本论文是在加有边界场的一维横场伊辛模型上，通过周期性脉冲驱动左边界场h_L(t)来研究时间晶体。

本论文在一维横场伊辛链的两端分别加上一个不受横场g影响的自旋，并且设置左边界场h_L(t)为正值，右边界场h_R(t)为负值，相邻自旋格点间的最近邻相互作用为1。我们首先通过Jordan-Wigner变换将一维横场伊辛链从泡利表象转换到Fock空间中，也就是将自旋系统转换成无自旋的费米子系统，然后再用Bogoliubov变换将无自旋的费米子系统的哈密顿量进行对角化，这样就将问题转化成为了一个求三对角矩阵本征值和本征态的问题，然后由本征值和本征态来求出体系的磁化强度。

在加入左右边界场h_L(t) 、h_R(t)后，可以来研究系统的相变。在h_L(t) ，hR< 时，我们首先将右边界场h_R(t)固定，然后通过不断改变左边界场h_L(t) 的值来研究其相变，在不断改变左边界场h_L(t) 的过程中，发现边界磁化强度m₁ 的正负值发生了跃变现象，跃变的位置是在hL=-h_R处，并且在此处区分了“正”相和“负”相，因此“正”相和“负”相之间的相变是一级相变；在h_Lh_R<0且h_R>1-g时，我们首先将右边界场h_R固定，然后通过不断改变左边界场h_L 的值来研究其相变，发现不同尺寸N下边界磁化强度m₁在相变点h_w附近处是连续的，但是边界磁化率χ₁₁在相变点h_w附近处是发散的，因此在这个区域中是二级相变，并且磁化强度为0的界面会从边界移动到中间，所以该区域的连续相变也称为浸润相变。

在确定一维横场伊辛链的相变以后，进一步研究了周期性脉冲驱动左边界场h_L(t)下的非平衡态行为。在确定初始状态以后，只需要知道一个周期内的状态演化规律，就可以得到任意一个时刻下系统所有的信息。首先通过基矢变换将对角形式哈密顿量的基态间相互表示出来，从而得到系统状态随时间的演化规律，进而得到边界磁化强度随着时间演化的规律。在本文中，结果显示系统状态在第一激发态和第二激发态之间相互转换，转换周期是驱动边界场h_L周期的2倍，边界磁化强度的演化周期也是驱动边界场h_L周期的2倍。在长时间作用下，可以将演化规律分成两类，一种是边界磁化强度随时间在两个值之间永远以正弦方式振荡，另一种是以正弦衰减方式振荡，最终在一个固定值处永远振荡下去。

In this paper, we study the time crystal on the Ising model of one-dimensional transverse field with boundary field by driving the left boundary field hL with periodic pulse.

In this paper, a spin that is not affected by the transverse field is added to both ends of the Ising chain of one-dimensional transverse field, and the left boundary field hL  is set to be positive, the right boundary field hR  is set to be negative, and the nearest neighbor interaction between adjacent spin lattice points is set to 1 . We first transform the one-dimensional Ising chain of transverse field from Pauli representation to Fock space by Jordan-Wigner  transformation, that is transform the spin system into the spin-free Fermi subsystem, and then diagonalize the Hamiltonian of the spin-free Fermi subsystem by Bogoliubov  transformation. In this way, the problem is transformed into a problem of finding the eigenvalues and eigenstates of the tridiagonal matrix, and then the magnetization of the system can be obtained from the eigenvalues and eigenstates.

The phase transition of the system can be studied by adding the left and right boundary fields hL and hR . In hL , hR< , We first fix the right boundary field hR , and then investigated the phase transition by constantly changing the value of the left boundary field hL , In the process of changing the left boundary field hL , it is found that the positive and negative values of the boundary magnetization m1  have a jump phenomenon. The jump is at hL=hR , and the "positive" phase and "negative" phase are differentiated here. Therefore, the phase transition between the "positive" phase and the "negative" phase is a first-order phase transition;In hLhR<0  and hR>1-g , We first fix the right boundary field hR , and then investigated the phase transition by constantly changing the value of the left boundary field hL ,It is found that the boundary magnetization m1  with different size N  is continuous near the phase transition point hw , but the boundary susceptibility χ11  is diverging near the phase transition point hw . Therefore, it is a second-order phase transition in this region, and the interface with magnetization 0  will move from the boundary to the middle, so the continuous phase transition in this region is also called the infiltration phase transition.

After determining the phase transition of the Ising chain in one-dimensional transverse field, the nonequilibrium behavior of the left boundary field hL  driven by periodic pulses is further studied. After the initial state is determined, all the information of the system at any time can be obtained only by knowing the state evolution law within a period. Firstly, the ground states of the diagonal Hamiltonian are represented by the basis vector transformation, and the evolution law of the system state with time is obtained, and then the evolution law of boundary magnetization with time is obtained. In this paper, the results show that the system state transitions between the first excited state and the second excited state, and the transition period is twice as long as the hL  period of the driving boundary field, and the evolution period of the boundary magnetization is also twice as long as the hL  period of the driving boundary field. In the case of long time evolution, the evolution law can be divided into two categories: one is that the boundary magnetization oscillates sinusoidally between the two values over time; the other is that the boundary magnetization oscillates sinusoidally, and finally oscillates forever at a fixed value.