Time crystal

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

In condensed matter physics, a time crystal is a quantum system of particles whose lowest-energy state is one in which the particles are in repetitive motion. The system cannot lose energy to the environment and come to rest because it is already in its quantum ground state. Because of this the motion of the particles does not really represent kinetic energy like other motion, it has "motion without energy". Time crystals were first proposed theoretically by Frank Wilczek in 2012 as a time-based analogue to common crystals — whereas the atoms in crystals are arranged periodically in space, the atoms in a time crystal are arranged periodically in both space and time.[1] Several different groups have demonstrated matter with stable periodic evolution in systems that are periodically driven.[2][3][4][5] In terms of practical use, time crystals may one day be used as quantum memories.[6]

The existence of crystals in nature is a manifestation of spontaneous symmetry breaking, which occurs when the lowest-energy state of a system is less symmetrical than the equations governing the system. In the crystal ground state, the continuous translational symmetry in space is broken and replaced by the lower discrete symmetry of the periodic crystal. As the laws of physics are symmetrical under continuous translations in time as well as space, the question arose in 2012 as to whether it is possible to break symmetry temporally, and thus create a "time crystal" resistant to entropy.[1]

If a discrete time translation symmetry is broken (which may be realized in periodically driven systems), then the system is referred to as a discrete time crystal. A discrete time crystal never reaches thermal equilibrium, as it is a type (or phase) of non-equilibrium matter. Breaking of time symmetry can only occur in non-equilibrium systems.[5] Discrete time crystals have in fact been observed in physics laboratories as early as 2016 (published in 2017). One example of a time crystal, which demonstrates non-equilibrium, broken time symmetry is a constantly rotating ring of charged ions in an otherwise lowest-energy state.[6]

History[edit]

The idea of a quantized time crystal was theorized in 2012 by Frank Wilczek,[7][8] a Nobel laureate and professor at MIT. In 2013, Xiang Zhang, a nanoengineer at University of California, Berkeley, and his team proposed creating a time crystal in the form of a constantly rotating ring of charged ions.[9][10]

In response to Wilczek and Zhang, Patrick Bruno (European Synchrotron Radiation Facility) and Masaki Oshikawa (University of Tokyo) published several articles stating that space-time crystals were impossible.[11][12]

Subsequent work developed more precise definitions of time translation symmetry-breaking, which ultimately led to the Watanabe-Oshikawa "no-go" statement that quantum space-time crystals in equilibrium are not possible.[13][14] Later work restricted the scope of Watanabe and Oshikawa: strictly speaking, they showed that long-range order in both space and time is not possible in equilibrium, but breaking of time translation symmetry alone is still possible.[15][16]

Several realizations of time crystals, which avoid the equilibrium no-go arguments, were later proposed.[17] In 2014 Krzysztof Sacha [pl] at Jagiellonian University in Krakow predicted the behaviour of discrete time crystals in a periodically driven system with "an ultracold atomic cloud bouncing on an oscillating mirror."[18][19]

In 2016, research groups at Princeton and at Santa Barbara independently suggested that periodically driven quantum spin systems could show similar behaviour.[20] Also in 2016, Norman Yao at Berkeley and colleagues proposed a different way to create discrete time crystals in spin systems.[21] These ideas were successfully and independently realized by two experimental teams: a group led by Harvard's Mikhail Lukin[22] and a group led by Christopher Monroe at University of Maryland.[23] Both experiments were published in the same issue of Nature in March 2017.

Later, dissipative time crystals, which are many body quantum systems that break continuous time translation symmetry by coupling to an external bath have also been proposed in several ways[24][25][26][27] and realized experimentally in a Bose-Einstein condensate in strongly dissipative optical cavity[28] by a group lead by Tilman Esslinger at ETH Zurich.

In 2019 physicists Valerii Kozin and Oleksandr Kyriienko proved that, in theory, a permanent quantum time crystal can exist as an isolated system if the system contains unusual long-range multiparticle interactions. The original "no-go" argument only holds in the presence of typical short-range fields that decay as quickly as rα for some α>0. Kozin and Kyriienko instead analyzed a spin-1/2 many-body Hamiltonian with long-range multispin interactions, and showed it broke continuous time-translational symmetry. Certain spin correlations in the system oscillate in time, despite the system being closed and in a ground energy state. However, demonstrating such a system in practice might be prohibitively difficult,[29][30] and concerns about the physicality of the long-range nature of the model have been raised.[31]

Time translation symmetry[edit]

Symmetries in nature lead directly to conservation laws, something which is precisely formulated by the Noether theorem.[32]

The basic idea of time-translation symmetry is that a translation in time has no effect on physical laws, i.e. that the laws of nature that apply today were the same in the past and will be the same in the future.[33] This symmetry implies the conservation of energy.[34]

Broken symmetry in normal crystals[edit]

Normal process (N-process) and Umklapp process (U-process). While the N-process conserves total phonon momentum, the U-process changes phonon momentum.

Common crystals exhibit broken translation symmetry: they have repeated patterns in space and are not invariant under arbitrary translations or rotations. The laws of physics are unchanged by arbitrary translations and rotations. However, if we hold fixed the atoms of a crystal, the dynamics of an electron or other particle in the crystal depend on how it moves relative to the crystal, and particle momentum can change by interacting with the atoms of a crystal — for example in Umklapp processes.[35] Quasimomentum, however, is conserved in a perfect crystal.[36]

Time crystals show a broken symmetry analogous to a discrete space-translation symmetry breaking. For example,[citation needed] the molecules of a liquid freezing on the surface of a crystal can align with the molecules of the crystal, but with a pattern less symmetric than the crystal: it breaks the initial symmetry. This broken symmetry exhibits three important characteristics:[citation needed]

  • the system has a lower symmetry than the underlying arrangement of the crystal,
  • the system exhibits spatial and temporal long-range order (unlike a local and intermittent order in a liquid near the surface of a crystal),
  • it is the result of interactions between the constituents of the system, which align themselves relative to each other.

Broken symmetry in discrete time crystals (DTC)[edit]

Time crystals seem to break time-translation symmetry and have repeated patterns in time even if the laws of the system are invariant by translation of time. The time crystals that are experimentally realized show discrete time-translation symmetry breaking, not the continuous one: they are periodically driven systems oscillating at a fraction of the frequency of the driving force. (According to Philip Ball, DTC are so-called because "their periodicity is a discrete, integer multiple of the driving period."[37])

The initial symmetry, which is the discrete time-translation symmetry () with , is spontaneously broken to the lower discrete time-translation symmetry with , where is time, the driving period, an integer.[38]

Many systems can show behaviors of spontaneous time translation symmetry breaking but may not be discrete (or Floquet) time crystals: convection cells, oscillating chemical reactions, aerodynamic flutter, and subharmonic response to a periodic driving force such as the Faraday instability, NMR spin echos, parametric down-conversion, and period-doubled nonlinear dynamical systems.[38]

However, discrete (or Floquet) time crystals are unique in that they follow a strict definition of discrete time-translation symmetry breaking:[39]

  • it is a broken symmetry – the system shows oscillations with a period longer than the driving force,
  • the system is in crypto-equilibrium – these oscillations generate no entropy, and a time-dependent frame can be found in which the system is indistinguishable from an equilibrium when measured stroboscopically[39] (which is not the case of convection cells, oscillating chemical reactions and aerodynamic flutter),
  • the system exhibits long-range order – the oscillations are in phase (synchronized) over arbitrarily long distances and time.

Moreover, the broken symmetry in time crystals is the result of many-body interactions: the order is the consequence of a collective process, just like in spatial crystals.[38] This is not the case for NMR spin echos.

These characteristics makes discrete time crystals analogous to spatial crystals as described above and may be considered a novel type or phase of nonequilibrium matter.[38]

Thermodynamics[edit]

Time crystals do not violate the laws of thermodynamics: energy in the overall system is conserved, such a crystal does not spontaneously convert thermal energy into mechanical work, and it cannot serve as a perpetual store of work. But it may change perpetually in a fixed pattern in time for as long as the system can be maintained. They possess "motion without energy"[40]—their apparent motion does not represent conventional kinetic energy.[41]

It has been proven that a time crystal cannot exist in thermal equilibrium.[14] Note, however, that Khemani et al. pointed out that this proof has a subtle error which renders it invalid. See Appendix A of Khemani, Moessner, and Sondhi (2019).[16] Recent experimental advances in probing discrete time crystals in their periodically driven nonequilibrium states have led to the beginning exploration of novel phases of nonequilibrium matter.[38]

Time crystals do not evade the Second Law of Thermodynamics, as shown in a study by Google researchers and physicists from Princeton, Stanford and other universities.[42] Time crystals are also the first objects to spontaneously break “time-translation symmetry,” the usual rule that a stable object will remain the same throughout time, but they don't don't break any law of thermodynamics: entropy, understood as a measure of disorder in the system, remains stationary over time, marginally satisfying the second law of thermodynamics by not decreasing.[43] [44]A time crystal is both stable and ever-changing, with special moments that come at periodic intervals in time.[45]

Experiments[edit]

In October 2016, Christopher Monroe at the University of Maryland claimed to have created the world's first discrete time crystal. Using the ideas proposed by Yao et al.,[21] his team trapped a chain of 171Yb+ ions in a Paul trap, confined by radio-frequency electromagnetic fields. One of the two spin states was selected by a pair of laser beams. The lasers were pulsed, with the shape of the pulse controlled by an acousto-optic modulator, using the Tukey window to avoid too much energy at the wrong optical frequency. The hyperfine electron states in that setup, 2S1/2 |F = 0, mF = 0⟩ and |F = 1, mF = 0⟩, have very close energy levels, separated by 12.642831 GHz. Ten Doppler-cooled ions were placed in a line 0.025 mm long and coupled together.

The researchers observed a subharmonic oscillation of the drive. The experiment showed "rigidity" of the time crystal, where the oscillation frequency remained unchanged even when the time crystal was perturbed, and that it gained a frequency of its own and vibrated according to it (rather than only the frequency of the drive). However, once the perturbation or frequency of vibration grew too strong, the time crystal "melted" and lost this subharmonic oscillation, and it returned to the same state as before where it moved only with the induced frequency.[23]

Also in 2016, Mikhail Lukin at Harvard also reported the creation of a driven time crystal. His group used a diamond crystal doped with a high concentration of nitrogen-vacancy centers, which have strong dipole–dipole coupling and relatively long-lived spin coherence. This strongly interacting dipolar spin system was driven with microwave fields, and the ensemble spin state was determined with an optical (laser) field. It was observed that the spin polarization evolved at half the frequency of the microwave drive. The oscillations persisted for over 100 cycles. This subharmonic response to the drive frequency is seen as a signature of time-crystalline order.[22]

On August 17, 2020 Nature Materials published a letter from Aalto University saying that for the first time they were able to observe interactions and the flow of constituent particles between two time crystals in a Helium-3 superfluid cooled to within one ten thousandth of a degree from absolute zero (0.0001 K or -273.15 °C).[46]

In February 2021 a team at Max Planck Institute for Intelligent Systems described the creation of time crystal consisting of magnons and probed them under scanning transmission X-ray microscopy to capture the recurring periodic magnetization structure in the first known video record of such type.[47][48]

In July 2021 a collaboration consisting of Google and physicists from multiple universities reported the observation of a discrete time crystal on Google's Sycamore processor. A chip of 20 qubits was used to obtain a many body localization configuration of up and down spins and then stimulated with a laser to achieve a periodically driven "Floquet" system where all up spins are flipped for down and viceversa in periodic cycles which are multiples of the laser's. No energy is absorbed from the laser so the system remains in a protected eigenstate order.[49][50]

References[edit]

  1. ^ a b Zakrzewski, Jakub (15 October 2012). "Viewpoint: Crystals of Time". physics.aps.org. APS Physics. Archived from the original on 2 February 2017.
  2. ^ Sacha, Krzysztof (2015). "Modeling spontaneous breaking of time-translation symmetry". Physical Review A. 91 (3): 033617. arXiv:1410.3638. Bibcode:2015PhRvA..91c3617S. doi:10.1103/PhysRevA.91.033617. ISSN 1050-2947. S2CID 118627872.
  3. ^ Khemani et al. (2016)
  4. ^ Else et al. (2016).
  5. ^ a b Richerme, Phil (January 18, 2017). "How to Create a Time Crystal". Physics. American Physical Society. 10: 5. Bibcode:2017PhyOJ..10....5R. doi:10.1103/Physics.10.5. Retrieved 5 April 2021.
  6. ^ a b "Physicists Create World's First Time Crystal".
  7. ^ Wilczek, Frank (2012). "Quantum Time Crystals". Physical Review Letters. 109 (16): 160401. arXiv:1202.2539. Bibcode:2012PhRvL.109p0401W. doi:10.1103/PhysRevLett.109.160401. ISSN 0031-9007. PMID 23215056. S2CID 1312256.
  8. ^ Shapere, Alfred; Wilczek, Frank (2012). "Classical Time Crystals". Physical Review Letters. 109 (16): 160402. arXiv:1202.2537. Bibcode:2012PhRvL.109p0402S. doi:10.1103/PhysRevLett.109.160402. ISSN 0031-9007. PMID 23215057. S2CID 4506464.
  9. ^ See Li et al. (2012a, 2012b).
  10. ^ Wolchover, Natalie (25 April 2013). "Perpetual Motion Test Could Amend Theory of Time". quantamagazine.org. Simons Foundation. Archived from the original on 2 February 2017.
  11. ^ See Bruno (2013a) and Bruno (2013b).
  12. ^ Thomas, Jessica (15 March 2013). "Notes from the Editors: The Aftermath of a Controversial Idea". physics.aps.org. APS Physics. Archived from the original on 2 February 2017.
  13. ^ See Nozières (2013), Yao et al. (2017), p. 1 and Volovik (2013).
  14. ^ a b Watanabe, Haruki; Oshikawa, Masaki (2015). "Absence of Quantum Time Crystals". Physical Review Letters. 114 (25): 251603. arXiv:1410.2143. Bibcode:2015PhRvL.114y1603W. doi:10.1103/PhysRevLett.114.251603. ISSN 0031-9007. PMID 26197119. S2CID 312538.
  15. ^ Medenjak, Marko; Buča, Berislav; Jaksch, Dieter (2020-07-20). "Isolated Heisenberg magnet as a quantum time crystal". Physical Review B. 102 (4): 041117. arXiv:1905.08266. Bibcode:2020PhRvB.102d1117M. doi:10.1103/physrevb.102.041117. ISSN 2469-9950. S2CID 160009779.
  16. ^ a b Khemani, Vedika; Moessner, Roderich; Sondhi, S. L. (23 October 2019). "A Brief History of Time Crystals". arXiv:1910.10745 [cond-mat.str-el].
  17. ^ See Wilczek (2013b) and Yoshii et al. (2015).
  18. ^ Sacha, Krzysztof (2015). "Modeling spontaneous breaking of time-translation symmetry". Physical Review A. 91 (3): 033617. arXiv:1410.3638. Bibcode:2015PhRvA..91c3617S. doi:10.1103/PhysRevA.91.033617. ISSN 1050-2947. S2CID 118627872. We show that an ultracold atomic cloud bouncing on an oscillating mirror can reveal spontaneous breaking of a discrete time-translation symmetry
  19. ^ Sacha, Krzysztof (2020). Time Crystals. Springer Series on Atomic, Optical, and Plasma Physics. 114. Springer. doi:10.1007/978-3-030-52523-1. ISBN 978-3-030-52522-4.
  20. ^ See Khemani et al. (2016) and Else et al. (2016)
  21. ^ a b Yao, N. Y.; Potter, A. C.; Potirniche, I.-D.; Vishwanath, A. (2017). "Discrete Time Crystals: Rigidity, Criticality, and Realizations". Physical Review Letters. 118 (3): 030401. arXiv:1608.02589. Bibcode:2017PhRvL.118c0401Y. doi:10.1103/PhysRevLett.118.030401. ISSN 0031-9007. PMID 28157355. S2CID 206284432.
  22. ^ a b Choi, Soonwon; Choi, Joonhee; Landig, Renate; Kucsko, Georg; Zhou, Hengyun; Isoya, Junichi; Jelezko, Fedor; Onoda, Shinobu; Sumiya, Hitoshi; Khemani, Vedika; von Keyserlingk, Curt; Yao, Norman Y.; Demler, Eugene; Lukin, Mikhail D. (2017). "Observation of discrete time-crystalline order in a disordered dipolar many-body system". Nature. 543 (7644): 221–225. arXiv:1610.08057. Bibcode:2017Natur.543..221C. doi:10.1038/nature21426. ISSN 0028-0836. PMC 5349499. PMID 28277511.
  23. ^ a b Zhang, J.; Hess, P. W.; Kyprianidis, A.; Becker, P.; Lee, A.; Smith, J.; Pagano, G.; Potirniche, I.-D.; Potter, A. C.; Vishwanath, A.; Yao, N. Y.; Monroe, C. (2017). "Observation of a discrete time crystal" (PDF). Nature. 543 (7644): 217–220. arXiv:1609.08684. Bibcode:2017Natur.543..217Z. doi:10.1038/nature21413. PMID 28277505. S2CID 4450646.
  24. ^ Iemini, Fernando; Russomanno, Angelo; Keeling, Jonathan; Schirò, Marco; Dalmonte, Marcello; Fazio, Rosario (16 July 2018). "Boundary time crystals". Phys. Rev. Lett. 121 (35301): 035301. arXiv:1708.05014. Bibcode:2018PhRvL.121c5301I. doi:10.1103/PhysRevLett.121.035301. PMID 30085780. S2CID 51683292.
  25. ^ Gong, Zongping; Hamazaki, Ryusuke; Ueda, Masahito (25 January 2018). "Discrete Time-Crystalline Order in Cavity and Circuit QED Systems". Phys. Rev. Lett. 120 (40404): 040404. arXiv:1708.01472. Bibcode:2018PhRvL.120d0404G. doi:10.1103/PhysRevLett.120.040404. PMID 29437420. S2CID 206307409.
  26. ^ Filippo Maria, Gambetta; Carollo, Federico; Marcuzzi, Matteo; Garrahan, Juan P.; Lesanovsky, Igor (8 January 2019). "Discrete Time Crystals in the Absence of Manifest Symmetries or Disorder in Open Quantum Systems". Phys. Rev. Lett. 122 (15701): 015701. arXiv:1807.10161. Bibcode:2019PhRvL.122a5701G. doi:10.1103/PhysRevLett.122.015701. PMID 31012672. S2CID 119187766.
  27. ^ Buča, Berislav; Tindall, Joseph; Jaksch, Dieter (2019-04-15). "Non-stationary coherent quantum many-body dynamics through dissipation". Nature Communications. 10 (1): 1730. arXiv:1804.06744. Bibcode:2019NatCo..10.1730B. doi:10.1038/s41467-019-09757-y. ISSN 2041-1723. PMC 6465298. PMID 30988312.
  28. ^ Dogra, Nishant; Landini, Manuele; Kroeger, Katrin; Hruby, Lorenz; Donner, Tobias; Esslinger, Tilman (2019-12-20). "Dissipation-induced structural instability and chiral dynamics in a quantum gas". Science. 366 (6472): 1496–1499. arXiv:1901.05974. Bibcode:2019Sci...366.1496D. doi:10.1126/science.aaw4465. ISSN 0036-8075. PMID 31857481. S2CID 119283814.
  29. ^ Cho, Adrian (27 November 2019). "Back to the future: The original time crystal makes a comeback". Science. doi:10.1126/science.aba3793. Retrieved 19 March 2020.
  30. ^ Kozin, Valerii K.; Kyriienko, Oleksandr (2019-11-20). "Quantum Time Crystals from Hamiltonians with Long-Range Interactions". Physical Review Letters. 123 (21): 210602. arXiv:1907.07215. Bibcode:2019PhRvL.123u0602K. doi:10.1103/PhysRevLett.123.210602. ISSN 0031-9007. PMID 31809146. S2CID 197431242.
  31. ^ Khemani, Vedika; Moessner, Roderich; Sondhi, S. L. (2020). "Comment on "Quantum Time Crystals from Hamiltonians with Long-Range Interactions"". arXiv:2001.11037 [cond-mat.str-el].
  32. ^ Cao, Tian Yu (25 March 2004). Conceptual Foundations of Quantum Field Theory. Cambridge: Cambridge University Press. ISBN 978-0-521-60272-3. See p. 151.
  33. ^ Wilczek, Frank (16 July 2015). A Beautiful Question: Finding Nature's Deep Design. Penguin Books Limited. ISBN 978-1-84614-702-9. See Ch. 3.
  34. ^ Feng, Duan; Jin, Guojun (2005). Introduction to Condensed Matter Physics. singapore: World Scientific. ISBN 978-981-238-711-0. See p. 18.
  35. ^ Sólyom, Jenö (19 September 2007). Fundamentals of the Physics of Solids: Volume 1: Structure and Dynamics. Springer. ISBN 978-3-540-72600-5. See p. 193.
  36. ^ Sólyom, Jenö (19 September 2007). Fundamentals of the Physics of Solids: Volume 1: Structure and Dynamics. Springer. ISBN 978-3-540-72600-5. See p. 191.
  37. ^ Ball, Philip (July 17, 2018). "In search of time crystals". Physics World. 31 (7): 29. Bibcode:2018PhyW...31g..29B. doi:10.1088/2058-7058/31/7/32. Retrieved September 6, 2021. The “discrete” comes from the fact that their periodicity is a discrete, integer multiple of the driving period.
  38. ^ a b c d e Else, D. W.; Monroe, C.; Nayak, C.; Yao, N. Y. (March 2020). "Discrete Time Crystals". Annual Review of Condensed Matter Physics. 11: 467–499. doi:10.1146/annurev-conmatphys-031119-050658. S2CID 173188223.
  39. ^ a b Yao; Nayak (2018). "Time crystals in periodically driven systems". Physics Today. 71 (9): 40–47. arXiv:1811.06657. Bibcode:2018PhT....71i..40Y. doi:10.1063/PT.3.4020. ISSN 0031-9228. S2CID 119433979.
  40. ^ Crew, Bec. "Time Crystals Might Exist After All – And They Could Break Space-Time Symmetry". ScienceAlert. Retrieved 2017-09-21.
  41. ^ ""Time Crystals" Could Be a Legitimate Form of Perpetual Motion". archive.is. 2017-02-02. Archived from the original on 2017-02-02. Retrieved 2017-09-21.CS1 maint: bot: original URL status unknown (link)
  42. ^ "Google May Have Created an Unruly New State of Matter: Time Crystals". Popular Mechanics. Retrieved 4 August 2021.
  43. ^ Kubota, Taylor; University, Stanford. "Physicists create time crystals with quantum computers". phys.org. Retrieved 2021-12-03.
  44. ^ Mi, Xiao; Ippoliti, Matteo; Quintana, Chris; Greene, Ami; Chen, Zijun; Gross, Jonathan; Arute, Frank; Arya, Kunal; Atalaya, Juan; Babbush, Ryan; Bardin, Joseph C. (2021-11-30). "Time-Crystalline Eigenstate Order on a Quantum Processor". Nature: 1–1. doi:10.1038/s41586-021-04257-w. ISSN 1476-4687.
  45. ^ "Eternal Change for No Energy: A Time Crystal Finally Made Real". Quanta Magazine. Retrieved 4 August 2021.
  46. ^ See Autti, S., Heikkinen, P.J., Mäkinen, J.T. et al. AC Josephson effect between two superfluid time crystals. Nat. Mater. (2020). https://doi.org/10.1038/s41563-020-0780-y
  47. ^ Träger, Nick; Gruszecki, Paweł; Lisiecki, Filip; Groß, Felix; Förster, Johannes; Weigand, Markus; Głowiński, Hubert; Kuświk, Piotr; Dubowik, Janusz; Schütz, Gisela; Krawczyk, Maciej (2021-02-03). "Real-Space Observation of Magnon Interaction with Driven Space-Time Crystals". Physical Review Letters. 126 (5): 057201. arXiv:1911.13192. Bibcode:2021PhRvL.126e7201T. doi:10.1103/PhysRevLett.126.057201. PMID 33605763. S2CID 208512720.
  48. ^ Williams, Jon. "World's first video recording of a space-time crystal". Max Planck Institute for Intelligent Systems. Retrieved 2021-08-07.
  49. ^ Mi, Xiao; Ippoliti, Matteo; Quintana, Chris; Greene, Amy; Chen, Zijun; Gross, Jonathan; Arute, Frank; Arya, Kunal; Atalaya, Juan; Babbush, Ryan; Bardin, Joseph C. (2021-07-28). "Observation of Time-Crystalline Eigenstate Order on a Quantum Processor". arXiv:2107.13571 [quant-ph].
  50. ^ Wolchover, Natalie (2021-07-30). "Eternal Change for No Energy: A Time Crystal Finally Made Real". Quanta Magazine. Retrieved 2021-07-30.

Academic articles[edit]

Books[edit]

Press[edit]

External links[edit]