- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Applied

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Tilted Dirac-like microwave cones in superconducting circuit lattices

Ahmad Motavassel1,*, Amir Youssefi2,1,*, Shingo Kono2,3, Seyed Akbar Jafari4,†, and Tobias J. Kippenberg2,3

  • 1EDWATEC SA, Lausanne, Switzerland
  • 2Laboratory of Photonics and Quantum Measurement, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland
  • 3Center for Quantum Science and Engineering, EPFL, Lausanne, Switzerland
  • 4II. Physikalisches Institut C, RWTH Aachen University, 52074 Aachen, Germany

  • *These authors contributed equally.
  • †Contact author: akbar.jafari@rwth-aachen.de
PDF

Phys. Rev. Applied 25, 044061 – Published 22 April, 2026

DOI: https://doi.org/10.1103/84bq-l88b

Abstract

Dirac-like band crossings are paradigms in condensed-matter systems to emulate high-energy physics phenomena. They are associated with two aspects: gap and tilting. The ability to design a sign-changing gap gives rise to band topology, whereas the tilting of band crossings, which is a gateway for large gravitylike effects remains uncharted. In this work, we introduce an experimental platform to realize tilted Dirac-like microwave cone in large-scale superconducting circuit lattices. The direction and magnitude of the tilt can be controlled by engineering the axially preferred second-neighbor coupling. We demonstrate three lattices with 731-site LC resonator featuring tilt values of up to 59% of relative difference in the opposite-direction group velocities. This is obtained by reconstructing the density of states (DOS) of measured microwave resonance frequencies. Harnessing the tilt of Dirac-like band crossings lays the foundation for weaving the fabric of an emergent solid-state spacetime.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (73)

  1. Wolfgang Rindler, Relativity: Special, General, and Cosmological (Oxford University Press, New York, 2006), 2nd ed.
  2. John D. Joannopoulos, Steven G. Johnson, Joshua N. Winn, and Robert D. Meade, Photonic Crystals: Molding The Flow of Light (Princeton University Press, Princeton, NJ, 2008), 2nd ed.
  3. Lewis Ryder, Introduction to General Relativity (Cambridge University Press, New York, 2009).
  4. Saswati Roy and Asoke Kumar Sen, Study of gravitational deflection of light ray, J. Phys.: Conf. Ser. 1330, 012002 (2019).
  5. Ofek Bengyat, Andrea Di Biagio, Markus Aspelmeyer, and Marios Christodoulou, Gravity mediated entanglement between oscillators as quantum superposition of geometries, Phys. Rev. D 110, 056046 (2024).
  6. F. G. Friedlander, The Wave Equation on a Curved Space-Time (Cambridge University Press, Cambridge, 1975).
  7. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge University Press, New York, 1982).
  8. R. V. Pound and G. A. Rebka, Apparent weight of photons, Phys. Rev. Lett. 4, 337 (1960).
  9. Sean M. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Cambridge University Press, San Francisco, CA, 2019).
  10. Hideo Aoki and Mildred S. Dresselhaus, eds. Physics of Graphene (Springer International Publishing, Cham, Switzerland, 2014).
  11. Mikhail I. Katsnelson, The Physics of Graphene (Cambridge University Press, Cambridge, 2020).
  12. N. P. Armitage, E. J. Mele, and Ashvin Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  13. Binghai Yan and Claudia Felser, Topological materials: Weyl semimetals, Annu. Rev. Condens. Matter Phys. 8, 337 (2017).
  14. S. A. Jafari, Moving frame theory of zero-bias photocurrent on the surface of topological insulators, Phys. Rev. Res. 6, 033006 (2024).
  15. N. Ogawa, R. Yoshimi, K. Yasuda, A. Tsukazaki, M. Kawasaki, and Y. Tokura, Zero-bias photocurrent in ferromagnetic topological insulator, Nat. Commun. 7, 12246 (2016).
  16. Ahmad Motavassal and S. A. Jafari, Circuit realization of a tilted Dirac cone: Platform for fabrication of curved spacetime geometry on a chip, Phys. Rev. B 104, L241108 (2021).
  17. Yasin Yekta, Hanif Hadipour, and Seyed Akbar Jafari, Tunning the tilt of the Dirac cone by atomic manipulations in 8Pmmn borophene, Commun. Phys. 6, 46 (2023).
  18. Koji Kajita, Yutaka Nishio, Naoya Tajima, Yoshikazu Suzumura, and Akito Kobayashi, Molecular Dirac fermion systems—theoretical and experimental approaches—, J. Phys. Soc. Jpn. 83, 072002 (2014).
  19. Hong-Yan Lu, Armindo S. Cuamba, Shih-Yang Lin, Lei Hao, Rui Wang, Hai Li, YuanYuan Zhao, and C. S. Ting, Tilted anisotropic Dirac cones in partially hydrogenated graphene, Phys. Rev. B 94, 195423 (2016).
  20. Alejandro Lopez-Bezanilla and Peter B. Littlewood, Electronic properties of 8Pmmn borophene, Phys. Rev. B 93, 241405 (2016).
  21. Xiang-Feng Zhou, Xiao Dong, Artem R. Oganov, Qiang Zhu, Yongjun Tian, and Hui-Tian Wang, Semimetallic two-dimensional boron allotrope with massless Dirac fermions, Phys. Rev. Lett. 112, 085502 (2014).
  22. L. L. Tao and Evgeny Y. Tsymbal, Two-dimensional type-II Dirac fermions in a LaAlO3/LaNiO3/LaAlO3 quantum well, Phys. Rev. B 98, 121102 (2018).
  23. H. Mirhosseini, M. Flieger, and J. Henk, Dirac-cone-like surface state in W(110): Dispersion, spin texture and photoemission from first principles, New J. Phys. 15, 033019 (2013).
  24. K. Miyamoto, A. Kimura, K. Kuroda, T. Okuda, K. Shimada, H. Namatame, M. Taniguchi, and M. Donath, Spin-polarized Dirac-cone-like surface state with d character at W(110), Phys. Rev. Lett. 108, 066808 (2012).
  25. A. Varykhalov, D. Marchenko, J. Sánchez-Barriga, E. Golias, O. Rader, and G. Bihlmayer, Tilted Dirac cone on W(110) protected by mirror symmetry, Phys. Rev. B 95, 245421 (2017).
  26. S. K. Firoz Islam and A. M. Jayannavar, Signature of tilted Dirac cones in Weiss oscillations of 8Pmmn borophene, Phys. Rev. B 96, 235405 (2017).
  27. Anirban Kundu, Zhuo Bin Siu, Hyunsoo Yang, and Mansoor B. A Jalil, Magnetotransport of Weyl semimetals with tilted Dirac cones, New J. Phys. 22, 083081 (2020).
  28. Parijat Sengupta, Yaohua Tan, Enrico Bellotti, and Junxia Shi, Anomalous heat flow in 8Pmmn borophene with tilted Dirac cones, J. Phys.: Condens. Matter 30, 435701 (2018).
  29. Yoshikazu Suzumura, Igor Proskurin, and Masao Ogata, Reflectance of Dirac electrons in organic conductor, J. Phys.: Conf. Ser. 603, 012011 (2015).
  30. Maximilian Trescher, Björn Sbierski, Piet W. Brouwer, and Emil J. Bergholtz, Quantum transport in Dirac materials: Signatures of tilted and anisotropic Dirac and Weyl cones, Phys. Rev. B 91, 115135 (2015).
  31. Andrew Wild, Eros Mariani, and Mikhail E. Portnoi, Optical absorption in two-dimensional materials with tilted Dirac cones, Phys. Rev. B 105, 205306 (2022).
  32. Zhao-Kun Yang, Jing-Rong Wang, and Guo-Zhu Liu, Effects of Dirac cone tilt in a two-dimensional Dirac semimetal, Phys. Rev. B 98, 195123 (2018).
  33. Ahmadreza Moradpouri, Mahdi Torabian, and Seyed Akbar Jafari, Electron currents from gradual heating in tilted Dirac cone materials, SciPost Phys. Core 6, 010 (2023).
  34. Saber Rostamzadeh, Sevval Tasdemir, Mustafa Sarisaman, S. A. Jafari, and Mark-Oliver Goerbig, Tilt-induced vortical response and mixed anomaly in inhomogeneous Weyl matter, Phys. Rev. B 107, 075155 (2023).
  35. A. Moradpouri, S. A. Jafari, and Mahdi Torabian, Holographic hydrodynamics of tilted Dirac materials, J. High Energy Phys. 2023, 5 (2023).
  36. Koji Hashimoto and Yoshinori Matsuo, Escape from black hole analogs in materials: Type-II Weyl semimetals and generic edge states, Phys. Rev. B 102, 195128 (2020).
  37. Huaqing Huang, Kyung-Hwan Jin, and Feng Liu, Black-hole horizon in the Dirac semimetal Zn2In2S5, Phys. Rev. B 98, 121110 (2018).
  38. G. E. Volovik, Black hole and hawking radiation by type-II Weyl fermions, JETP. Lett. 104, 645 (2016).
  39. G. E. Volovik and K. Zhang, Lifshitz transitions, type-II Dirac and Weyl fermions, event horizon and all that, J. Low Temp. Phys. 189, 276 (2017).
  40. Mikhail Zubkov, Analogies between the black hole interior and the type II Weyl semimetals, Universe 4, 135 (2018).
  41. Juan L. Mañes, Fernando deJuan, Mauricio Sturla, and Marña A. H. Vozmediano, Generalized effective hamiltonian for graphene under nonuniform strain, Phys. Rev. B 88, 155405 (2013).
  42. T. Farajollahpour, Z. Faraei, and S. A. Jafari, Solid-state platform for space-time engineering: The 8Pmmn borophene sheet, Phys. Rev. B 99, 235150 (2019).
  43. Iacopo Carusotto, Andrew A. Houck, Alicia J. Kollár, Pedram Roushan, David I. Schuster, and Jonathan Simon, Photonic materials in circuit quantum electrodynamics, Nat. Phys. 16, 268 (2020).
  44. Alexandra S. Sheremet, Mihail I. Petrov, Ivan V. Iorsh, Alexander V. Poshakinskiy, and Alexander N. Poddubny, Waveguide quantum electrodynamics: Collective radiance and photon-photon correlations, Rev. Mod. Phys. 95, 015002 (2023).
  45. Ehud Altman, Kenneth R. Brown, Giuseppe Carleo, Lincoln D. Carr, Eugene Demler, Cheng Chin, Brian DeMarco, Sophia E. Economou, Mark A. Eriksson, Kai-Mei C. Fu et al. Quantum simulators: Architectures and opportunities, PRX Quantum 2, 017003 (2021).
  46. Sebastian Schmidt and Jens Koch, Circuit QED lattices: Towards quantum simulation with superconducting circuits, Ann. Phys. 525, 395 (2013).
  47. Eunjong Kim, Xueyue Zhang, Vinicius S. Ferreira, Jash Banker, Joseph K. Iverson, Alp Sipahigil, Miguel Bello, Alejandro González-Tudela, Mohammad Mirhosseini, and Oskar Painter, Quantum electrodynamics in a topological waveguide, Phys. Rev. X 11, 011015 (2021).
  48. Rosario Fazio and Herre Van Der Zant, Quantum phase transitions and vortex dynamics in superconducting networks, Phys. Rep. 355, 235 (2001).
  49. Marco Scigliuzzo, Giuseppe Calajò, Francesco Ciccarello, Daniel Perez Lozano, Andreas Bengtsson, Pasquale Scarlino, Andreas Wallraff, Darrick Chang, Per Delsing, and Simone Gasparinetti, Controlling atom-photon bound states in an array of Josephson-junction resonators, Phys. Rev. X 12, 031036 (2022).
  50. Amir Youssefi, Shingo Kono, Andrea Bancora, Mahdi Chegnizadeh, Jiahe Pan, Tatiana Vovk, and Tobias J. Kippenberg, Topological lattices realized in superconducting circuit optomechanics, Nature 612, 666 (2022).
  51. Alicia J. Kollár, Mattias Fitzpatrick, and Andrew A. Houck, Hyperbolic lattices in circuit quantum electrodynamics, Nature 571, 45 (2019).
  52. Devin L. Underwood, Will E. Shanks, Jens Koch, and Andrew A. Houck, Low-disorder microwave cavity lattices for quantum simulation with photons, Phys. Rev. A 86, 023837 (2012).
  53. D. L. Underwood, W. E. Shanks, Andy C. Y. Li, Lamia Ateshian, Jens Koch, and Andrew A. Houck, Imaging photon lattice states by scanning defect microscopy, Phys. Rev. X 6, 021044 (2016).
  54. Aaron Bostwick, Taisuke Ohta, Thomas Seyller, Karsten Horn, and Eli Rotenberg, Quasiparticle dynamics in graphene, Nat. Phys. 3, 36 (2006).
  55. B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, NJ, 2013).
  56. M. O. Goerbig, J.-N. Fuchs, G. Montambaux, and F. Piéchon, Tilted anisotropic Dirac cones in quinoid-type graphene and α-(BEDT-TTF)2i3, Phys. Rev. B 78, 045415 (2008).
  57. S. M. Girvin, Circuit QED: Superconducting Qubits Coupled to Microwave Photons (Oxford University Press, Oxford, 2014), p. 113.
  58. G. E. Volovik, Type-II Weyl semimetal versus gravastar, JETP Lett. 114, 236 (2021).
  59. T. Farajollahpour and S. A. Jafari, Synthetic non-Abelian gauge fields and gravitomagnetic effects in tilted Dirac cone systems, Phys. Rev. Res. 2, 023410 (2020).
  60. S. A. Jafari, Electric field assisted amplification of magnetic fields in tilted Dirac cone systems, Phys. Rev. B 100, 045144 (2019).
  61. Z. Jalali-Mola and S. A. Jafari, Polarization tensor for tilted Dirac fermion materials: Covariance in deformed Minkowski spacetime, Phys. Rev. B 100, 075113 (2019).
  62. Aaron D. O’Connell, M. Ansmann, Radoslaw C. Bialczak, Max Hofheinz, Nadav Katz, Erik Lucero, C. McKenney, Matthew Neeley, Haohua Wang, Eva Maria Weig et al. Microwave dielectric loss at single photon energies and millikelvin temperatures, Appl. Phys. Lett. 92, 112903 (2008).
  63. Jiayue Yang, Niayesh Afshordi, Mahdi Torabian, Seyed Akbar Jafari, and G. Baskaran, Smart holes: Analogue black holes with the right temperature and entropy, J. High Energy Phys. 2025, 226 (2024).
  64. Karl Martel and Eric Poisson, Regular coordinate systems for Schwarzschild and other spherical spacetimes, Am. J. Phys. 69, 476 (2001).
  65. Abolfath Hosseinzadeh and S. A. Jafari, Semiclassical transport in two-dimensional Dirac materials with spatially variable tilt, Ann. Phys. (N. Y.) 471, 169838 (2024).
  66. Markus Aspelmeyer, How to avoid the appearance of a classical world in gravity experiments, ArXiv:2203.05587.
  67. Marios Christodoulou, Andrea Di Biagio, Markus Aspelmeyer, Časlav Brukner, Carlo Rovelli, and Richard Howl, Locally mediated entanglement in linearized quantum gravity, Phys. Rev. Lett. 130, 100202 (2023).
  68. Uroš Delić, Manuel Reisenbauer, Kahan Dare, David Grass, Vladan Vuletić, Nikolai Kiesel, and Markus Aspelmeyer, Cooling of a levitated nanoparticle to the motional quantum ground state, Science 367, 892 (2020).
  69. https://doi.org/10.5281/zenodo.19109817
  70. D. M. Pozar, Microwave Engineering (Wiley, Hoboken, NJ, 2011).
  71. Abraham Savitzky and M. J. E. Golay, Smoothing and differentiation of data by simplified least squares procedures, Anal. Chem. 36, 1627 (1964).
  72. Boris Luk’yanchuk, Nikolay I. Zheludev, Stefan A. Maier, Naomi J. Halas, Peter Nordlander, Harald Giessen, and Chong Tow Chong, The Fano resonance in plasmonic nanostructures and metamaterials, Nat. Mater. 9, 707 (2010).
  73. Mikhail F. Limonov, Mikhail V. Rybin, Alexander N. Poddubny, and Yuri S. Kivshar, Fano resonances in photonics, Nat. Photonics 11, 543 (2017).

Outline

Information

Phys. Rev. Applied 25, 044061– Published 22 April, 2026
  • Received 20 April 2025
  • Revised 23 December 2025
  • Accepted 2 March 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation