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Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene
Open accessAbstract
The extraordinary electronic properties of Dirac materials, the two-dimensional partners of Weyl semimetals, arise from the linear crossings in their band structure.When the dispersion around the Dirac points is tilted, one can predict the emergence of intricate transport phenomena such as modified Klein tunneling, intrinsic anomalous Hall effects, and ferrimagnetism.However, Dirac materials are rare, particularly with tilted Dirac cones.Recently, artificial materials whose building blocks present orbital degrees of freedom have appeared as promising candidates for the engineering of exotic Dirac dispersions.Here we take advantage of the orbital structure of photonic resonators arranged in a honeycomb lattice to implement photonic lattices with semi-Dirac, tilted, and, most interestingly, type-III Dirac cones that combine flat and linear dispersions.Type-III Dirac cones emerge from the touching of a flat and a parabolic band when synthetic photonic strain is introduced in the lattice, and they possess a nontrivial topological charge.This photonic realization provides a recipe for the synthesis of orbital Dirac matter with unconventional transport properties and, in combination with polariton nonlinearities, opens the way to study Dirac superfluids in topological landscapes.
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TL;DR
This paper demonstrates photonic orbital graphene in a honeycomb lattice of polariton micropillars, where strain in the orbital p bands produces semi-Dirac, tilted, and type-III Dirac cones. It reports the first experimental observation of type-III Dirac cones, formed by the touching of a flat band and a linear band, and explains their emergence via topological Lifshitz transitions.
Key findings
- Type-III Dirac cones were observed for synthetic strain β = 1.5, combining a flat band and a linear band; measured velocities were |v0x| = (0.29 ± 0.03) d|tL| and |vx| = (0.35 ± 0.03) d|tL|, in agreement with tight-binding prediction |v0x| = |vx| = 0.37 d|tL|. Along the perpendicular direction, |v0y| = (0.00 ± 0.04) d|tL| and |vy| = (0.46 ± 0.04) d|tL|, matching predictions of 0 and 0.48 d|tL|.
- A semi-Dirac dispersion was directly observed at β = 0.5, where two Dirac cones merge, with parabolic touching along ky and linear dispersion along kx.
- Tilted type-I Dirac cones emerge from the flat-parabolic band touching at Γ for β < 1, with measured tilt v0y agreeing with tight-binding calculations.
- Topological analysis shows the β = 1 flat-parabolic touching has winding W = 2; for β ≠ 1 it evolves into a pair of Dirac cones with W = 1, while the semi-Dirac merging has W = 0. Type-III cones are robust for 1 < β < 2 and merge at β = 2.
Methods
- The platform is a honeycomb lattice of coupled micropillars etched from an AlGaAs semiconductor microcavity with twelve GaAs quantum wells, operating in the strong-coupling regime at 10 K to form exciton-polaritons. Micropillar diameter is 2.75 μm (2.60 μm for type-III samples), with center-to-center distances varied between 2.40 and 2.72 μm, or 2.40 and 2.10 μm for type-III.
- Angle-resolved photoluminescence under nonresonant continuous-wave excitation at 745 nm was used to reconstruct the band dispersion. The p-orbital bands arise from coupling of first excited modes, with longitudinal hopping tL and transverse hopping tT.
- Synthetic uniaxial strain was implemented by changing the center-to-center distance d′ along one link direction, defining β = t′L/tL. Experimental dispersions were fitted with a nearest-neighbor tight-binding model, and effective Dirac velocities were extracted from linear fits near the Dirac points.
Limitations
- The paper states that the β = 2 merging of type-III Dirac cones could not be observed experimentally because engineered hoppings in compressed structures were limited to β = 1.5.
- The paper notes that tT is not strictly zero in the experiment, producing faint emission between horizontal links; simulations reproduce this with tT = −0.1tL on strained links.
- The paper states that the uppermost band deviates from a flat band due to coupling to higher-energy bands from photonic d orbitals, and that polarization-related splitting is neglected because it is much smaller than the bandwidth.
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In the paper
“The orbital bands present a flat band that touches a parabolic band.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > I. INTRODUCTION
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In the paper
“The result of the compression in the polariton lattice is shown in Fig. 5(c) for β ¼ 1.5:In the direction k x parallel to that along which the center-to-center distance d 0 is reduced, two new Dirac points emerge from the flat-parabolic band touching.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > III. TILTED AND SEMI-DIRAC CONES
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In the paper
“The richness of this multiband system allows, in addition, the observation of semi-Dirac cones, which combine massless and massive dispersions.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > I. INTRODUCTION
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“they possess a nontrivial topological charge”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > Abstract
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In the paper
“Type-III Dirac cones emerge from the touching of a flat and a parabolic band when synthetic photonic strain is introduced in the lattice”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > Abstract
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In the paper
“Decreasing β, that is, emulating the stretching of the lattice, brings the Dirac cones closer together in the k y direction [Figs.3(a) and 3(b)] until they merge at β ¼ 0.5 in a single band touching [Fig.3(c)].”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > III. TILTED AND SEMI-DIRAC CONES
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In the paper
“we provide the first experimental observation of type-III Dirac cones”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > I. INTRODUCTION
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Rests on an assumption: Type-III Dirac points have not yet been reported experimentally.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“When asymmetry in the hopping is introduced, which simulates uniaxial strain in solid-state graphene, the flat-parabolic band touching evolves into tilted and type-III Dirac cones.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > I. INTRODUCTION
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“If we account for different hoppings t L and t 0 L for different links, Hamiltonian (B2) is still valid with the same expressions for f 1 and g, and f 2 replaced by f 2 ¼ β þ 1 4 ðe ika 1 þ e ika 2 Þ.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > VI. CONCLUSION
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In the paper
“The touching of a flat and a dispersive band with winding W ¼ 2 evolves into two Dirac cones in the presence of strain.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > VI. CONCLUSION
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In the paper
“As a function of β ¼ t 0 L =t L , its spectrum ϵðkÞ takes the form ϵðkÞ ¼ Æ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 0 ðkÞ Æ EðkÞ p ;ðB3Þ”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > VI. CONCLUSION
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In the paper
“By analyzing their topological charge, i.e., the winding of the Hamiltonian around each Dirac point, we show that the semi-Dirac, tilted, and type-III Dirac cones emerge as a consequence of topological Lifshitz transitions induced by strain in the orbital bands.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > I. INTRODUCTION
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In the paper
“The fact that they emerge from a flat band makes type-III Dirac cones particularly robust to changes in the hopping, as we show below.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > I. INTRODUCTION
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“The coupling of the p x , p y orbitals in the isotropic honeycomb lattice can be described by a tight-binding Hamiltonian of the form [36,43]”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > Additional information
What the sources say
- Not matchedcites [36]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [43]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The photonic realization reported here demonstrates the flexibility of orbital bands to implement unconventional Dirac points.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > VI. CONCLUSION
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- It may state two separate claims as one. 18
- Probably not a substantial claim. 4
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In the paper
“At the merging point [β ¼ 0.5, Fig. 3(c)], we provide the direct observation of a semi-Dirac dispersion, with the touching of two parabolic bands along the k y direction and a linear dispersion along k x [shown in the left inset of Fig. 3(c)].”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > III. TILTED AND SEMI-DIRAC CONES
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In the paper
“Four bands are observed: The lowest one is flat, while the two central bands present two type-I Dirac points at the K and K' points and touch the flat band at the Γ point [36].”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > II. PHOTONIC ORBITAL LATTICE
What the sources say
- Not matchedcites [36]
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In the paper
“while for β > 1, they are E D ¼ E 0 - 3 2 jt L j; cosðK x Þ ¼ 2 -β 2 β ; K y ¼ 0:ðB7Þ”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > VI. CONCLUSION
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In the paper
“After converting the slopes to effective Weyl velocities [Eq.( 1)], we obtain jv 0y j ¼ 0.07 Æ 0.03, close to the expected value of 0, and jv y j ¼ 0.72 Æ 0.03 in units of djt L j.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > APPENDIX C: MEASUREMENT AND CALCULATION OF THE EFFECTIVE DIRAC VELOCITIES
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In the paper
“At 10 K, the temperature of the experiments, the excitonic and photonic resonances are in the strong coupling regime, with a Rabi splitting of 15 meV.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > Additional information
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In the paper
“For β < 1, the energy E D and momentum-space coordinates of the Dirac points emerging at the bottom bands are, respectively, E D ¼ E 0 - jt L j 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 þ 6β 2 q ; K x ¼ 0; cosðK y Þ ¼ β;ðB6Þ”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > VI. CONCLUSION
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In the paper
“The single band touching at β ¼ 1 is described by a winding W ¼ 2. When β ≷ 1, the band touching evolves into a pair of Dirac cones (tilted type-I or type-III) with winding W ¼ 1.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > V. TOPOLOGICAL INVARIANTS OF TILTED AND TYPE-III DIRAC CONES
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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“From the experimental photoluminescence of Fig. 5(c), we measure jv 0x j ¼ ð0.29 Æ 0.03Þ djt L j and jv x j ¼ ð0.35 Æ 0.03Þ djt L j [with d ¼ ð2d 0 þ dÞ=3], which is in good agreement with the tight-binding prediction for β ¼ 1.5 (jv 0x j ¼ jv x j ¼ 0.37 djt L j, see Appendix C).”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > III. TILTED AND SEMI-DIRAC CONES
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In the paper
“For 1 < β < 2, we have (same units as above) jv 0x j ¼ jv x j ¼ 1 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðβ 2 -1Þð4 -β 2 Þ q; jv 0y j ¼ 0; jv y j ¼ ffiffi ffi 3 p 4 ffiffiffiffiffiffiffiffiffiffiffiffi ffi β 2 -1 q :”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > APPENDIX C: MEASUREMENT AND CALCULATION OF THE EFFECTIVE DIRAC VELOCITIES
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In the paper
“The type-III Dirac cones are thus present for all values of β between 1 and 2, proving their robustness against variations of the strain parameter.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > III. TILTED AND SEMI-DIRAC CONES
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“From the tight-binding Hamiltonian, the effective Dirac velocities can be obtained from the derivative along k x and k y of the eigenvalues of Eqs.(B3)-(B5) at the position of the emergent Dirac cones [Eqs.(B6) and (B7)], resulting in the analytical expressions for β < 1 (normalized to djt L j, where we have assumed that, in the experiment, d ≈ d 0 ):”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > APPENDIX C: MEASUREMENT AND CALCULATION OF THE EFFECTIVE DIRAC VELOCITIES
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- The authors state this as an assumption: In the experiment, d ≈ d'.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“jv 0x j ¼ 0; jv x j ¼ jv 0y j ¼ 3β 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 -β 2 1 þ 2β 2 s ; jv y j ¼ ffiffi ffi 3 p 4 ffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 -β 2 q :”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > APPENDIX C: MEASUREMENT AND CALCULATION OF THE EFFECTIVE DIRAC VELOCITIES
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In the paper
“At β ¼ 2, actually, the Dirac cones merge at the M point and annihilate, opening a gap for β > 2.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > V. TOPOLOGICAL INVARIANTS OF TILTED AND TYPE-III DIRAC CONES
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In the paper
“In the etched structures, the lateral defects induced during the microfabrication process reduce the lifetime to about 5-15 ps at the considered photon-exciton detunings.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > Additional information
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The polariton lifetime measured in a similar unetched microcavity is of the order of 30 ps.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > Additional information
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“Remarkably, the orbital p bands show the two possible universal scenarios for the merging of Dirac cones in two dimensions [50]: (i) Two Dirac cones with W ¼ 1 emerge from a point with W ¼ 2, which we have just discussed, and it is reported in the lower part of the spectra of Fig. 3; (ii) two Dirac cones with opposite winding W ¼ Æ1 merge in a semi-Dirac cone with W ¼ 0, reported in the spectra in”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > V. TOPOLOGICAL INVARIANTS OF TILTED AND TYPE-III DIRAC CONES
What the sources say
- Not matchedcites [50]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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- Not matched
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In the paper
“they are also sensitive to external magnetic fields, allowing the implementation of quantum Hall phases [54,55]”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > VI. CONCLUSION
What the sources say
- Not matchedcites [54]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [55]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- Not matched
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In the paper
“Note that for type-I Dirac cones, v 0x ¼ v 0y ¼ 0, and the absolute value of the two slopes should be identical and equal to v y .”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > APPENDIX C: MEASUREMENT AND CALCULATION OF THE EFFECTIVE DIRAC VELOCITIES
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In the paper
“Thanks to their excitonic component, they present significant repulsive interactions in the high-density regime, which can be achieved when using a resonant excitation scheme [53]”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > VI. CONCLUSION
What the sources say
- Not matchedcites [53]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
- Not matched
In the paper
“This behavior has been predicted for other lattice geometries [45]”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > VI. CONCLUSION
What the sources say
- Not matchedcites [45]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Note that the fact that jv 0x j ¼ jv x j (with opposite sign: v 0x ¼ -v x ) for β > 1 is the signature of the type-III Dirac cone, and it results in a flat band along the k x direction.”
§ Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene > APPENDIX C: MEASUREMENT AND CALCULATION OF THE EFFECTIVE DIRAC VELOCITIES
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper relies on this without support.
Type-III Dirac points have not yet been reported experimentally.
“Type-III Dirac points have not yet been reported experimentally”
The authors state this as an assumption.
In the nearest-neighbor tight-binding calculation, the coupling t_T is assumed to be zero.
“assuming t T ¼ 0”
The authors state this as an assumption.
The on-site energy for the p_x, p_y orbitals is 0.
“where we have assumed that the on-site energy for the p x , p y orbitals is 0,”
The authors state this as an assumption.
In the experiment, d ≈ d'.
“where we have assumed that, in the experiment, d ≈ d 0”
The authors state this as an assumption.
The hopping of orbitals oriented perpendicular to the links is negligible (t_T = 0).
“If the hopping of orbitals oriented perpendicular to the links is negligible (t T ¼ 0)”
Citations
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BackgroundTopological Insulators and Topological Superconductors
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Tarruell, L., Greif, D., Uehlinger, T., Jotzu, G., & Esslinger, T. (2012) · Nature · doi:10.1038/nature10871 · Cited by 972
Observation of tunable band gap and anisotropic Dirac semimetal state in black phosphorus
Kim, J., Baik, S. S., Ryu, S. H., Sohn, Y., Park, S., Park, B.-G., Denlinger, J. D., Yi, Y., Choi, H. J., & Kim, K. S. (2015) · Science · doi:10.1126/science.aaa6486 · Cited by 938 · cited 1×
“This case is a topological Lifshitz transition in which two Dirac cones with opposite topological charge merge and annihilate, as predicted for standard s band graphene [4,45] and reported in photonic [7,8] and atomic [5,6] honeycomb lattices and in black phosphorus [9].”
BackgroundLasing in topological edge states of a one-dimensional lattice
St-Jean, P., Goblot, V., Galopin, E., Lemaître, A., Ozawa, T., Le Gratiet, L., Sagnes, I., Bloch, J., & Amo, A. (2017) · Nature Photonics · doi:10.1038/s41566-017-0006-2 · Cited by 892
Self-citationExperimental observation of topological Fermi arcs in type-II Weyl semimetal MoTe2
Deng, K., Wan, G., Deng, P., Zhang, K., Ding, S., Wang, E., Yan, M., Huang, H., Zhang, H., Xu, Z., et al. (2016) · Nature Physics · doi:10.1038/nphys3871 · Cited by 794 · cited 1×
“Type-III Dirac points have not yet been reported experimentally, and tilted type-I and type-II Dirac points have been challenging to synthesize [13,18,20] because”
BackgroundArtificial flat band systems: from lattice models to experiments
Leykam, D., Andreanov, A., & Sergej Flach, A. (2018) · Advances in Physics X · doi:10.1080/23746149.2018.1473052 · Cited by 536 · cited 1×
“conducting gap in Weyl semimetals [23], and they provide a new platform for the study of correlated phases with a flat band [24].”
BackgroundExciton-polariton topological insulator
Klembt, S., Harder, T. H., Egorov, O. A., Winkler, K., Ge, R., Bandres, M. A., Emmerling, M., Worschech, L., Liew, T. C. H., Segev, M., et al. (2018) · Nature · doi:10.1038/s41586-018-0601-5 · Cited by 535 · cited 1×
“Polariton orbital bands of the kind reported here open exciting perspectives for the study of topological lasers [56,57] and of Dirac superfluids in the mean-field interacting regime [58].”
Background
Flagged first, then by citations
Cited by
Sorted by citations
Ozawa, T., Price, H. M., Amo, A., Goldman, N., Hafezi, M., Lu, L., Rechtsman, M. C., Schuster, D., Simon, J., Zilberberg, O., et al. (2019) · Reviews of Modern Physics · doi:10.1103/revmodphys.91.015006 · Cited by 3,722
“Interestingly, homogeneous strain has been employed to synthesize critically tilted Dirac cones in the orbital p bands of lattices of coupled micropillars (Milićević et al., 2018).”
MethodSelf-citationNi, X., Yves, S., Krasnok, A., & Alù, A. (2023) · Chemical Reviews · doi:10.1021/acs.chemrev.2c00800 · Cited by 166
Photonic flat‐band lattices and unconventional light localization
Tang, L., Song, D., Xia, S., Xia, S., Ma, J., Yan, W., Hu, Y., Xu, J., Leykam, D., & Chen, Z. (2020) · Nanophotonics · doi:10.1515/nanoph-2020-0043 · Cited by 126
Li, J., Yang, X., Ma, C., et al. (2021) · Applied Catalysis B: Environmental · doi:10.1016/j.apcatb.2021.120053 · Cited by 96
Direct observation of photonic Landau levels and helical edge states in strained honeycomb lattices
Jamadi, O., Rozas, E., Salerno, G., Milićević, M., Ozawa, T., Sagnes, I., Lemaître, A., Gratiet, L. L., Harouri, A., Carusotto, I., et al. (2020) · Light Science & Applications · doi:10.1038/s41377-020-00377-6 · Cited by 97
“[33] M.”
Self-citation
Metadata
- Authors
- Milićević, M., Montambaux, G., Ozawa, T., Jamadi, O., Real, B., Sagnes, I., Lemaître, A., Le Gratiet, L., Harouri, A., Bloch, J., & Amo, A.
- Affiliations
- Université Paris-Sud (FR)
- RIKEN (JP)
- Centre National de la Recherche Scientifique (FR)
- Venue
- Physical Review X
- Publisher
- American Physical Society (APS)
- Published
- 2019-07-23
- Volume
- 9
- Issue
- 3
- ISSN
- 2160-3308 (ISSN-L)
- OpenAlex
- W2884840542
- Semantic Scholar
- 38b2bab9af1d5a17dfd7279a962bf7883f5d587f
- Cited by
- 171
- References
- 55
- Citations by year
- 2026: 8 · 2025: 27 · 2024: 28 · 2023: 22 · 2022: 27 · 2021: 30 · 2020: 25 · 2019: 4
- FWCI
- 9.54
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- Physical Review X (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- HAL (Le Centre pour la Communication Scientifique Directe) (submittedVersion, other-oa)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- Figshare (submittedVersion)
- arXiv (Cornell University)
- Greater South Information System
- Funders
- H2020 European Research Council
- Agence Nationale de la Recherche (ANR-16-CE30-0021, ANR-11-LABX-0007, ANR-10-LABX-0035)
- Japan Society for the Promotion of Science (JP18H05857)
- RIKEN
- French RENATECH network
- CPER Photonics for Society P4S
- I-Site ULNE
- Métropole Européenne de Lille
- Topics
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Topological Materials and Phenomena
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Strong Light-Matter Interactions
- Keywords
- tilted Dirac cones, flat bands, photonic lattices, honeycomb lattice, photonic resonators, topological charge, Klein tunneling, anomalous Hall effect, orbital degrees of freedom
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-23, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-23 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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