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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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“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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“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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“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.
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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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“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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“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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“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
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“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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“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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“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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“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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“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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“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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“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
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- 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 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
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“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
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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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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.
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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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“In two dimensions, Klein tunneling, weak antilocalization, unconventional Landau levels, or bulk pseudoconfinement appear as someof theirmost remarkable features [1,2].”
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“Polaritons are particularly well suited to study these scenarios: 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]; they are also sensitive to external magnetic fields, allowing the implem…”
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“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
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Self-citationExperimental observation of topological Fermi arcs in type-II Weyl semimetal MoTe2
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“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
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“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].”
BackgroundMethodInfluentialSelf-citationSpectroscopic evidence for a type II Weyl semimetallic state in MoTe2
Huang, L., McCormick, T. M., Ochi, M., Zhao, Z., Suzuki, M., Arita, R., Wu, Y., Mou, D., Cao, H. B., Yan, J., et al. (2016) · Nature Materials · doi:10.1038/nmat4685 · Cited by 498
Sun, K., Yao, H., Fradkin, E., & Kivelson, S. A. (2009) · Physical Review Letters · doi:10.1103/physrevlett.103.046811 · Cited by 438 · cited 1×
“This behavior has been predicted for other lattice geometries [45], and it presents a natural playground to investigate the transition between different topological phases when particle interactions are present [51,52] or when time-reversal symmetry is broken.”
BackgroundPressure-Induced Zero-Gap Semiconducting State in Organic Conductor α-(BEDT-TTF)2I3Salt
Katayama, S., Kobayashi, A., & Suzumura, Y. (2006) · Journal of the Physical Society of Japan · doi:10.1143/jpsj.75.054705 · Cited by 419
Tilted anisotropic Dirac cones in quinoid-type graphene and α−(BEDT-TTF)2I3
Goerbig, M. O., Fuchs, J.-N., Montambaux, G., & Piéchon, F. (2008) · Physical Review B · doi:10.1103/physrevb.78.045415 · Cited by 405
Self-citationMerging of Dirac points in a two-dimensional crystal
Montambaux, G., Piéchon, F., Fuchs, J.-N., & Goerbig, M. O. (2009) · Physical Review B · doi:10.1103/physrevb.80.153412 · Cited by 360 · cited 2×
“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].”
BackgroundSelf-citationIdeal Weyl points and helicoid surface states in artificial photonic crystal structures
Yang, B., Guo, Q., Tremain, B., Liu, R., Barr, L. E., Yan, Q., Gao, W., Liu, H., Xiang, Y., Chen, J., et al. (2018) · Science · doi:10.1126/science.aaq1221 · Cited by 348 · cited 1×
“Using more sophisticated approaches, evidence of type-II Weyl points has recently been reported in microwave metamaterials [33,34] and via conical diffraction in laser-written waveguides with elaborate couplings [35].”
MethodPhotocurrents in Weyl semimetals
Chan, C.-K., Lindner, N. H., Refael, G., & Lee, P. A. (2017) · Physical Review B · doi:10.1103/physrevb.95.041104 · Cited by 327 · cited 1×
“The interest of materials presenting this dispersion is its nonisotropic transport properties, which can be used for valley filtering in p-n junctions [14] or for the generation of a photocurrent [15].”
BackgroundKarzig, T., Bardyn, C.-E., Lindner, N. H., & Refael, G. (2015) · Physical Review X · doi:10.1103/physrevx.5.031001 · Cited by 322 · cited 1×
“Polaritons are particularly well suited to study these scenarios: 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]; they are also sensitive to external magnetic fields, allowing the implem…”
BackgroundExperimental observation of optical Weyl points and Fermi arc-like surface states
Noh, J., Huang, S., Leykam, D., Chong, Y. D., Chen, K. P., & Rechtsman, M. C. (2017) · Nature Physics · doi:10.1038/nphys4072 · Cited by 314 · cited 1×
“Using more sophisticated approaches, evidence of type-II Weyl points has recently been reported in microwave metamaterials [33,34] and via conical diffraction in laser-written waveguides with elaborate couplings [35].”
MethodTopological Creation and Destruction of Edge States in Photonic Graphene
Rechtsman, M. C., Plotnik, Y., Zeuner, J. M., Song, D., Chen, Z., Szameit, A., & Segev, M. (2013) · Physical Review Letters · doi:10.1103/physrevlett.111.103901 · Cited by 302 · 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].”
BackgroundExperimental Realization of Type-II Dirac Fermions in a PdTe 2 Superconductor
Noh, H.-J., Jeong, J., Cho, E.-J., Kim, K., Min, B. I., & Park, B. (2017) · Physical Review Letters · doi:10.1103/physrevlett.119.016401 · Cited by 298
Structured Weyl Points in Spin-Orbit Coupled Fermionic Superfluids
Xu, Y., Zhang, F., & Zhang, C. (2015) · Physical Review Letters · doi:10.1103/physrevlett.115.265304 · Cited by 295 · cited 1×
“larger than 1, a type-II Dirac point [10,16,17] is formed [Fig.”
BackgroundPolariton Z Topological Insulator
Nalitov, A. V., Solnyshkov, D. D., & Malpuech, G. (2015) · Physical Review Letters · doi:10.1103/physrevlett.114.116401 · Cited by 273
Towards polariton blockade of confined exciton–polaritons
Delteil, A., Fink, T., Schade, A., Höfling, S., Schneider, C., & İmamoğlu, A. (2019) · Nature Materials · doi:10.1038/s41563-019-0282-y · Cited by 254 · cited 1×
“Going a step further, recent experimental reports have shown evidence of singlepolariton blockade effects in a single confined site [59,60], demonstrating the great potentiality of the polariton platform to study strongly correlated phases in elaborate landscapes such as those provided by orbital polariton lattices.”
BackgroundBrillouin-zone-selection effects in graphite photoelectron angular distributions
Shirley, E. L., Terminello, L. J., Santoni, A., et al. (1995) · Physical review. B, Condensed matter · doi:10.1103/physrevb.51.13614 · Cited by 248 · cited 1×
“To avoid destructive interference effects along high-symmetric crystallographic directions, characteristic of bipartite lattices [42], we record the emission as a function of ky for kx 1⁄4 4π=3d, passing through the K, Γ, and K’ points in the second Brillouin zone [dashed line in Fig.”
Methodp x , y -orbital counterpart of graphene: Cold atoms in the honeycomb optical lattice
Wu, C., & Sarma, S. D. (2008) · Physical Review B · doi:10.1103/physrevb.77.235107 · Cited by 213
Creating and probing electron whispering-gallery modes in graphene
Zhao, Y., Wyrick, J., Natterer, F. D., Rodriguez-Nieva, J. F., Lewandowski, C., Watanabe, K., Taniguchi, T., Levitov, L. S., Zhitenev, N. B., & Stroscio, J. A. (2015) · Science · doi:10.1126/science.aaa7469 · Cited by 212 · cited 1×
“In two dimensions, Klein tunneling, weak antilocalization, unconventional Landau levels, or bulk pseudoconfinement appear as someof theirmost remarkable features [1,2].”
BackgroundAtom-Photon Interactions: Basic Processes and Applications
Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G., & Meystre, P. (1992) · Physics Today · doi:10.1063/1.2809840 · Cited by 190
Topological Transition of Dirac Points in a Microwave Experiment
Bellec, M., Kuhl, U., Montambaux, G., & Mortessagne, F. (2013) · Physical Review Letters · doi:10.1103/physrevlett.110.033902 · Cited by 189 · 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].”
BackgroundSelf-citationEmergence of quantum correlations from interacting fibre-cavity polaritons
Muñoz‐Matutano, G., Wood, A., Johnsson, M., Vidal, X., Baragiola, B. Q., Reinhard, A., Lemaître, A., Bloch, J., Amo, A., Nogues, G., et al. (2019) · Nature Materials · doi:10.1038/s41563-019-0281-z · Cited by 188 · cited 1×
“Going a step further, recent experimental reports have shown evidence of singlepolariton blockade effects in a single confined site [59,60], demonstrating the great potentiality of the polariton platform to study strongly correlated phases in elaborate landscapes such as those provided by orbital polariton lattices.”
BackgroundSelf-citationSpin-Orbit Coupling for Photons and Polaritons in Microstructures
Sala, V. G., Solnyshkov, D. D., Carusotto, I., Jacqmin, T., Lemaître, A., Terças, H., Nalitov, A., Abbarchi, M., Galopin, E., Sagnes, I., et al. (2015) · Physical Review X · doi:10.1103/physrevx.5.011034 · Cited by 172 · cited 1×
“2(b), the pmodes show a small splitting arising from small anisotropy in the shape of the pillar and from polarization-momentum coupling characteristic of semiconductor microcavities [38,39].”
BackgroundSelf-citationExciton-polaritons in lattices: A non-linear photonic simulator
Amo, A., & Bloch, J. (2016) · Comptes Rendus. Physique · doi:10.1016/j.crhy.2016.08.007 · Cited by 157 · cited 1×
“tonic lattices represent an opportunity to explore the physics of unconventional Dirac points thanks to the at-will control of on-site energies and hoppings [29].”
BackgroundSelf-citationBlack hole and hawking radiation by type-II Weyl fermions
Воловик, Г. Е. (2016) · Journal of Experimental and Theoretical Physics Letters · doi:10.1134/s0021364016210050 · Cited by 147 · cited 1×
“This is an asset for the engineering of photonic materials that combine different types of Dirac dispersions, a promising configuration for the study of analogue black holes in photonics [21,25,26].”
BackgroundUnconventional Flatband Line States in Photonic Lieb Lattices
Xia, S., Ramachandran, A., Xia, S., Li, D., Liu, X., Tang, L., Hu, Y., Song, D., Xu, J., Leykam, D., et al. (2018) · Physical Review Letters · doi:10.1103/physrevlett.121.263902 · Cited by 147 · cited 1×
“Similar one-dimensional plaquette states have recently been reported related to the two-dimensional flat band of a Lieb lattice [48].”
BackgroundEffective theory of quadratic degeneracies
Chong, Y. D., Wen, X.-G., & Soljačić, M. (2008) · Physical Review B · doi:10.1103/physrevb.77.235125 · Cited by 146
Hirata, M., Ishikawa, K., Miyagawa, K., Tamura, M., Berthier, C., Basko, D. M., Kobayashi, A., Matsuno, G., & Kanoda, K. (2016) · Nature Communications · doi:10.1038/ncomms12666 · Cited by 138 · cited 2×
“In two– dimensional materials, this kind of tilted Dirac dispersion has been predicted to appear in quinoid-type [10] and hydrogenated graphene [11], and it has been indirectly evidenced in the organic semiconductor [10,12,13] α − ðBEDT − TTFÞ2I3.”
BackgroundPhotoinduced Nonequilibrium Topological States in Strained Black Phosphorus
Liu, H., Sun, J. P., Cheng, C., Liu, F., & Meng, S. (2018) · Physical Review Letters · doi:10.1103/physrevlett.120.237403 · Cited by 137 · cited 1×
“dispersion has been labeled a type-III Dirac cone [21,22].”
Bloch-Zener Oscillations across a Merging Transition of Dirac Points
Lim, L.-K., Fuchs, J.-N., & Montambaux, G. (2012) · Physical Review Letters · doi:10.1103/physrevlett.108.175303 · Cited by 134 · 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].”
BackgroundSelf-citationQuantum dimer calculations on the spin-1/2 kagome ´ Heisenberg antiferromagnet
Zeng, C., & Elser, V. (1995) · Physical review. B, Condensed matter · doi:10.1103/physrevb.51.8318 · Cited by 123
Polariton condensation in photonic molecules.
Galbiati, M., Ferrier, L., Solnyshkov, D., Tanese, D., Wertz, E. A., Amo, A., Abbarchi, M., Senellart, P., Sagnes, I., Lemaître, A., et al. (2011) · Physical Review Letters · doi:10.1103/physrevlett.108.126403 · Cited by 118 · cited 1×
“2(d), while keeping the other two couplings tL constant [44].”
BackgroundSelf-citationOrbital Edge States in a Photonic Honeycomb Lattice
Milićević, M., Ozawa, T., Montambaux, G., Carusotto, I., Galopin, E., Lemaître, A., Le Gratiet, L., Sagnes, I., Bloch, J., & Amo, A. (2017) · Physical Review Letters · doi:10.1103/physrevlett.118.107403 · Cited by 117 · cited 2×
“discrete photonic spectrum of a single micropillar, with the lowest-energy s mode being cylindrically symmetric, and the first excited modes formed by two px, py orbitals with lobes oriented 90° from each other and a node at the center of the pillar [37].”
BackgroundMethodSelf-citationSpin Textures of Exciton-Polaritons in a Tunable Microcavity with Large TE-TM Splitting
Dufferwiel, S., Li, F., Cancellieri, E., Giriunas, L., Trichet, A. A. P., Whittaker, D. M., Walker, P. M., Fras, F., Clarke, E., Smith, J. M., et al. (2015) · Physical Review Letters · doi:10.1103/physrevlett.115.246401 · Cited by 110 · cited 1×
“2(b), the pmodes show a small splitting arising from small anisotropy in the shape of the pillar and from polarization-momentum coupling characteristic of semiconductor microcavities [38,39].”
BackgroundBlack-hole horizon in the Dirac semimetal Zn2In2S5
Huang, H., Jin, K., & Liu, F. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.98.121110 · Cited by 109
Edge states in polariton honeycomb lattices
Milićević, M., Ozawa, T., Andreakou, P., Carusotto, I., Jacqmin, T., Galopin, E., Lemaître, A., Le Gratiet, L., Sagnes, I., Bloch, J., et al. (2015) · 2D Materials · doi:10.1088/2053-1583/2/3/034012 · Cited by 76 · cited 1×
“2(e) and studied in previous works [36,40].”
Self-citationKlein tunneling and electron optics in Dirac-Weyl fermion systems with tilted energy dispersion
Nguyễn, V. H., & Charlier, J. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.97.235113 · Cited by 73 · cited 1×
“The interest of materials presenting this dispersion is its nonisotropic transport properties, which can be used for valley filtering in p-n junctions [14] or for the generation of a photocurrent [15].”
BackgroundWang, H.-X., Chen, Y., Hang, Z. H., Kee, H.-Y., & Jiang, J.-H. (2017) · npj Quantum Materials · doi:10.1038/s41535-017-0058-z · Cited by 70 · cited 1×
“Current schemes are based on the design of long-distant coupling of photons in lattices of resonators [30,31] and the use of screw symmetries [32].”
BackgroundManipulation of Dirac points in graphene-like crystals
de Gail, R., Fuchs, J.-N., Goerbig, M. O., et al. (2012) · Physica B Condensed Matter · doi:10.1016/j.physb.2012.01.072 · Cited by 58
Self-citationManipulating type-I and type-II Dirac polaritons in cavity-embedded honeycomb metasurfaces
Mann, C.-R., Sturges, T. J., Weick, G., Barnes, W. L., & Mariani, E. (2018) · Nature Communications · doi:10.1038/s41467-018-03982-7 · Cited by 58
Topological Varma Superfluid in Optical Lattices
Liberto, M. F. D., Hemmerich, A., & Smith, C. M. (2016) · Physical Review Letters · doi:10.1103/physrevlett.117.163001 · Cited by 57 · 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].”
BackgroundEffect of the type-I to type-II Weyl semimetal topological transition on superconductivity
Li, D., Rosenstein, B., Shapiro, B. Y., & Shapiro, I. (2017) · Physical review. B./Physical review. B · doi:10.1103/physrevb.95.094513 · Cited by 55 · 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].”
BackgroundEmergence of Type-II Dirac Points in Graphynelike Photonic Lattices
Pyrialakos, G. G., Nye, N. S., Kantartzis, N. V., & Christodoulides, D. N. (2017) · Physical Review Letters · doi:10.1103/physrevlett.119.113901 · Cited by 54 · cited 2×
“In most of these cases, the tilted Dirac cones appear as a consequence of band inversions [30,31], a situation in which the magnitude of the tilt cannot be easily controlled.”
BackgroundOccurrence of nematic, topological, and Berry phases when a flat and a parabolic band touch
Dóra, B., Herbut, I. F., & Moessner, R. (2014) · Physical Review B · doi:10.1103/physrevb.90.045310 · Cited by 53
Tilted anisotropic Dirac cones in partially hydrogenated graphene
Lu, H., Cuamba, A. S., Lin, S., Hao, L., Wang, R., Li, H., Zhao, Y., & Ting, C. S. (2016) · Physical review. B./Physical review. B · doi:10.1103/physrevb.94.195423 · Cited by 53 · cited 1×
“In two– dimensional materials, this kind of tilted Dirac dispersion has been predicted to appear in quinoid-type [10] and hydrogenated graphene [11], and it has been indirectly evidenced in the organic semiconductor [10,12,13] α − ðBEDT − TTFÞ2I3.”
BackgroundWinding Vector: How to Annihilate Two Dirac Points with the Same Charge
Montambaux, G., Lim, L.-K., Fuchs, J.-N., & Piéchon, F. (2018) · Physical Review Letters · doi:10.1103/physrevlett.121.256402 · Cited by 53
Self-citationType-II Dirac Photons at Metasurfaces
Hu, C., Li, Z., Tong, R., Wu, X., Xia, Z., Wang, L., Li, S., Huang, Y., Wang, S., Hou, B., et al. (2018) · Physical Review Letters · doi:10.1103/physrevlett.121.024301 · Cited by 47 · cited 1×
“Using more sophisticated approaches, evidence of type-II Weyl points has recently been reported in microwave metamaterials [33,34] and via conical diffraction in laser-written waveguides with elaborate couplings [35].”
MethodFermionic analogue of black hole radiation with a super high Hawking temperaturePreprint
Liu, H., Sun, J., Huang, H., Liu, F., & Meng, S. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1809.00479 · Cited by 7
Topological transition in stratified fluids
Perrot, M., Delplace, P., & Venaille, A. (2018) · arXiv · doi:10.48550/arxiv.1810.03328
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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
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