# The world's simplest theory of time crystals.

In this article, I will explain the time crystal. This is a system in which the entire system is filled with a fixed unit structure by discrete translational and rotational operations in the time direction rather than the space direction. A crystal is a system that can transition to any unit structure in the whole system without duplication by repeating translation and rotation operations on the unit structure a finite or infinite number of times. For example, consider a two-dimensional equilateral triangular lattice as shown in Figure 1. The unit structure in this lattice is the red part of a rhombus-shaped equilateral triangle connected vertically. This unit structure can be moved to another unit structure with a different position by a 60-degree rotation about the axis of the atom and a movement along the length of one side.

Fig. 1 Equilateral triangular two-dimensional lattice and its unit structure.

A time crystal has a unit structure in the time direction. In other words, periodic structural changes occur. However, this is a spontaneous phenomenon, not like the Rabi oscillation, which occurs only when an external laser is applied. This phenomenon can be realized by alternately multiplying two or more Hamiltonians with a constant period ﻿$T=T_1+T_2$﻿ .

﻿$H_1=\hbar g(1-\epsilon)\sum_l\sigma_l^x (0﻿

﻿$H_2=\hbar\sum_{m}J_{lm}^z\sigma_l^z\sigma_m^z+\hbar\sum_lB_l^z\sigma_l^z(T_1﻿

The first and second terms in H2 are sometimes multiplied separately, and in some systems only the first term is used. In particular, the first term of H2 determines the duration of the time crystal. This Hamiltonian is called the Floquet lattice, which can be multiplied to produce a time crystal oscillation with a period of T/2. Time crystals are a quantum mechanical phenomenon, but in fact they are no different from the conserved quantity systems found in mechanics and electromagnetism textbooks. In a time crystal, it is itself a resonant circuit, the transition between the states of the time crystal oscillation is a conserved quantity corresponding to the electromagnetic field in the resonant circuit, the first term of H2 is the power generation, and H1 is the transformer adjacent to the resonant circuit. This phenomenon can occur in any quantum system, and has been experimentally realized in a one-dimensional system using ytterbium ions [1] and nitrogen defects in diamond [2]. Furthermore, research has been done to grow entanglement structures by using the time variation of the time crystal itself [3]. The development of this field is still in its infancy. I am looking forward to the future development of this field.