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In this video (at this time in the video) the vlogger seems to make a statement that the only way time crystals can be achieved is if some of the atoms (at a periodic distance from each other) are entangled. But I can't tell if I trust him considering his background. In fact, even he states earlier in the video that he might be wrong.

So is he right? Are time crystals only achieved via entanglement?

https://youtu.be/ucwmGZ51X7E?t=126

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A quantum time crystal does not from my reading of Wilczek and others appear to require entanglement, but the idea is interesting. It does seems plausible that a quantum time crystal model could be developed with entanglement.

A quantum time crystal is just a periodicity in a system in time that has a lattice structure. An elementary quantum time crystal is then just a chain that is periodic in time. This chain would then be a measure of some periodicity in a system. Wilzcek's time crystal is a curious system then that exhibits dynamics in the ground state. Normally a ground state is where the Hamiltonian acts trivially. However if time translation symmetry is violated then some type of motion in the ground state is possible. This comes very close to being a form of perpetual motion machine. Breaking time symmetry may though be involved with the arrow of time.

The Wilczek time crystal assumes a charge $q$ confined to a ring of unit radius contains a magnetic flux $2\pi\alpha/q$ with the gauge covariant momentum $\pi_\pi~=~\dot\phi~+~\alpha$, for $\phi$ an angle around the ring and $-i\partial/\partial\phi$ a generator of angular momentum. The Lagrangian is then $$ L~=~\frac{1}{2}\dot\phi^2~+~\alpha\dot\phi $$ and the Hamiltonian $$ H~=~\frac{1}{2}(\pi_\phi~-~\alpha)^2. $$ For states defined as $|\ell\rangle~=~|e^{iL\phi}\rangle$ it is not hard to see that $\langle\dot\phi\rangle~-~\ell~-~\alpha$ and even for the ground state with $\ell~=~0$ there is the expectation $\langle\ell_0|\dot\phi|\ell_0\rangle~=~-\alpha$

The Page-Wooters model has two Hilbert spaces $H_1$ and $H_2$ with an entanglement of their states given by the product states of these Hamiltonians $H_1\otimes I_2~+~ I_1\otimes H_2$. A state then of the form $$ |\Psi\rangle~ =~\sum_{ij}c_{ij}(|\psi_i\rangle|\phi_j\rangle~+~ |\psi_j\rangle|\phi_i\rangle), $$ $|\psi_i\rangle~\in~H_1$ and $|\phi_j\rangle~\in~H_2$ is an entanglement of state from these two Hamiltonians. Now we take an arbitrary state of the for $|\chi\rangle~=~ \sum_ka_i|\psi_i\rangle~\in~H_1$ and project onto $|\Psi\rangle$ with $$ \langle\chi|\Psi\rangle~=~\sum_{ijk}a^*_k c_{ij}(\langle\psi_k|\rangle|\psi_i\rangle|\phi_j\rangle~+~\langle\psi_k|\psi_j\rangle|\phi_i\rangle), $$ $$ ~=~\sum_{ijk}a^*_k c_{ij}(\delta_{ik}|\phi_j\rangle~+~\delta_{jk}|\phi_i\rangle) $$ $$ ~=~\sum_{ij}(a^*_ic_{ij}|\phi_j\rangle~+~a^*_jc_{ij}|\phi_i\rangle). $$ Since $c_{ij}~=~a^*_ia_j$ this is then written as $$ \langle\chi|\Psi\rangle ~=~ \sum_{ij}(|a_i|^2a_j|\phi_j\rangle ~+~ |a_j|^2a_i|\phi_i\rangle) ~=~ 2\sum_{ij}|a_i|^2a_j|\phi_j\rangle $$ The matrix element $c_{ij} ~=~ a^*_ia_j$ is a relative phase term $c_{ij} ~=~ exp(i\theta_i)exp(-i\theta'_j)$ and the difference in this relative phase $\theta_i ~-~\theta'_j~=~\omega t$. This projection does here then is a way of measuring the phase of one system relative to another.

This relative phase definition of time holds for a system with different ground states for $H_1$ and $H_2$ We then have something analogous to a time crystal. The main interest with the Page-Wooters model is to define time within the Wheeler-DeWitt equation $H\Psi[g]~=~0$. The occurrence of time may then be a relative phase with entangled states, which have analogues to a time crystal. It has though been shown that time crystals are defined on an approximate vacuum, and hold for a Floquet oscillator. This means they are quasi-stable. This is certainly an interesting area to study, and I offer here only cursory observations

F. Wilczek, "Quantum Time Crystals" PRL $\bf 109$ 16 (2012). https://arxiv.org/abs/1202.2539v2

D. V. Else, B. Bauer, C. Nayak, "Floquet Time Crystals, Phys. Rev. Lett. $\bf 117$, 090402 (2016) https://arxiv.org/abs/1603.08001

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It depends entirely on your exact definitions. Many times, someone has proposed a "classical time crystal", but then someone else has argued that that isn't a "true" time crystal because it's time crystality doesn't originate from the quantum effects that the latter person thinks make the effect really interesting. Some people consider that to be a "no true Scotsman" fallacy though.

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