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REVIEW 3 major objections 4 minor 72 references

Topology and interactions in the photonic Creutz and Creutz-Hubbard ladders

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that on a rungless Creutz ladder with strong repulsive interactions, two bosons form a doublon that Aharonov-Bohm cages at half the single-particle flux, and that a synthetic-dimension waveguide lattice can realize this…

desk verdict The doublon-caging and edge-state physics is sound and worth a serious referee; the photonic implementation is a sketch, not a blueprint, and the paper oversells how ready it is for experiment. read the letter →

arxiv 1909.00987 v2 pith:RTNURPFG submitted 2019-09-03 quant-ph cond-mat.mes-hall

Juan Zurita , Charles Creffield , Gloria Platero This is my paper · ORCID
classification quant-phcond-mat.mes-hall
keywords CreutzladderAharonov-BohmcagingdoublonsflatbandstopologicaledgestatesphotonicwaveguidelatticessyntheticdimensionsCreutz-Hubbardmodel
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 aims to establish that the Creutz ladder, a ladder-shaped lattice whose single-particle bands become flat at special magnetic fluxes, can serve as a photonic platform where topology, flat-band localization, and interactions meet. For one particle it explains how Aharonov-Bohm caging and the ladder's two-site edge states could be observed in waveguide lattices, either through a rhombus-chain mapping or through a synthetic dimension. For two particles it makes the central claim that a strong repulsive Hubbard interaction binds them into a doublon whose effective magnetic flux is doubled, so at $\varphi=\pm\pi/2$ the doublon is Aharonov-Bohm caged while single particles remain mobile. It also identifies stationary doublon edge states that can form NOON states, and proposes a three-dimensional waveguide geometry, built from the Cartesian product of two ladders with one synthetic dimension, to implement the two-boson model. If the proposal holds, interacting and topological effects could be studied in the same photonic device rather than in separate setups.

What carries the argument

The load-bearing object is the Schrieffer-Wolff effective doublon Hamiltonian for the two-boson Creutz-Hubbard model: in the large-$U$ limit it replaces each doubly occupied site by a doublon quasiparticle that hops with amplitude $J^2/U$ and sees twice the gauge phase of a single particle, so the caging condition moves from $\varphi=\pm\pi$ to $\varphi=\pm\pi/2$. The companion mechanism is the mapping of the two-particle Fock space onto a quasi-2D lattice, the Cartesian product of two Creutz ladders, which is then rearranged into a 3D waveguide geometry whose fourth 'leg-index' dimension is synthetic; this converts a two-body problem into single-particle hopping on a higher-dimensional lattice that a photonic array can in principle simulate.

What would settle it

In a real or simulated waveguide implementation of the rungless two-boson Creutz-Hubbard model at $\varphi=\pi/2$ with $U\gg J$, inject two photons into one site and propagate for one effective doublon period ($\sim \pi U/(2J^2)$); the doublon-caging claim requires that the two-body occupation stays inside the initial cage, so any substantial probability on sites outside the cage, or any splitting of the pair onto different rungs, would refute it. Injecting a single photon under the same conditions should instead show free propagation, giving a sharp within-device contrast.

Watch

Extended reading notes

Core claim

The central discovery is that interactions move Aharonov-Bohm caging to a new value of the magnetic flux. In the noninteracting rungless ladder ($m=0$) the flat bands and single-particle caging occur at $\varphi=\pm\pi$; a strong repulsive Hubbard term $U$ then produces, through a Schrieffer-Wolff transformation, an effective doublon Hamiltonian whose hopping amplitudes are renormalized from $J$ to $J^2/U$ and whose accumulated gauge phases are doubled. The doublon thus behaves as a particle of charge $2q$, so its flat-band condition shifts to $\varphi=\pm\pi/2$ (and, by periodicity, $\pm 3\pi/2$), precisely the fluxes where single particles are not localized. Numerical time evolution of an initial doublon at $\varphi=\pi/2$ confirms it stays inside its cage, while at other fluxes it propagates or, when $U\simeq J$, collapses into two independent particles. The paper further shows that doublon edge states of the form $(|2_{1,A}\rangle-e^{i\varphi}|2_{1,B}\rangle)/\sqrt{2}$ are stationary and that combining the two ends yields a NOON state. Its photonic proposal maps the two-particle Fock space onto a quasi-2D lattice that is the Cartesian product of two Creutz ladders, then rearranges that lattice into a 3D waveguide geometry with one synthetic dimension, making the interacting topological model experimentally addressable.

Load-bearing premise

The proposal's load-bearing premise is that the quasi-2D effective lattice can be physically built as a 3D waveguide array with the required nonlocal hopping amplitudes (the dotted links in its Figure 6b): the paper cites an existing oscillating-column setup as a template but supplies no quantitative coupling-strength design or tolerance analysis for those specific hops.

Editorial extensions

If this is right

  • A waveguide experiment with two injected photons should see a doublon remain confined to its cage at $\varphi=\pm\pi/2$, while a single photon injected alone propagates freely in the same device.
  • Because the doublon hopping amplitude is $J^2/U$, doublon dynamics is roughly an order of magnitude slower than single-particle dynamics, setting the propagation length and coherence needed for observation.
  • The $m=0$ edge states are AB-localized in two sites for any flux, and their doublon counterparts are stationary; the superposition of left and right doublon edge states forms a photonic NOON state usable in metrology.
  • By periodicity the doublon caging also occurs at $\varphi=\pm 3\pi/2$, so the effect survives beyond a single fine-tuned flux.
  • Realizing the proposed lattice would be the first waveguide-based simulation of a quasi-1D topological insulator with interactions, adding interacting physics to a platform previously limited to topologically trivial 1D systems.

Reading between the lines

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

  • An extension the paper leaves implicit: by the same effective-flux logic, an $n$-boson bound state should cage at $\varphi=\pi/n$ (mod $2\pi/n$), so tuning the flux alone would select which cluster size is frozen while others move.
  • The practical hinge of the photonic proposal is the set of nonlocal hopping amplitudes in the 3D rearrangement; if those couplings cannot be engineered precisely, the simulator fails even though the underlying two-body physics remains correct.
  • A sharper experimental test than site-intensity snapshots is the two-photon correlation function: for a caged doublon all $G^{(2)}$ weight stays inside the cage even as the single-particle density spreads, cleanly separating AB caging from ordinary Anderson localization.
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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

3 major / 4 minor

Summary. The paper examines the noninteracting Creutz ladder and its two-boson interacting extension. For the single-particle model it re-derives Aharonov-Bohm caging at φ = π in the rungless case, gives closed-form edge states for m = 0, and reviews possible photonic implementations via synthetic dimensions. For the two-particle Creutz-Hubbard model it proposes a Schrieffer-Wolff effective doublon Hamiltonian, predicts doublon caging at φ = ±π/2 (where single particles remain mobile), doublon edge states, and doublon collapse at intermediate U; exact time-evolution simulations in Fig. 5 support these claims. It then maps the two-particle problem to a single-particle tight-binding model on a quasi-2D lattice and sketches a 3D waveguide-lattice implementation with one synthetic dimension.

Significance. The predicted coexistence of caged doublons and mobile single particles at φ = π/2 is a concrete, falsifiable many-body effect that extends flat-band caging beyond the single-particle sector. The paper's exact two-particle numerics and analytic edge-state formulas are internally consistent and are the strongest part of the manuscript. If the waveguide scheme could be made quantitative, the proposal would be a meaningful step toward interacting topological photonic simulators. That said, the experimental section as written remains a qualitative sketch rather than a complete implementation scheme.

major comments (3)
  1. [Section III B, Fig. 6b] The photonic implementation is asserted without a quantitative design. The text states that three nonlocal hoppings per unit cell (two for the rungless ladder) are necessary and that a setup 'could be similar' to Ref. [37], but it does not specify how the required amplitudes and Peierls phases of the effective model (J_α^2/U, J^2/U, and the diagonal U potential in Eq. (12)) are generated by the oscillating-column scheme, nor does it give modulation parameters or a tolerance analysis. Because the photonic implementation is one of the two central deliverables named in the abstract, this gap is load-bearing for the experimental claim; the two-particle dynamics in Fig. 5 are unaffected.
  2. [Section III A, Eq. (12)] The Schrieffer-Wolff effective Hamiltonian is stated without derivation and without a quantitative criterion for 'U ≫ J, m'. The prediction of doublon caging at φ = ±π/2 rests on treating the doublon as a charge-2q particle hopping on a renormalized Creutz ladder with doubled Peierls phases. Without a stated validity bound, the analytical argument alone does not specify the parameter window in which the effective model is quantitative. The exact numerical simulations in Fig. 5b support the prediction, but the effective-Hamiltonian claim needs either a derivation/estimate or an explicit statement that it is heuristic.
  3. [Equations (1) and (12)] The Creutz model is defined with two apparently identical horizontal hopping terms, 'J_α c†_{j+1,α} c_{j,α} + J c†_{j+1,α} c_{j,α}', with no coupling between opposite legs; the same duplication appears in Eq. (12). The second term should couple to the opposite leg (c_{j,α} replaced by c_{j,bar α}) to represent the diagonal crossings shown in Fig. 1a. As written, these equations do not define the Creutz ladder described in the text and figures; this is a load-bearing definitional error that must be corrected.
minor comments (4)
  1. [Equation (11)] The relation between first- and second-quantized basis states is misprinted: the symmetrized combination should read |1_{i,α}1_{j,β}⟩ = (|i,α;j,β⟩ + |j,β;i,α⟩)/√2 for (i,α) ≠ (j,β).
  2. [Section II, after Eq. (1)] The phrase 'with σ_A/B = ±1 and A = B and vice versa' appears to be a garbled statement of σ_A = -σ_B; please clarify the index conventions.
  3. [Section III B] The sentence 'If the synthetic dimension is periodic, no nonlocal hoppings are needed' is not explained. If this route avoids the nonlocal-hopping problem of Fig. 6b, it deserves a concrete description rather than a parenthetical remark.
  4. [Figure 5 and text] The numerical simulations report timesteps and system sizes but not the integration method; for reproducibility, state in the text or appendix whether the time evolution was obtained by exact exponentiation of the full Hamiltonian.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the doublon-caging prediction follows from a derived effective Hamiltonian and is independently verified by exact two-particle numerics.

full rationale

The central results are self-contained. The single-particle AB caging at φ=π is derived from the interference phases in Eq. (1) and confirmed by exact numerics (Fig. 2d). The doublon prediction at φ=±π/2 is obtained in Sec. III A from the Schrieffer-Wolff effective Hamiltonian, Eq. (12), where the doublon hopping amplitudes are J_α^2/U and J^2/U. Since J_α = J exp(i σ_α φ/2), the doublon hopping carries phase exp(i σ_α φ), so the effective doublon Creutz model at flux φ is the single-particle model at φ/2; the doublon-caging flux π/2 follows analytically rather than being inserted by hand. Crucially, the time evolutions in Fig. 5 are computed from the full two-particle Hamiltonian (9), not from the effective model, so the caging and edge states (Eqs. (13)–(14)) are independently checked. The prior same-group works [16,61,62] are cited for context and for related doublon effective models, but the present derivation supplies its own Schrieffer-Wolff transformation and numerics, so these citations are not load-bearing. The photonic proposal in Sec. III B uses the standard Fock-space-to-quasi-2D-lattice mapping (Appendix B) and cites [37] only for a possible experimental platform; this is a feasibility suggestion, not a circular argument. No fitted parameter is renamed as a prediction, and no uniqueness or ansatz is imported solely through self-citation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central physics rests on the standard tight-binding Creutz ladder, the two-particle bosonic Hubbard model, and Schrieffer-Wolff perturbation theory. No free parameters are fitted to experimental data; the numbers J, U, m, φ, L, and the time steps are chosen simulation parameters. The main unsupported element is the feasibility assumption for the photonic implementation: engineering nonlocal hoppings in a 3D waveguide lattice. No new particles, forces, or entities are postulated.

assumptions (4)
  • domain assumption Paraxial propagation of classical light in waveguide lattices follows the single-particle Schrodinger equation, with propagation distance as time.
    Invoked in Section II B to justify photonic simulation of the Creutz ladder and is the basis for the whole photonic implementation.
  • domain assumption The Creutz-Hubbard Hamiltonian preserves particle number and the two-particle subspace is sufficient to capture the interaction effects of interest.
    Section III restricts to two bosons because the Hamiltonian conserves particle number; this is exact for the model but limits the claimed regime to sparse filling.
  • domain assumption The Schrieffer-Wolff transformation is valid for U much greater than J and m, producing the effective doublon Hamiltonian Eq. (12).
    Section III A states the effective Hamiltonian relies on decoupling of doubly occupied states for large U; no quantitative bound is given.
  • domain assumption A doublon behaves as a charge-2q quasiparticle, halving the flux required for Aharonov-Bohm caging.
    Section III A uses this to predict doublon caging at φ=±π/2; it is supported by the renormalized hopping phases in the effective Hamiltonian.

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Cite this review

Pith. "Pith review of Topology and interactions in the photonic Creutz and Creutz-Hubbard ladders." pith.science (2026). https://pith.science/paper/RTNURPFG

@misc{pith2026190900987,
  author       = {Pith},
  title        = {Pith review of: Topology and interactions in the photonic Creutz and Creutz-Hubbard ladders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RTNURPFG}},
  note         = {Machine review of arXiv:1909.00987}
}
read the original abstract

The latest advances in the field of photonics have enabled the simulation of an increasing number of quantum models in photonic systems, turning them into an important tool for realizing exotic quantum phenomena. In this paper we suggest different ways in which these systems can be used to study the interplay between flat band dynamics, topology and interactions in a well-known quasi-1D topological insulator: the Creutz ladder. Firstly, a simple experimental protocol is proposed to observe the Aharonov-Bohm localization in the noninteracting system, and the different experimental setups that might be used for this are reviewed. We then consider the inclusion of a repulsive Hubbard-type interaction term, which can give rise to repulsively bound pairs termed doublons. The dynamics of these quasiparticles are studied for different points of the phase diagram, including a regime in which pairs are localized and particles are free to move. Finally, a scheme for the photonic implementation of a two-particle bosonic Creutz-Hubbard model is presented.

Figures

Figures reproduced from arXiv: 1909.00987 by the authors.

Figure 1
Figure 1. FIG. 1. a) Geometry of the Creutz ladder. The coordinates [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. a) AB caging. In the system with [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Wavefunctions for both edge states corresponding to [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. a) Creutz ladder in one spatial and one synthetic [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Numerical time evolution of the two-particle Creutz [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. a) Quasi-2D lattice for the effective model, result of the Cartesian product of two Creutz ladders (left and bottom). [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Quasi-2D plots for the simulations in Figure [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

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Reference graph

Works this paper leans on

72 extracted references · 50 canonical work pages

  1. [16]

    C. E. Creffield and G. Platero, Phys. Rev. Lett. 105, 086804 (2010)

    work page 2010

  2. [37]

    Four modes in a square lattice would suffice to implement our system

    In it, oscillating columns of waveguides act as sites in a 2D lattice, and the synthetic dimension is represented by the modes of oscillation of the column. Four modes in a square lattice would suffice to implement our system. In Figure 7, the quasi-2D representations of the sim- ulations in Figure 5 are shown. This way of presenting them carries more infor...

    work page

  3. [1]

    R. P. Feynman, International Journal of Theoretical Physics 21, 467 (1982)

    1982

  4. [2]

    Preskill, arXiv e-prints (2012), arXiv:1203.5813 [quant-ph]

    J. Preskill, arXiv e-prints (2012), arXiv:1203.5813 [quant-ph]

    arXiv 2012

  5. [3]

    Quantum supremacy with analog quantum processors for material science and machine learning

    J. Tangpanitanon, S. Thanasilp, M.-A. Lemonde, and D. G. Angelakis, arXiv preprint arXiv:1906.03860 (2019)

    work page Pith review arXiv 2019

  6. [4]

    I. M. Georgescu, S. Ashhab, and F. Nori, Rev. Mod. Phys. 86, 153 (2014)

    2014

  7. [5]

    Asb´ oth, L

    J. Asb´ oth, L. Oroszl´ any, and A. P´ alyi,A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions , Lecture Notes in Physics (Springer International Publishing, 2016)

    work page 2016

  8. [6]

    W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979)

    1979

Show all 72 references
  1. [7]

    Creutz, Phys

    M. Creutz, Phys. Rev. Lett. 83, 2636 (1999)

    1999

  2. [8]

    Vidal, R

    J. Vidal, R. Mosseri, and B. Dou¸ cot, Phys. Rev. Lett. 81, 5888 (1998)

    1998

  3. [9]

    Vidal, B

    J. Vidal, B. Dou¸ cot, R. Mosseri, and P. Butaud, Phys. Rev. Lett. 85, 3906 (2000)

    2000

  4. [10]

    C. C. Abilio, P. Butaud, T. Fournier, B. Pannetier, J. Vi- dal, S. Tedesco, and B. Dalzotto, Phys. Rev. Lett. 83, 5102 (1999)

    1999

  5. [11]

    doublons

    ultracold atom lattices [12] and, most recently, in a photonic system. [13] As the single-particle kinetic energy is zero in a flat band, even extremely weak interactions can have an enor- mous effect on the system’s properties. Introducing a contact interaction of Hubbard-type ...

    1909 arXiv

  6. [12]

    Shinohara, T

    K. Shinohara, T. Aoki, and A. Morinaga, Phys. Rev. A 66, 042106 (2002)

    2002

  7. [13]

    Mukherjee, M

    S. Mukherjee, M. Di Liberto, P. ¨Ohberg, R. R. Thomson, and N. Goldman, Phys. Rev. Lett. 121, 075502 (2018)

    2018

  8. [14]

    C. Naud, G. Faini, and D. Mailly, Phys. Rev. Lett. 86, 5104 (2001)

    2001

  9. [15]

    Tovmasyan, E

    M. Tovmasyan, E. P. L. van Nieuwenburg, and S. D. Huber, Phys. Rev. B 88, 220510 (2013)

    2013

  10. [17]

    Takayoshi, H

    S. Takayoshi, H. Katsura, N. Watanabe, and H. Aoki, Phys. Rev. A 88, 063613 (2013)

    2013

  11. [18]

    Leykam, A

    D. Leykam, A. S. Solntsev, A. A. Sukhorukov, and A. S. Desyatnikov, Physical Review A 92 (2015)

    2015

  12. [19]

    Mittal, E

    S. Mittal, E. A. Goldschmidt, and M. Hafezi, Nature 561, 502 (2018)

    2018

  13. [20]

    Ozawa, H

    T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zil- berberg, and I. Carusotto, Rev. Mod. Phys. 91, 015006 (2019)

    2019

  14. [21]

    Longhi, Opt

    S. Longhi, Opt. Lett. 38, 3716 (2013)

    2013

  15. [22]

    Slobozhanyuk, S

    A. Slobozhanyuk, S. H. Mousavi, X. Ni, D. Smirnova, Y. S. Kivshar, and A. B. Khanikaev, Nature Photonics 11, 130136 (2016)

    2016

  16. [23]

    Wang, X.-L

    Y. Wang, X.-L. Pang, Y.-H. Lu, J. Gao, Y.-J. Chang, L.-F. Qiao, Z.-Q. Jiao, H. Tang, and X.-M. Jin, Optica 6, 955 (2019)

    2019

  17. [24]

    S. Liu, W. Gao, Q. Zhang, S. Ma, L. Zhang, C. Liu, Y. J. Xiang, T. J. Cui, S. Zhang, et al., Research 2019, 8609875 (2019)

    2019

  18. [25]

    Hadad, J

    Y. Hadad, J. C. Soric, A. B. Khanikaev, and A. Al` u, Nature Electronics 1, 178 (2018)

    2018

  19. [26]

    C. H. Lee, S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling, and R. Thomale, Communi- cations Physics 1 (2018)

    2018

  20. [27]

    Di Liberto, S

    M. Di Liberto, S. Mukherjee, and N. Goldman, Phys. Rev. A 100, 043829 (2019)

    2019

  21. [28]

    N. A. Olekhno, E. I. Kretov, A. A. Stepanenko, D. S. Filonov, V. V. Yaroshenko, B. Cappello, L. Matekovits, and M. A. Gorlach, arXiv e-prints (2019), arXiv:1907.01016 [cond-mat.mes-hall]

    2019 arXiv

  22. [29]

    Serra-Garcia, R

    M. Serra-Garcia, R. S¨ usstrunk, and S. D. Huber, Phys. Rev. B 99, 020304 (2019)

    2019

  23. [30]

    D. O. Krimer and R. Khomeriki, Phys. Rev. A84, 041807 (2011)

    2011

  24. [31]

    Corrielli, A

    G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and R. Osellame, Nature Communications 4 (2013)

    2013

  25. [32]

    Longhi, Opt

    S. Longhi, Opt. Lett. 36, 3248 (2011)

    2011

  26. [33]

    Mukherjee, M

    S. Mukherjee, M. Valiente, N. Goldman, A. Spracklen, E. Andersson, P. ¨Ohberg, and R. R. Thomson, Phys. Rev. A 94, 053853 (2016)

    2016

  27. [34]

    Zhong, Y

    H. Zhong, Y. Zhang, Y. Zhu, D. Zhang, C. Li, Y. Zhang, F. Li, M. R. Beli, and M. Xiao, Annalen der Physik 529, 1600258 (2017)

    2017

  28. [35]

    A. Rai, C. Lee, C. Noh, and D. G. Angelakis, Scientific Reports 5, 8438 (2015)

    2015

  29. [36]

    Zhong, R

    H. Zhong, R. Wang, F. Ye, J. Zhang, L. Zhang, Y. Zhang, M. R. Beli, and Y. Zhang, Results in Physics 12, 996 (2019)

    2019

  30. [38]

    Four topological phases can be distinguished. The FIG. 2. a) AB caging. In the system with φ = π, the time evolution of the state |j,α⟩ never allows the particle to leave the AB cage centered on that site (shaded in orange). The numbers inside each site and the color code used...

  31. [39]

    J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P. Chen, T. L. Hughes, and M. C. Rechtsman, Nature Pho- tonics 12, 408 (2018)

    2018

  32. [40]

    Lustig, S

    E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, Nature 567, 356 (2019)

    2019

  33. [41]

    Li and S

    L. Li and S. Chen, Phys. Rev. B 92, 085118 (2015)

    2015

  34. [42]

    Jafari, H

    R. Jafari, H. Johannesson, A. Langari, and M. A. Martin-Delgado, Phys. Rev. B 99, 054302 (2019)

    2019

  35. [43]

    Altland and M

    A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 (1997)

    1997

  36. [44]

    J¨ unemann, A

    J. J¨ unemann, A. Piga, S.-J. Ran, M. Lewenstein, M. Rizzi, and A. Bermudez, Phys. Rev. X 7, 031057 (2017)

    2017

  37. [45]

    Gholizadeh, M

    S. Gholizadeh, M. Yahyavi, and B. Het´ enyi, Europhysics Letters 122, 27001 (2018)

    2018

  38. [46]

    C. G. Velasco and B. Paredes, arXiv preprint arXiv:1907.11460 (2019)

    2019 arXiv

  39. [47]

    T. L. Hughes, E. Prodan, and B. A. Bernevig, Phys. Rev. B 83, 245132 (2011)

    2011

  40. [48]

    Y. Kuno, T. Orito, and I. Ichinose, arXiv e-prints (2019), arXiv:1904.03463 [cond-mat.stat-mech]

    2019 arXiv

  41. [49]

    J. L. Movilla and J. Planelles, Phys. Rev. B 84, 195110 (2011)

    2011

  42. [50]

    Lazarides and G

    N. Lazarides and G. P. Tsironis, Phys. Rev. B 96, 054305 (2017)

    2017

  43. [51]

    Sun and L.-K

    N. Sun and L.-K. Lim, Phys. Rev. B 96, 035139 (2017)

    2017

  44. [52]

    L. Yuan, Q. Lin, M. Xiao, and S. Fan, Optica 5, 1396 (2018)

    2018

  45. [53]

    Boada, A

    O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, Phys. Rev. Lett. 108, 133001 (2012)

    2012

  46. [54]

    A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeli¯ unas, and M. Lewenstein, Phys. Rev. Lett. 112, 043001 (2014)

    2014

  47. [55]

    Hyoun Kang, J

    J. Hyoun Kang, J. H. Han, and Y.-i. Shin, arXiv e-prints (2019), arXiv:1902.10304 [cond-mat.quant-gas]

    2019 arXiv

  48. [56]

    X. Li, E. Zhao, and W. Vincent Liu, Nature Communi- cations 4, 1523 (2013)

    2013

  49. [57]

    Kremer, I

    M. Kremer, I. Petrides, E. Meyer, M. Heinrich, O. Zil- berberg, and A. Szameit, arXiv e-prints (2018), 13 arXiv:1805.05209 [cond-mat.mes-hall]

    2018 arXiv

  50. [58]

    Sticlet, L

    D. Sticlet, L. Seabra, F. Pollmann, and J. Cayssol, Phys. Rev. B 89, 115430 (2014)

    2014

  51. [59]

    Tovmasyan, S

    M. Tovmasyan, S. Peotta, P. T¨ orm¨ a, and S. D. Huber, Phys. Rev. B 94, 245149 (2016)

    2016

  52. [60]

    Tovmasyan, S

    M. Tovmasyan, S. Peotta, L. Liang, P. T¨ orm¨ a, and S. D. Huber, Phys. Rev. B 98, 134513 (2018)

    2018

  53. [61]

    Bello, G

    M. Bello, G. Platero, and S. Kohler, Phys. Rev. B 96, 045408 (2017)

    2017

  54. [62]

    Bravyi, D

    S. Bravyi, D. P. DiVincenzo, and D. Loss, Annals of Physics 326, 2793 (2011)

    2011

  55. [63]

    Hofmann and M

    F. Hofmann and M. Potthoff, Phys. Rev. B 85, 205127 (2012)

    2012

  56. [64]

    Bello, C

    M. Bello, C. E. Creffield, and G. Platero, Scientific Re- ports 6, 22562 EP (2016)

    2016

  57. [65]

    Bello, C

    M. Bello, C. E. Creffield, and G. Platero, Phys. Rev. B 95, 094303 (2017)

    2017

  58. [66]

    M. A. Gorlach and A. N. Poddubny, Phys. Rev. A 95, 053866 (2017)

    2017

  59. [67]

    Salerno, M

    G. Salerno, M. Di Liberto, C. Menotti, and I. Carusotto, Phys. Rev. A 97, 013637 (2018)

    2018

  60. [68]

    I. Afek, O. Ambar, and Y. Silberberg, Science 328, 879 (2010)

    2010

  61. [69]

    M. A. Gorlach and A. N. Poddubny, Phys. Rev. A 95, 033831 (2017)

    2017

  62. [70]

    Longhi and G

    S. Longhi and G. D. Valle, Opt. Lett. 36, 4743 (2011)

    2011

  63. [71]

    Di Liberto, A

    M. Di Liberto, A. Recati, I. Carusotto, and C. Menotti, Phys. Rev. A 94, 062704 (2016)

    2016

  64. [72]

    B. A. Bell, K. Wang, A. S. Solntsev, D. N. Neshev, A. A. Sukhorukov, and B. J. Eggleton, Optica 4, 1433 (2017)

    2017

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