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Time Crystal

In condensed matter physics, a time crystal is a phase of matter characterized by the breaking of symmetries in time, in contrast to conventional crystals that break spatial symmetry. In crystals like snowflakes or diamonds, atoms or molecules are arranged in a repeating lattice, creating a periodic structure in space. Time crystals, however, break time-translation symmetry, resulting in observables that repeat periodically over time, forming patterns in the temporal dimension.

This temporal symmetry breaking allows time crystals to exhibit self-sustained oscillations that, in theory, could persist indefinitely without energy input. While time crystals have been popularly described as a form of perpetual motion^[1] that can evade the laws of thermodynamics,^[2] they cannot exist in thermal equilibrium. Time crystals rely on many-body localization (MBL), a phenomenon where interactions in disordered quantum systems prevent thermalization. MBL stabilizes the system, preserving the oscillatory behavior and preventing it from degrading into a trivial, equilibrium state.

The idea of time crystals was first proposed in 2012 by physicist Frank Wilczek, who theorized that certain systems could spontaneously break time-translation symmetry.^[3] Subsequent experiments have demonstrated the existence of time crystals in systems that are periodically driven, where the breaking of discrete time-translation symmetry can be observed. Such systems, which create a discrete time crystal (DTC), do not reach thermal equilibrium and are a form of non-equilibrium matter.^[4]

In addition to breaking time symmetry, time crystals can exhibit topological order, a phenomenon associated with long-range quantum entanglement. This property enhances their potential for applications in quantum computing and quantum thermodynamics by increasing resilience against the loss of quantum coherence. As a result time crystals are likely to have applications in computing and sensing technologies. Beyond practical applications, time crystals may also offer insights into the nature of time, potentially bridging the divide between the theories of general relativity, which unifies both space and time as spacetime, and quantum mechanics, where time is treated as separate from spatial dimensions.

Concept

Ordinary (non-time) crystals form through spontaneous symmetry breaking related to a spatial symmetry. Such processes can produce materials with interesting properties, such as diamonds, salt crystals, and ferromagnetic metals. By analogy, a time crystal arises through the spontaneous breaking of a time-translation symmetry. A time crystal can be informally defined as a time-periodic self-organizing structure. While an ordinary crystal is periodic (has a repeating structure) in space, a time crystal has a repeating structure in time. A time crystal is periodic in time in the same sense that the pendulum in a pendulum-driven clock is periodic in time. Unlike a pendulum, a time crystal "spontaneously" self-organizes into robust periodic motion (breaking a temporal symmetry).^[5]

Time-translation symmetry

Symmetries in nature lead directly to conservation laws, something which is precisely formulated by Noether's theorem.^[6]

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.^[7] This symmetry implies the conservation of energy.^[8]

Broken symmetry in normal crystals

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.^[9] Quasimomentum, however, is conserved in a perfect crystal.^[10]

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.

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. Many time crystals that are experimentally realized show discrete time-translation symmetry breaking rather than "continuous". they are periodically driven systems oscillating at a fraction of the frequency of the driving force. DTC's are so-called because "their periodicity is a discrete, integer multiple of the driving period".^[11]

The initial symmetry, which is the discrete time-translation symmetry ( $t\to t+nT$ ) with $n=1$ , is spontaneously broken to the lower discrete time-translation symmetry with $n>1$ , where $t$ is time, $T$ the driving period, $n$ an integer.^[12]

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.^[12]

However, discrete (or Floquet) time crystals are unique in that they follow a strict definition of discrete time-translation symmetry breaking:^[13]

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^[13] (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.^[12] 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.^[12]

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"^{[citation needed]}—their apparent motion does not represent conventional kinetic energy.^[14] 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.^[12]

Time crystals do not evade the second law of thermodynamics,^{[citation needed]} although they spontaneously break "time-translation symmetry", the usual rule that a stable object will remain the same throughout time. In thermodynamics, a time crystal's entropy, understood as a measure of disorder in the system, remains stationary over time, marginally satisfying the second law of thermodynamics by not decreasing.^[15]

History

The idea of a quantized time crystal was theorized in 2012 by Frank Wilczek,^[16] 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.^[17]

In response to Wilczek and Zhang, Patrick Bruno (European Synchrotron Radiation Facility)^[18] and Masaki Oshikawa (University of Tokyo) published several articles stating that space–time crystals were impossible.

Subsequent work developed more precise definitions of time-translation symmetry-breaking, which ultimately led to the Watanabe–Oshikawa^[19] "no-go" statement that quantum space–time crystals in equilibrium are not possible.^[20] 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.^[21]

Several realizations of time crystals, which avoid the equilibrium no-go arguments, were later proposed.^[22] In 2014 Krzysztof Sacha at Jagiellonian University in Kraków predicted the behaviour of discrete time crystals in a periodically driven system with "an ultracold atomic cloud bouncing on an oscillating mirror".^[23]

In 2016, research groups at Princeton and at Santa Barbara independently suggested that periodically driven quantum spin systems could show similar behaviour.^[24] Also in 2016, Norman Yao at Berkeley and colleagues proposed a different way to create discrete time crystals in spin systems.^[25] These ideas were successful and independently realized by two experimental teams: a group led by Harvard's Mikhail Lukin^[26] and a group led by Christopher Monroe at University of Maryland.^[27] Both experiments were published in the same issue of Nature in March 2017.

Later, time crystals in open systems, so called dissipative time crystals, were proposed in several platforms breaking a discrete^[28] and a continuous^[29] time-translation symmetry. A dissipative time crystal was experimentally realized for the first time in 2021 by the group of Andreas Hemmerich at the Institute of Laser Physics at the University of Hamburg.^[30] The researchers used a Bose–Einstein condensate strongly coupled to a dissipative optical cavity and the time crystal was demonstrated to spontaneously break discrete time-translation symmetry by periodically switching between two atomic density patterns.^[30]

In an earlier experiment in 2019, conducted by the group of Tilman Esslinger at ETH Zurich,^[31] limit cycle dynamics^[32] were observed, but evidence of robustness against perturbations and the spontaneous character of the time-translation symmetry breaking were not addressed.

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.,^[25] 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, ²S_1/2 |F = 0, m_F = 0⟩ and |F = 1, m_F = 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.^[27]

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.^[26]

In May 2018, a group in Aalto University reported that they had observed the formation of a time quasicrystal and its phase transition to a continuous time crystal in a Helium-3 superfluid cooled to within one ten thousandth of a kelvin from absolute zero (0.0001 K).^[33] On August 17, 2020 Nature Materials published a letter from the same group saying that for the first time they were able to observe interactions and the flow of constituent particles between two time crystals.^[34]

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,^[35] and concerns about the physicality of the long-range nature of the model have been raised.^[36]

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.^[37]

In July 2021, a team led by Andreas Hemmerich at the Institute of Laser Physics at the University of Hamburg presented the first realization of a time crystal in an open system, a so-called dissipative time crystal using ultracold atoms coupled to an optical cavity. The main achievement of this work is a positive application of dissipation – actually helping to stabilise the system's dynamics.^[30]

In November 2021, a collaboration between Google and physicists from multiple universities reported the observation of a discrete time crystal on Google's Sycamore processor, a quantum computing device. 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 vice-versa in periodic cycles which are multiples of the laser's frequency. While the laser is necessary to maintain the necessary environmental conditions, no energy is absorbed from the laser, so the system remains in a protected eigenstate order.^[38]

Previously in June and November 2021 other teams had obtained virtual time crystals based on floquet systems under similar principles to those of the Google experiment, but on quantum simulators rather than quantum processors: first a group at the University of Maryland obtained time crystals on trapped-ions qubits using high frequency driving rather than many-body localization^[39] and then a collaboration between TU Delft and TNO in the Netherlands called Qutech created time crystals from nuclear spins in carbon-13 nitrogen-vacancy (NV) centers on a diamond, attaining longer times but fewer qubits.^[40]

In February 2022, a scientist at UC Riverside reported a dissipative time crystal akin to the system of July 2021 but all-optical, which allowed the scientist to operate it at room temperature. In this experiment injection locking was used to direct lasers at a specific frequency inside a microresonator creating a lattice trap for solitons at subharmonic frequencies.^[41]

In March 2022, a new experiment studying time crystals on a quantum processor was performed by two physicists at the University of Melbourne, this time using IBM's Manhattan and Brooklyn quantum processors observing a total of 57 qubits.^[42]

In June 2022, the observation of a continuous time crystal was reported by a team at the Institute of Laser Physics at the University of Hamburg, supervised by Hans Keßler and Andreas Hemmerich. In periodically driven systems, time-translation symmetry is broken into a discrete time-translation symmetry due to the drive. Discrete time crystals break this discrete time-translation symmetry by oscillating at a multiple of the drive frequency. In the new experiment, the drive (pump laser) was operated continuously, thus respecting the continuous time-translation symmetry. Instead of a subharmonic response, the system showed an oscillation with an intrinsic frequency and a time phase taking random values between 0 and 2π, as expected for spontaneous breaking of continuous time-translation symmetry. Moreover, the observed limit cycle oscillations were shown to be robust against perturbations of technical or fundamental character, such as quantum noise and, due to the openness of the system, fluctuations associated with dissipation. The system consisted of a Bose–Einstein condensate in an optical cavity, which was pumped with an optical standing wave oriented perpendicularly with regard to the cavity axis and was in a superradiant phase localizing at two bistable ground states between which it oscillated.^[43]

In August 2022, a study was published by a team at ETH Zurich using a time crystal as a self-oscillating quantum pump, without any external driving.^[44]

In February 2024, a team from Dortmund University in Germany built a time crystal from indium gallium arsenide that lasted for 40 minutes, nearly 10 million times longer than the previous record of around 5 milliseconds. In addition, the lack of any decay suggests the crystal could have lasted even longer, stating that it could last "at least a few hours, perhaps even longer".^[45]

References

Footnotes

^
For examples see:
- "a system with spontaneous breaking of time translation symmetry in its ground state must have some sort of motion in its ground state, and is therefore perilously close to fitting the definition of a perpetual motion machine." Wilczek (2012)
- “In quantum physics, a perpetual motion machine is fine as long as we keep our eyes closed, and it must only start slowing down if we observe the motion,” Sutter (2022)
- "In 2012 Nobel-prize-winning physicist Frank Wilczek first proposed a scheme for realizing a perpetual periodic oscillation ... He suggested threading a small magnetic field through a superconducting ring, which, in response, would form a current that can circulate indefinitely." Roushan (2022)
- "...making it a bit like perpetual motion machine. But others soon proved a "no-go theorem" that said such a thing was impossible—and replaced it with a less fantastical definition of a time crystal that researchers soon demonstrated in the lab. But now, two physicists have shown that the original notion of a time crystal is possible after all" Cho (2019)
^
For example, see:
- “The consequence is amazing: You evade the second law of thermodynamics,” said Roderich Moessner, director of the Max Planck Institute for the Physics of Complex Systems in Dresden, Germany, and a co-author on the Google paper. That’s the law that says disorder always increases. Wolchover (2021)
- "the reason that this completely gets around the second law of thermodynamics — I don’t want to say “breaks it” it’s just a setting in which the law of thermodynamics doesn’t apply. And that’s because the second law of thermodynamics tells you that systems reach entropy-maximizing equilibrium states... So we’re talking about quantum systems that can remain out of equilibrium. Which means that all of our usual notions of how to think about phases of matter constrained by the laws of equilibrium thermodynamics should be revisited." Vedika Khemani. Strogatz (2023), Event occurs at 19:56
^ Wilczek (2012); Sacha & Zakrzewski (2018), p. 2
^ Cf. Zhang et al. (2017); Choi et al. (2017); for comment, see Richerme (2017).
^ Sacha & Zakrzewski (2018).
^ Cao (2004), p. 151.
^ Wilczek (2015), p. 279.
^ Feng & Jin (2005), p. 18.
^ Sólyom (2007), p. 193.
^ Sólyom (2007), p. 191.
^ Ball (2018).
^ ^a ^b ^c ^d ^e Else et al. (2020).
^ ^a ^b Yao & Nayak (2018).
^ Cowen (2012).
^ Kubota (2021).
^ Cf. Wilczek (2012); Shapere & Wilczek (2012); for comment, see Zakrzewski (2012).
^ See Li et al. (2012a, 2012b); for comment, see Aron (2012); Wolchover (2013).
^ Bruno (2013a, 2013b); for comment, see Thomas (2013).
^ Watanabe & Oshikawa (2015).
^ See Nozières (2013), Yao et al. (2017), p. 1 and Volovik (2013).
^ Medenjak, Buča & Jaksch (2020); Uhrich et al. (2020); Khemani, Moessner & Sondhi (2019).
^ Cf. Wilczek (2013b); Yoshii et al. (2015); For comment, see Watanabe & Oshikawa (2015), p. 1.
^ Sacha (2015a); Sacha (2020).
^ See Khemani et al. (2016) and Else et al. (2016).
^ ^a ^b Yao et al. (2017).
^ ^a ^b Choi et al. (2017).
^ ^a ^b Zhang et al. (2017).
^ Cf. Iemini et al. (2018); Gong, Hamazaki & Ueda (2018); Gambetta et al. (2019); Buča & Jaksch (2019).
^ Cf. Iemini et al. (2018); Buča, Tindall & Jaksch (2019).
^ ^a ^b ^c Keßler et al. (2021); for comment see Gong & Ueda (2021); Ball (2021a).
^ Dogra et al. (2019), as cited in Defenu et al. (2023), p. 69; Zaletel et al. (2023), pp. 17, 19. For relation to Dreon et al. (2022), see ETH Zurich (2022)
^ Piazza & Ritsch (2015), as cited in Dogra et al. (2019)
^ Autti, Eltsov & Volovik (2018); for comment, see Ball (2018), Aalto University (2018)
^ Autti et al. (2021); for comment see Lancaster University (2020).
^ Kozin & Kyriienko (2019); for comment, see Cho (2019).
^ Khemani, Moessner & Sondhi (2020).
^ Träger et al. (2021); for comment, see Chen (2021); Starr (2021).
^ Mi et al. (2022); for comment, see Ball (2021b); Wolchover (2021); Delbert (2021)
^ Kyprianidis et al. (2021); for comment, see Sanders (2021).
^ Randall et al. (2021); for comment, see Boerkamp (2021).
^ Taheri et al. (2022); for comment, see Starr (2022).
^ Frey & Rachel (2022); for comment, see Cho (2022).
^ Kongkhambut et al. (2022); for comment, see LeBlanc (2022)
^ Dreon et al. (2022); for comment, see ETH Zurich (2022).
^ Greilich et al. (2024); for comment, see Orf (2024); McRae (2024)

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Sacha, K. (2015a). "Modeling Spontaneous Breaking of Time-Translation Symmetry". Physical Review A. 91 (3): 033617. arXiv:1410.3638. doi:10.1103/PhysRevA.91.033617.

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General reference

Cao, T. Y. (2004). Conceptual Foundations of Quantum Field Theory. Cambridge: Cambridge University Press. ISBN 978-0-521-60272-3. OCLC 726825884.

Feng, D.; Jin, G. (2005). Introduction to Condensed Matter Physics. Singapore: World Scientific. ISBN 978-981-238-711-0. OCLC 633571932.

Lee, T. D. (1981). Particle Physics and Introduction to Field Theory. Contemporary concepts in physics. Vol. 1. Chur, Switzerland: CRC Press. ISBN 978-3-7186-0032-8. OCLC 7735531 – via Internet Archive.

Sólyom, J. (2007). Fundamentals of the Physics of Solids: Volume 1, Structure and Dynamics. Berlin, New York: Springer Science & Business Media. doi:10.1007/978-3-540-72600-5. ISBN 978-3-540-72600-5. OCLC 184985012.

Wilczek, F. (2015). "Symmetry III: Emmy Noether – Time, Energy, and Sanity". A Beautiful Question: Finding Nature's Deep Design. New York: Penguin Press. ISBN 978-1-84614-702-9. OCLC 907292308.

Multimedia

Fridman, L. (2021-05-29). "Frank Wilczek: Physics of Quarks, Dark Matter, Complexity, Life & Aliens, Lex Fridman Podcast #187". YouTube (Podcast). Lex Fridman Podcast.

Johnston, H. (2022-02-17). "Physics World Weekly Podcast: Time Crystals Go Large, Challenges of Quantum Optimization, Physics Careers in 2022". Physics World (Podcast). IOP Publishing.

Strogatz, S. (2023-05-03). "The Joy of Why: Is Perpetual Motion Possible at the Quantum Level?". Quanta Magazine (Podcast). Simons Foundation.

External links

Time Crystal 2

History

Concept

Broken symmetry in normal crystals

Broken time translation symmetry

Symmetries in physics^[1]
Symmetry	Transformation	Unobservable	Conservation law
Space-translation	$\mathbf {r} \rightarrow \mathbf {r} +\delta \mathbf {r}$	absolute position in space	momentum
Time-translation	$t\rightarrow t+\delta t$	absolute time	energy
Rotation	$\mathbf {r} \rightarrow \mathbf {r} '$	absolute direction in space	angular momentum
Space inversion	$\mathbf {r} \rightarrow -\mathbf {r}$	absolute left or right	parity
Time-reversal	$t\rightarrow -t$	absolute sign of time	Kramers degeneracy
Sign reversion of charge	$e\rightarrow -e$	absolute sign of electric charge	charge conjugation
Particle substitution		distinguishability of identical particles	Bose or Fermi statistics
Gauge transformation	$\psi \rightarrow e^{iN\theta }\psi$	relative phase between different normal states	particle number

^ Feng & Jin (2005), p. 18; Lee (1981), pp. 179–180

[Perpetual-1] For examples see:
"a system with spontaneous breaking of time translation symmetry in its ground state must have some sort of motion in its ground state, and is therefore perilously close to fitting the definition of a perpetual motion machine." Wilczek (2012)

“In quantum physics, a perpetual motion machine is fine as long as we keep our eyes closed, and it must only start slowing down if we observe the motion,” Sutter (2022)

"In 2012 Nobel-prize-winning physicist Frank Wilczek first proposed a scheme for realizing a perpetual periodic oscillation ... He suggested threading a small magnetic field through a superconducting ring, which, in response, would form a current that can circulate indefinitely." Roushan (2022)

"...making it a bit like perpetual motion machine. But others soon proved a "no-go theorem" that said such a thing was impossible—and replaced it with a less fantastical definition of a time crystal that researchers soon demonstrated in the lab. But now, two physicists have shown that the original notion of a time crystal is possible after all" Cho (2019)

[2] "a system with spontaneous breaking of time translation symmetry in its ground state must have some sort of motion in its ground state, and is therefore perilously close to fitting the definition of a perpetual motion machine." Wilczek (2012)

[3] “In quantum physics, a perpetual motion machine is fine as long as we keep our eyes closed, and it must only start slowing down if we observe the motion,” Sutter (2022)

[4] "In 2012 Nobel-prize-winning physicist Frank Wilczek first proposed a scheme for realizing a perpetual periodic oscillation ... He suggested threading a small magnetic field through a superconducting ring, which, in response, would form a current that can circulate indefinitely." Roushan (2022)

[5] "...making it a bit like perpetual motion machine. But others soon proved a "no-go theorem" that said such a thing was impossible—and replaced it with a less fantastical definition of a time crystal that researchers soon demonstrated in the lab. But now, two physicists have shown that the original notion of a time crystal is possible after all" Cho (2019)

[second_law-2] For example, see:
“The consequence is amazing: You evade the second law of thermodynamics,” said Roderich Moessner, director of the Max Planck Institute for the Physics of Complex Systems in Dresden, Germany, and a co-author on the Google paper. That’s the law that says disorder always increases. Wolchover (2021)

"the reason that this completely gets around the second law of thermodynamics — I don’t want to say “breaks it” it’s just a setting in which the law of thermodynamics doesn’t apply. And that’s because the second law of thermodynamics tells you that systems reach entropy-maximizing equilibrium states... So we’re talking about quantum systems that can remain out of equilibrium. Which means that all of our usual notions of how to think about phases of matter constrained by the laws of equilibrium thermodynamics should be revisited." Vedika Khemani. Strogatz (2023), Event occurs at 19:56

[7] “The consequence is amazing: You evade the second law of thermodynamics,” said Roderich Moessner, director of the Max Planck Institute for the Physics of Complex Systems in Dresden, Germany, and a co-author on the Google paper. That’s the law that says disorder always increases. Wolchover (2021)

[8] "the reason that this completely gets around the second law of thermodynamics — I don’t want to say “breaks it” it’s just a setting in which the law of thermodynamics doesn’t apply. And that’s because the second law of thermodynamics tells you that systems reach entropy-maximizing equilibrium states... So we’re talking about quantum systems that can remain out of equilibrium. Which means that all of our usual notions of how to think about phases of matter constrained by the laws of equilibrium thermodynamics should be revisited." Vedika Khemani. Strogatz (2023), Event occurs at 19:56

[3] Wilczek (2012); Sacha & Zakrzewski (2018), p. 2

[4] Cf. Zhang et al. (2017); Choi et al. (2017); for comment, see Richerme (2017).

[FOOTNOTESachaZakrzewski2018-5] Sacha & Zakrzewski (2018).

[FOOTNOTECao2004151-6] Cao (2004), p. 151.

[FOOTNOTEWilczek2015279-7] Wilczek (2015), p. 279.

[FOOTNOTEFengJin200518-8] Feng & Jin (2005), p. 18.

[FOOTNOTESólyom2007193-9] Sólyom (2007), p. 193.

[FOOTNOTESólyom2007191-10] Sólyom (2007), p. 191.

[FOOTNOTEBall2018-11] Ball (2018).

[FOOTNOTEElseMonroeNayakYao2020-12] Else et al. (2020).

[FOOTNOTEYaoNayak2018-13] Yao & Nayak (2018).

[FOOTNOTECowen2012-14] Cowen (2012).

[FOOTNOTEKubota2021-15] Kubota (2021).

[Wilczek_2012-16] Cf. Wilczek (2012); Shapere & Wilczek (2012); for comment, see Zakrzewski (2012).

[17] See Li et al. (2012a, 2012b); for comment, see Aron (2012); Wolchover (2013).

[18] Bruno (2013a, 2013b); for comment, see Thomas (2013).

[FOOTNOTEWatanabeOshikawa2015-19] Watanabe & Oshikawa (2015).

[20] See Nozières (2013), Yao et al. (2017), p. 1 and Volovik (2013).

[21] Medenjak, Buča & Jaksch (2020); Uhrich et al. (2020); Khemani, Moessner & Sondhi (2019).

[22] Cf. Wilczek (2013b); Yoshii et al. (2015); For comment, see Watanabe & Oshikawa (2015), p. 1.

[23] Sacha (2015a); Sacha (2020).

[24] See Khemani et al. (2016) and Else et al. (2016).

[FOOTNOTEYaoPotterPotirnicheVishwanath2017-25] Yao et al. (2017).

[FOOTNOTEChoiChoiLandigKucsko2017-26] Choi et al. (2017).

[FOOTNOTEZhangHessKyprianidisBecker2017-27] Zhang et al. (2017).

[28] Cf. Iemini et al. (2018); Gong, Hamazaki & Ueda (2018); Gambetta et al. (2019); Buča & Jaksch (2019).

[29] Cf. Iemini et al. (2018); Buča, Tindall & Jaksch (2019).

[Keßler2021-30] Keßler et al. (2021); for comment see Gong & Ueda (2021); Ball (2021a).

[31] Dogra et al. (2019), as cited in Defenu et al. (2023), p. 69; Zaletel et al. (2023), pp. 17, 19. For relation to Dreon et al. (2022), see ETH Zurich (2022)

[32] Piazza & Ritsch (2015), as cited in Dogra et al. (2019)

[33] Autti, Eltsov & Volovik (2018); for comment, see Ball (2018), Aalto University (2018)

[34] Autti et al. (2021); for comment see Lancaster University (2020).

[35] Kozin & Kyriienko (2019); for comment, see Cho (2019).

[FOOTNOTEKhemaniMoessnerSondhi2020-36] Khemani, Moessner & Sondhi (2020).

[37] Träger et al. (2021); for comment, see Chen (2021); Starr (2021).

[38] Mi et al. (2022); for comment, see Ball (2021b); Wolchover (2021); Delbert (2021)

[39] Kyprianidis et al. (2021); for comment, see Sanders (2021).

[40] Randall et al. (2021); for comment, see Boerkamp (2021).

[41] Taheri et al. (2022); for comment, see Starr (2022).

[42] Frey & Rachel (2022); for comment, see Cho (2022).

[43] Kongkhambut et al. (2022); for comment, see LeBlanc (2022)

[44] Dreon et al. (2022); for comment, see ETH Zurich (2022).

[45] Greilich et al. (2024); for comment, see Orf (2024); McRae (2024)

[46] Feng & Jin (2005), p. 18; Lee (1981), pp. 179–180

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[1]

v t e Statistical mechanics
Theory	Principle of maximum entropy ergodic theory
Statistical thermodynamics	Ensembles partition functions equations of state thermodynamic potential: U H F G Maxwell relations
Models	Ferromagnetism models Ising Potts Heisenberg percolation Particles with force field depletion force Lennard-Jones potential
Mathematical approaches	Boltzmann equation H-theorem Vlasov equation BBGKY hierarchy stochastic process mean-field theory and conformal field theory
Critical phenomena	Phase transition Critical exponents correlation length size scaling
Entropy	Boltzmann Shannon Tsallis Rényi von Neumann
Applications	Statistical field theory elementary particle superfluidity Condensed matter physics Complex system chaos information theory Boltzmann machine

v t e States of matter (list)
State	Solid Liquid Gas / Vapor Supercritical fluid Plasma
Low energy	Bose–Einstein condensate Fermionic condensate Degenerate matter Quantum Hall Rydberg matter Strange matter Superfluid Supersolid Photonic molecule
High energy	QCD matter Quark–gluon plasma Color-glass condensate
Other states	Colloid Crystal Liquid crystal Time crystal Quantum spin liquid Exotic matter Programmable matter Dark matter Antimatter Magnetically ordered Antiferromagnet Ferrimagnet Ferromagnet String-net liquid Superglass
Phase transitions	Boiling Boiling point Condensation Critical line Critical point Crystallization Deposition Evaporation Flash evaporation Freezing Chemical ionization Ionization Lambda point Melting Melting point Recombination Regelation Saturated fluid Sublimation Supercooling Triple point Vaporization Vitrification
Quantities	Enthalpy of fusion Enthalpy of sublimation Enthalpy of vaporization Latent heat Latent internal energy Trouton's rule Volatility
Concepts	Baryonic matter Binodal Compressed fluid Cooling curve Equation of state Leidenfrost effect Macroscopic quantum phenomena Mpemba effect Order and disorder (physics) Spinodal Superconductivity Superheated vapor Superheating Thermo-dielectric effect

v t e Major branches of physics
Divisions	Pure Applied Engineering
Approaches	Experimental Theoretical Computational
Classical	Classical mechanics Newtonian Analytical Celestial Continuum Acoustics Classical electromagnetism Classical optics Ray Wave Thermodynamics Statistical Non-equilibrium
Modern	Relativistic mechanics Special General Nuclear physics Particle physics Quantum mechanics Atomic, molecular, and optical physics Atomic Molecular Modern optics Condensed matter physics Solid-state physics Crystallography
Interdisciplinary	Astrophysics Atmospheric physics Biophysics Chemical physics Geophysics Materials science Mathematical physics Medical physics Ocean physics Quantum information science
Related	History of physics Nobel Prize in Physics Philosophy of physics Physics education research Timeline of physics discoveries

User:QuantumKnots/Time crystal

Time Crystal

Concept

Time-translation symmetry

Broken symmetry in normal crystals

Discrete time crystals

Thermodynamics

History

Experiments

References

Footnotes

Bibliography

News Articles

Academic literature

General reference

Multimedia

Further reading

External links

Time Crystal 2

History

Concept

Broken symmetry in normal crystals

Broken time translation symmetry