In this article, I will present a paper on time crystals in open systems with stochastic automata reproduced by spin trains. The contents of this paper can be summarized as follows.

This time crystal is in contact with a heat bath.

A stochastic cellular automaton, which determines the direction of motion with probability and transitions the system, is used for the time crystal.

The space of stochastic cellular automata in a multidimensional system is reduced to the Langevin equation, which is developed in time, and then multiplied by the Hamiltonian to form a time crystal.

In this paper, the time crystal, which is originally realized in a closed system, is realized in an open system under thermal disturbance. A single process in a stochastic cellular automaton is determined by reducing the cellular automaton to a Hamiltonian as follows

1, Let the coordinate s in the cellular automaton be the coordinate Q(s) in the Hamiltonian.

2,Perform a majority flip between the adjacent cells in A and B, and change the state of the bit by changing the two potentials that vary periodically according to the result. One creates a wall between the 0 state and the 1 state, and the other drops the state to one of them.

3, Detect errors in the state of A based on B, which records the previous state of A. If there is an error, repeat step 2. If there is an error, redo step 2, if not, go to step 4.

If there is no error, go to step 4.

5,Convert the coordinates in the Hamiltonian back to the coordinates of the cellular automaton. This is done by transforming the physical coordinate system (p,q) to the cellular automaton coordinate system s so that the energy eigenvalues are the smallest according to S(q)=argmin|Q(s)-q|.

A and B are set to the same initial state. As a result, a time-crystal oscillation appears in the expectation value of the magnetic moment of the spin. The time crystals reported so far have been spin populations or one-dimensional arrays of spins. This is the first time that we have realized a multi-dimensional system. It is expected to be developed in the future.

[2110.00585] An absolutely stable open time crystal (arxiv.org)

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