# Time crystal

(Redirected from Space-time crystal)

In condensed matter physics, a time crystal refers to a system or subsystem whose lowest-energy states evolve periodically. This name was proposed theoretically by Frank Wilczek in 2012 as a temporal analog to common crystals, which are periodic spatially.[1] Experimental realization of matter with stable periodic evolution was demonstrated in 2016-2017, but not in the way theorized in 2012.[2] In terms of practical use, time crystals may one day be used as quantum memories.[3]

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]

A rigorous proof in 2015 appears to rule out such time crystals in the equilibrium systems as theorized in 2012. But a loophole remains: periodically driven nonequilibrium systems have discrete time translation symmetry which may be spontaneously broken. 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.[2]

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.[3]

## History

The idea of a quantized time crystal was theorized in 2012 by Frank Wilczek [4][5] 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.[6][7]

In response to Wilczek and Zhang, Patrick Bruno, a theorist at the European Synchrotron Radiation Facility in Grenoble, France, published several articles in 2013 claiming to show that space-time crystals were impossible. Also later Masaki Oshikawa from the University of Tokyo showed that time crystals would be impossible at their ground state; moreover, he implied that any matter cannot exist in non-equilibrium in its ground state.[8][9]

Subsequent work developed more precise definitions of time translation symmetry-breaking, which ultimately led to the Watanabe-Oshikawa "no-go" proof that quantum time crystals in equilibrium are not possible.[10][11]

Several realizations of time crystals, which avoid the equilibrium no-go arguments, were later proposed.[12] In 2014 Krzysztof Sacha at Jagiellonian University in Krakow predicted the behaviour of discrete time crystals in a periodically driven system of ultracold atoms.[13][14] Later works[15] suggested that periodically driven quantum spin systems could show similar behaviour.

In 2016 Norman Yao at Berkeley and colleagues proposed a different way to create discrete time crystals in spin systems.[16] His ideas were successfully and independently realized by two experimental teams: a group led by Harvard's Mikhail Lukin[17] and a group led by Christopher Monroe at University of Maryland.[18] Both experiments were published in the same issue of Nature in March 2017.

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 r−α 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,[19][20] and concerns about the physicality of the long-range nature of the model have been raised.[21]

## Time translation symmetry

Symmetries in nature lead directly to conservation laws, something which is precisely formulated by the Noether theorem.[22]

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.[23] This symmetry implies the conservation of energy.[24]

### Broken symmetry in normal crystals

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.[25] Quasimomentum, however, is conserved in a perfect crystal.[26]

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

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. The initial symmetry, which is the discrete time-translation symmetry (${\displaystyle t\to t+nT}$) with ${\displaystyle n=1}$, is spontaneously broken to the lower discrete time-translation symmetry with ${\displaystyle n>1}$, where ${\displaystyle t}$ is time, ${\displaystyle T}$ the driving period, ${\displaystyle n}$ an integer.[27]

Many systems can show behaviors of spontaneous time translation symmetry breaking but may not be discrete (or Floquet) time crytals: 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.[27]

However, discrete (or Floquet) time crystals are unique in that they follow a strict definition of discrete time-translation symmetry breaking:[28]

• 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-dependant frame can be found in which the system is indistinguishable from an equilibrium when measured stroboscopically[28] (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.[27] 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.[27]

## Thermodynamics

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"[29]—their apparent motion does not represent conventional kinetic energy.[30]

It has been proven that a time crystal cannot exist in thermal equilibrium.[11] 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.[27]

## Experiments

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.[16], 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.[18]

Later 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.[17]

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 (0.0001K or -273.15 °C)[31]

## Related concepts

• A similar idea called a choreographic crystal has been proposed.[32] By relaxing additional restrictions on the definition of time crystals, continuous time-translation symmetry breaking can be achieved in exceptional cases. For instance, if one allows the system to be open to an environment, but undriven, many-body systems with the appropriate algebraic structure can be time crystals.[33] Likewise, if one drops the requirement of long-range order in space, purely time-translation symmetry breaking is possible.[34]
• A new engineering concept of time crystal is explored recently on catalytic reaction cycles.[35] By considering each individual chemical reaction inside a catalytic reaction cycle as a single event, all the events could be connected by time-consuming intermediate states to convert a catalytic cycle into a time crystal. There, by simple changing certain conditions of a reaction cycle, we can selectively promote one of the certain reaction products from a catalytic reaction cycle. This protocol is named as time crystal engineering.

## References

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. ^ a b Richerme, Phil (January 18, 2017). "How to Create a Time Crystal". physics.aps.org. American Physical Society. Retrieved 5 April 2021.
3. ^ a b https://www.technologyreview.com/2016/10/04/157185/physicists-create-worlds-first-time-crystal/
4. ^ Wilczek, Frank (2012). "Quantum Time Crystals". Physical Review Letters. 109 (16): 160401. arXiv:1202.2539v2. Bibcode:2012PhRvL.109p0401W. doi:10.1103/PhysRevLett.109.160401. ISSN 0031-9007. PMID 23215056. S2CID 1312256.
5. ^ Shapere, Alfred; Wilczek, Frank (2012). "Classical Time Crystals". Physical Review Letters. 109 (16): 160402. arXiv:1202.2537v2. Bibcode:2012PhRvL.109p0402S. doi:10.1103/PhysRevLett.109.160402. ISSN 0031-9007. PMID 23215057. S2CID 4506464.
6. ^ See Li et al. (2012a, 2012b).
7. ^ 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.
8. ^ See Bruno (2013a) and Bruno (2013b).
9. ^ 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.
10. ^ See Nozières (2013), Yao et al. (2017), p. 1 and Volovik (2013).
11. ^ a b Watanabe, Haruki; Oshikawa, Masaki (2015). "Absence of Quantum Time Crystals". Physical Review Letters. 114 (25): 251603. arXiv:1410.2143v3. Bibcode:2015PhRvL.114y1603W. doi:10.1103/PhysRevLett.114.251603. ISSN 0031-9007. PMID 26197119. S2CID 312538.
12. ^ See Wilczek (2013b) and Yoshii et al. (2015).
13. ^ Sacha, Krzysztof (2015). "Modeling spontaneous breaking of time-translation symmetry". Physical Review A. 91 (3): 033617. arXiv:1410.3638v3. Bibcode:2015PhRvA..91c3617S. doi:10.1103/PhysRevA.91.033617. ISSN 1050-2947. S2CID 118627872.
14. ^ 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.
15. ^
16. ^ 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:1608.02589v2. Bibcode:2017PhRvL.118c0401Y. doi:10.1103/PhysRevLett.118.030401. ISSN 0031-9007. PMID 28157355. S2CID 206284432.
17. ^ 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:1610.08057v1. Bibcode:2017Natur.543..221C. doi:10.1038/nature21426. ISSN 0028-0836. PMC 5349499. PMID 28277511.
18. ^ 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.
19. ^ 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.
20. ^ 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.
21. ^ 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].
22. ^ 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.
23. ^ 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.
24. ^ Feng, Duan; Jin, Guojun (2005). Introduction to Condensed Matter Physics. singapore: World Scientific. ISBN 978-981-238-711-0. See p. 18.
25. ^ 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.
26. ^ 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.
27. Else, D. W.; Monroe, C.; Nayak, C.; Yao, N. Y. (March 2020). "Discrete Time Crystals". Annual Review of Condensed Matter Physics. 11: 467-499.
28. ^ 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.
29. ^ Crew, Bec. "Time Crystals Might Exist After All – And They Could Break Space-Time Symmetry". ScienceAlert. Retrieved 2017-09-21.
30. ^ ""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)
31. ^ 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
32. ^
33. ^ 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.
34. ^ 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.
35. ^ P. Sahoo et al. 2020,"Time Crystal Engineering in Catalytic Reaction Cycles". doi:10.1007/978-981-15-7253-1_4. Print ISBN 978-981-15-7252-4. Online ISBN 978-981-15-7253-1. Publisher Name: Springer, Singapore