, drawing from the fact that small perturbations often preserve topological features approximately in the Hermitian limit. revision: partial

standing simulated objections not resolved
  • Full perturbation analysis or numerical demonstration of the topological features under finite damping

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from standard Drude model to band topology

full rationale

The paper develops a quantization theory for screened plasmons starting from the lossless Drude conductivity of graphene, constructs the resulting band structure of the plasmonic crystal, and computes the geometric phase and edge states from that structure. This is a standard forward derivation from an explicit physical model (Drude conductivity) to eigenvalue spectra and topological invariants, with no reduction of the reported quantization or phase transition to a fitted parameter, self-citation chain, or redefinition of the target quantities. The central claims remain independent of the inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the Drude conductivity model for lossless graphene and the assumption of perfect screening by the metallic substrate; these are standard domain assumptions rather than new postulates.

free parameters (1)
  • modulation strength
    The amplitude of the periodic modulation is varied to induce the topological phase transition and edge-state merging.
axioms (2)
  • domain assumption Lossless graphene described by Drude conductivity
    Invoked to develop the quantization theory of screened plasmons.
  • domain assumption Perfect screening by metallic substrate
    Used to define the screened plasmon dispersion in the layered geometry.

pith-pipeline@v0.9.0 · 5664 in / 1286 out tokens · 37026 ms · 2026-05-21T18:27:50.901565+00:00 · methodology

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Reference graph

Works this paper leans on

90 extracted references · 90 canonical work pages

  1. [1]

    (1), the EM energy density has an electricu E and magneticu B contribution, such that u=u E +u B

    T otal Energy As alluded to above Eq. (1), the EM energy density has an electricu E and magneticu B contribution, such that u=u E +u B. In frequency space, we can express these contributions in the following form. For the magnetic part, uB = 1 µ0 Z t −∞ dt′ B· ∂B ∂t = 1 2µ0 Z Z dωdω ′ 2π e−i(ω−ω ′)tB⋆(r, ω′)B(r, ω).(A1) The electric contribution cannot be...

    work page

  2. [2]

    Orthogonality relations Using the generalized eigenvalue equation (8), which is a direct application of Maxwell’s curl equation ofHin matter, one can construct what we call an orthogonality relation for the mode functions. Forω q ±ω q′ ̸= 0, we find Z drA⋆ q′D+Aq = 0 (A6a) Z drAq′D−Aq = 0 (A6b) The procedure to find both of these relations is identical, a...

    work page

  3. [3]

    the electric field lies in the plane of incidence

    Periodic Modulation Let us fix the (generic) labelqto denote a 2D momentum label, and fix the polarization of the field top-polarized modes, i.e. the electric field lies in the plane of incidence. Here we consider the effect of introducing a modulation in the dielectric function in the expression for the total EM energy, ϵr(r, ω) =ϵ d + X G ϵG r (z, ω)eiG...

    work page

  4. [4]

    Polman and H

    A. Polman and H. A. Atwater, Plasmonics: Optics at the nanoscale, Materials Today8, 56 (2005)

    work page 2005

  5. [5]

    J. A. A. J. Perenboom, P. Wyder, and F. Meier, Elec- tronic properties of small metallic particles, Physics Re- ports78, 173 (1981)

    work page 1981

  6. [6]

    D. K. Gramotnev and S. I. Bozhevolnyi, Plasmonics be- yond the diffraction limit, Nature Photonics4, 83 (2010)

    work page 2010

  7. [7]

    W. L. Barnes, A. Dereux, and T. W. Ebbesen, Surface plasmon subwavelength optics, Nature424, 824 (2003)

    work page 2003

  8. [8]

    Polman, Plasmonics applied, Science322, 868 (2008)

    A. Polman, Plasmonics applied, Science322, 868 (2008)

    work page 2008

  9. [9]

    J. Liu, H. He, D. Xiao, S. Yin, W. Ji, S. Jiang, D. Luo, B. Wang, and Y. Liu, Recent advances of plasmonic nanoparticles and their applications, Materials11, 1833 (2018)

    work page 2018

  10. [10]

    X. Fan, W. Zheng, and D. J. Singh, Light scattering and surface plasmons on small spherical particles, Light: Sci- ence & Applications3, e179 (2014)

    work page 2014

  11. [11]

    M. V. Bashevoy, F. Jonsson, A. V. Krasavin, N. I. Zhe- ludev, Y. Chen, and M. I. Stockman, Generation of traveling surface plasmon waves by free-electron impact, Nano Letters6, 1113 (2006)

    work page 2006

  12. [12]

    H. Yoon, K. Y. M. Yeung, P. Kim, and D. Ham, Plas- monics with two-dimensional conductors, Philosophical transactions. Series A, Mathematical, physical, and en- gineering sciences372, 20130104 (2014)

    work page 2014

  13. [13]

    W. F. Andress, H. Yoon, K. Y. M. Yeung, L. Qin, K. West, L. Pfeiffer, and D. Ham, Ultra-subwavelength two-dimensional plasmonic circuits, Nano Letters12, 2272 (2012)

    work page 2012

  14. [14]

    Stern, Polarizability of a two-dimensional electron gas, Physical Review Letters18, 546 (1967)

    F. Stern, Polarizability of a two-dimensional electron gas, Physical Review Letters18, 546 (1967)

    work page 1967

  15. [15]

    Jablan, H

    M. Jablan, H. Buljan, and M. Soljaˇ ci´ c, Plasmonics in graphene at infrared frequencies, Physical Review B80, 245435 (2009)

    work page 2009

  16. [16]

    Bonaccorso, Z

    F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Graphene photonics and optoelectronics, Nature Photon- ics4, 611 (2010)

    work page 2010

  17. [17]

    Woessner, M

    A. Woessner, M. B. Lundeberg, Y. Gao, A. Prin- cipi, P. Alonso-Gonz´ alez, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hil- lenbrand, and F. H. L. Koppens, Highly confined low- loss plasmons in graphene–boron nitride heterostruc- tures, Nature Materials14, 421 (2015)

    work page 2015

  18. [18]

    Stauber, Plasmonics in Dirac systems: From graphene to topological insulators, Journal of Physics: Condensed Matter26, 123201 (2014)

    T. Stauber, Plasmonics in Dirac systems: From graphene to topological insulators, Journal of Physics: Condensed Matter26, 123201 (2014)

    work page 2014

  19. [19]

    Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, Gate- tuning of graphene plasmons revealed by infrared nano- imaging, Nature487, 82 (2012)

    work page 2012

  20. [20]

    J. Chen, M. Badioli, P. Alonso-Gonz´ alez, S. Thongrat- tanasiri, F. Huth, J. Osmond, M. Spasenovi´ c, A. Cen- teno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. Garc´ ıa de Abajo, R. Hillenbrand, and F. H. L. Koppens, Optical nano-imaging of gate-tunable graphene plasmons, Nature487, 77 (2012)

    work page 2012

  21. [21]

    Alcaraz Iranzo, S

    D. Alcaraz Iranzo, S. Nanot, E. J. C. Dias, I. Ep- stein, C. Peng, D. K. Efetov, M. B. Lundeberg, R. Par- ret, J. Osmond, J.-Y. Hong, J. Kong, D. R. Englund, N. M. R. Peres, and F. H. L. Koppens, Probing the ul- timate plasmon confinement limits with a van der Waals heterostructure, Science360, 291 (2018)

    work page 2018

  22. [22]

    Principi, R

    A. Principi, R. Asgari, and M. Polini, Acoustic plasmons and composite hole-acoustic plasmon satellite bands in graphene on a metal gate, Solid State Communications 151, 1627 (2011)

    work page 2011

  23. [23]

    L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, Graphene plasmonics for tunable terahertz metamaterials, Nature Nanotechnology6, 630 (2011)

    work page 2011

  24. [24]

    Takamura, N

    M. Takamura, N. Kumada, S. Wang, K. Kumakura, and Y. Taniyasu, Plasmon control driven by spatial carrier density modulation in graphene, ACS Photonics6, 947 (2019)

    work page 2019

  25. [25]

    A. N. Grigorenko, M. Polini, and K. S. Novoselov, Graphene plasmonics, Nature Photonics6, 749 (2012)

    work page 2012

  26. [26]

    Rodrigo, O

    D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. Garc´ ıa de Abajo, V. Pruneri, and H. Altug, Mid-infrared plasmonic biosensing with graphene, Science349, 165 (2015)

    work page 2015

  27. [27]

    Y. Li, H. Yan, D. B. Farmer, X. Meng, W. Zhu, R. M. Osgood, T. F. Heinz, and P. Avouris, Graphene plasmon enhanced vibrational sensing of surface-adsorbed layers, Nano Letters14, 1573 (2014)

    work page 2014

  28. [28]

    Vasi´ c, G

    B. Vasi´ c, G. Isi´ c, and R. Gaji´ c, Localized surface plas- mon resonances in graphene ribbon arrays for sensing of dielectric environment at infrared frequencies, Journal of Applied Physics113, 013110 (2013)

    work page 2013

  29. [29]

    Alonso-Gonz´ alez, A

    P. Alonso-Gonz´ alez, A. Y. Nikitin, Y. Gao, A. Woess- ner, M. B. Lundeberg, A. Principi, N. Forcellini, 13 W. Yan, S. V´ elez, Andreas. J. Huber, K. Watanabe, T. Taniguchi, F. Casanova, L. E. Hueso, M. Polini, J. Hone, F. H. L. Koppens, and R. Hillenbrand, Acoustic terahertz graphene plasmons revealed by photocurrent nanoscopy, Nature Nanotechnology12, 31 (2017)

    work page 2017

  30. [30]

    P. A. D. Gon¸ calves,Plasmonics and Light–Matter In- teractions in Two-Dimensional Materials and in Metal Nanostructures: Classical and Quantum Considera- tions, Springer Theses (Springer International Publish- ing, Cham, 2020)

    work page 2020

  31. [31]

    T. G. Rappoport, Y. V. Bludov, F. H. L. Koppens, and N. M. R. Peres, Topological graphene plasmons in a plasmonic realization of the Su–Schrieffer–Heeger model, ACS Photonics8, 1817 (2021)

    work page 2021

  32. [32]

    P. A. D. Gon¸ calves and N. M. R. Peres,An Introduction to Graphene Plasmonics(World Scientific, 2016)

    work page 2016

  33. [33]

    Rodr´ ıguez Echarri, J

    A. Rodr´ ıguez Echarri, J. D. Cox, and F. J. Garc´ ıa de Abajo, Quantum effects in the acoustic plasmons of atomically thin heterostructures, Optica6, 630 (2019)

    work page 2019

  34. [34]

    J. D. Joannopoulos, J. N. Winn, and S. G. Johnson,Pho- tonic Crystals: Molding the Flow of Light - Second Edi- tion(Princeton University Press, Princeton, NJ, 2011)

    work page 2011

  35. [35]

    A. Yu. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, Surface plasmon enhanced absorp- tion and suppressed transmission in periodic arrays of graphene ribbons, Physical Review B85, 081405 (2012)

    work page 2012

  36. [36]

    Zhao and Z

    B. Zhao and Z. M. Zhang, Strong plasmonic coupling between graphene ribbon array and metal gratings, ACS Photonics2, 1611 (2015)

    work page 2015

  37. [37]

    J. H. Strait, P. Nene, W.-M. Chan, C. Manolatou, S. Ti- wari, F. Rana, J. W. Kevek, and P. L. McEuen, Confined plasmons in graphene microstructures: Experiments and theory, Physical Review B87, 241410 (2013)

    work page 2013

  38. [38]

    T. G. Rappoport, I. Epstein, F. H. L. Koppens, and N. M. R. Peres, Understanding the electromagnetic re- sponse of graphene/metallic nanostructures hybrids of different dimensionality, ACS Photonics7, 2302 (2020)

    work page 2020

  39. [39]

    E. J. C. Dias and N. M. R. Peres, Controlling spoof plas- mons in a metal grating using graphene surface plasmons, ACS Photonics4, 3071 (2017)

    work page 2017

  40. [40]

    Bylinkin, E

    A. Bylinkin, E. Titova, V. Mikheev, E. Zhukova, S. Zhukov, M. Belyanchikov, M. Kashchenko, A. Mi- akonkikh, and D. Svintsov, Tight-binding terahertz plas- mons in chemical-vapor-deposited graphene, Physical Re- view Applied11, 054017 (2019)

    work page 2019

  41. [41]

    Guo and C

    T. Guo and C. Argyropoulos, Hybrid graphene-plasmon gratings, Journal of Applied Physics134, 050901 (2023)

    work page 2023

  42. [42]

    Xiong, C

    L. Xiong, C. Forsythe, M. Jung, A. S. McLeod, S. S. Sunku, Y. M. Shao, G. X. Ni, A. J. Sternbach, S. Liu, J. H. Edgar, E. J. Mele, M. M. Fogler, G. Shvets, C. R. Dean, and D. N. Basov, Photonic crystal for graphene plasmons, Nature Communications10, 4780 (2019)

    work page 2019

  43. [43]

    D. Jin, A. Kumar, K. Hung Fung, J. Xu, and N. X. Fang, Terahertz plasmonics in ferroelectric-gated graphene, Applied Physics Letters102, 201118 (2013)

    work page 2013

  44. [44]

    Baeumer, D

    C. Baeumer, D. Saldana-Greco, J. M. P. Martirez, A. M. Rappe, M. Shim, and L. W. Martin, Ferroelectrically driven spatial carrier density modulation in graphene, Nature Communications6, 6136 (2015)

    work page 2015

  45. [45]

    D. A. Miranda, Y. V. Bludov, N. A. Mortensen, and N. M. R. Peres, Topology in a one-dimensional plas- monic crystal: The optical approach, Journal of Optics 26, 125001 (2024)

    work page 2024

  46. [46]

    N. A. Mortensen, S. I. Bozhevolnyi, and A. Al` u, Topo- logical nanophotonics, Nanophotonics8, 1315 (2019)

    work page 2019

  47. [47]

    Z. Lan, M. L. Chen, F. Gao, S. Zhang, and W. E. Sha, A brief review of topological photonics in one, two, and three dimensions, Reviews in Physics9, 100076 (2022)

    work page 2022

  48. [48]

    Wang and J.-H

    H.-X. Wang and J.-H. Jiang, A short review of all- dielectric topological photonic crystals, Frontiers in Physics10, 866552 (2022)

    work page 2022

  49. [49]

    F. D. M. Haldane and S. Raghu, Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Physical Review Letters 100, 013904 (2008)

    work page 2008

  50. [50]

    Kim and J

    M. Kim and J. Rho, Topological edge and corner states in a two-dimensional photonic Su-Schrieffer-Heeger lattice, Nanophotonics9, 3227 (2020)

    work page 2020

  51. [51]

    Shindou, R

    R. Shindou, R. Matsumoto, S. Murakami, and J.-i. Ohe, Topological chiral magnonic edge mode in a magnonic crystal, Physical Review B87, 174427 (2013)

    work page 2013

  52. [52]

    P. A. McClarty, Topological magnons: A review, Annual Review of Condensed Matter Physics13, 171 (2022)

    work page 2022

  53. [53]

    Cuerda, J

    J. Cuerda, J. M. Taskinen, N. K¨ allman, L. Grabitz, and P. T¨ orm¨ a, Observation of quantum metric and non- Hermitian Berry curvature in a plasmonic lattice, Phys- ical Review Research6, L022020 (2024)

    work page 2024

  54. [54]

    Cuerda, J

    J. Cuerda, J. M. Taskinen, N. K¨ allman, L. Grabitz, and P. T¨ orm¨ a, Pseudospin-orbit coupling and non-Hermitian effects in the quantum geometric tensor of a plasmonic lattice, Physical Review B109, 165439 (2024)

    work page 2024

  55. [55]

    Di Pietro, M

    P. Di Pietro, M. Ortolani, O. Limaj, A. Di Gaspare, V. Giliberti, F. Giorgianni, M. Brahlek, N. Bansal, N. Koirala, S. Oh, P. Calvani, and S. Lupi, Observa- tion of Dirac plasmons in a topological insulator, Nature Nanotechnology8, 556 (2013)

    work page 2013

  56. [56]

    S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Gian- nini, Topological plasmonic chain with retardation and radiative effects, ACS Photonics5, 2271 (2018)

    work page 2018

  57. [57]

    Brey and H

    L. Brey and H. A. Fertig, Quantum band structure and topology in a one-dimensional modulated plasmonic crys- tal, Physical Review B111, 045430 (2025)

    work page 2025

  58. [58]

    D. Jin, T. Christensen, M. Soljaˇ ci´ c, N. X. Fang, L. Lu, and X. Zhang, Infrared topological plasmons in graphene, Physical Review Letters118, 245301 (2017)

    work page 2017

  59. [59]

    Wang, G.-Y

    H.-X. Wang, G.-Y. Guo, and J.-H. Jiang, Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology, New Journal of Physics21, 093029 (2019)

    work page 2019

  60. [60]

    A. Yu. Nikitin, F. Guinea, F. J. Garc´ ıa-Vidal, and L. Mart´ ın-Moreno, Edge and waveguide terahertz sur- face plasmon modes in graphene microribbons, Physical Review B84, 161407 (2011)

    work page 2011

  61. [61]

    B. A. Ferreira, B. Amorim, A. J. Chaves, and N. M. R. Peres, Quantization of graphene plasmons, Physical Re- view A101, 033817 (2020)

    work page 2020

  62. [62]

    J. S. Fakonas, A. Mitskovets, and H. A. Atwater, Path en- tanglement of surface plasmons, New Journal of Physics 17, 023002 (2015)

    work page 2015

  63. [63]

    R. W. Heeres, L. P. Kouwenhoven, and V. Zwiller, Quan- tum interference in plasmonic circuits, Nature Nanotech- nology8, 719 (2013)

    work page 2013

  64. [64]

    D. Xu, X. Xiong, L. Wu, X.-F. Ren, C. E. Png, G.-C. Guo, Q. Gong, and Y.-F. Xiao, Quantum plasmonics: New opportunity in fundamental and applied photonics, Advances in Optics and Photonics10, 703 (2018). 14

    work page 2018

  65. [65]

    M. S. Tame, K. R. McEnery, S ¸. K. ¨Ozdemir, J. Lee, S. A. Maier, and M. S. Kim, Quantum plasmonics, Na- ture Physics9, 329 (2013)

    work page 2013

  66. [66]

    D. Z. Manrique, J. W. You, H. Deng, F. Ye, and N. C. Panoiu, Quantum plasmon engineering with interacting graphene nanoflakes, The Journal of Physical Chemistry C121, 27597 (2017)

    work page 2017

  67. [67]

    Brey and H

    L. Brey and H. Fertig, Quantum plasmons in double layer systems, Physical Review B109, 045303 (2024)

    work page 2024

  68. [68]

    Z. Sun, D. N. Basov, and M. M. Fogler, Graphene as a source of entangled plasmons, Physical Review Research 4, 023208 (2022)

    work page 2022

  69. [69]

    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

  70. [70]

    Calaj´ o, P

    G. Calaj´ o, P. K. Jenke, L. A. Rozema, P. Walther, D. E. Chang, and J. D. Cox, Nonlinear quantum logic with colliding graphene plasmons, Physical Review Research 5, 013188 (2023)

    work page 2023

  71. [71]

    Gonzalez-Tudela, D

    A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Entanglement of two qubits mediated by one-dimensional plasmonic waveguides, Physical Review Letters106, 020501 (2011)

    work page 2011

  72. [72]

    M. G. Raymer, Quantum theory of light in a dispersive structured linear dielectric: A macroscopic Hamiltonian tutorial treatment, Journal of Modern Optics67, 196 (2020)

    work page 2020

  73. [73]

    R. D. L. Kronig and W. G. Penney, Quantum mechanics of electrons in crystal lattices, Proceedings of the Royal Society of London. Series A130, 499 (1931)

    work page 1931

  74. [74]

    D. J. Griffiths,Introduction to Electrodynamics, 5th ed. (Cambridge University Press, 2023)

    work page 2023

  75. [75]

    Archambault, F

    A. Archambault, F. Marquier, J.-J. Greffet, and C. Arnold, Quantum theory of spontaneous and stim- ulated emission of surface plasmons, Physical Review B 82, 035411 (2010)

    work page 2010

  76. [76]

    G. W. Hanson, S. A. Hassani Gangaraj, C. Lee, D. G. An- gelakis, and M. Tame, Quantum plasmonic excitation in graphene and loss-insensitive propagation, Physical Re- view A92, 013828 (2015)

    work page 2015

  77. [77]

    E. J. C. Dias, D. A. Iranzo, P. A. D. Gon¸ calves, Y. Hajati, Y. V. Bludov, A.-P. Jauho, N. A. Mortensen, F. H. L. Koppens, and N. M. R. Peres, Probing nonlocal effects in metals with graphene plasmons, Physical Review B 97, 245405 (2018)

    work page 2018

  78. [78]

    Fox,Quantum Optics: An Introduction(Oxford Uni- versity PressOxford, 2006)

    M. Fox,Quantum Optics: An Introduction(Oxford Uni- versity PressOxford, 2006)

    work page 2006

  79. [79]

    Goren, K

    T. Goren, K. Plekhanov, F. Appas, and K. Le Hur, Topo- logical Zak phase in strongly coupled LC circuits, Phys- ical Review B97, 041106 (2018)

    work page 2018

  80. [80]

    Vanderbilt,Berry Phases in Electronic Structure The- ory: Electric Polarization, Orbital Magnetization and Topological Insulators, 1st ed

    D. Vanderbilt,Berry Phases in Electronic Structure The- ory: Electric Polarization, Orbital Magnetization and Topological Insulators, 1st ed. (Cambridge University Press, 2018)

    work page 2018

Showing first 80 references.