本文主要研究有边界场的一维横场伊辛模型及XY模型的浸润相变点处的纠缠熵。

这两个模型的浸润相变已被胡坤进行了深入的研究，在相变点处，能隙、浸润层厚度、边界磁化率等物理量的临界行为也已被阐明。但浸润相变点处的纠缠熵的性质并没有得到研究，而纠缠熵是刻画量子多体系统和量子相变的重要工具。为了对这两个模型的浸润相变做更进一步的研究，我们使用数值方法研究了纠缠熵这一物理量的临界行为以得到了浸润相变的更加深入的性质。

我们使用“关联函数矩阵方法”计算子系统纠缠熵。首先，我们在自旋链两端各加上一个“幽灵格点”来等效边界场的作用，该格点仅与自旋链端点上的自旋发生作用而不受横场的影响，因此“幽灵格点”上的自旋算符与整个系统的哈密顿量对易。然后，我们通过 Jordan-Wigner 变换以及玻戈留波夫变换将对自旋链的求解转变为对无自旋的费米子系统的求解并且得到了对角化的哈密顿量。根据费米子算符的反对易关系，我们可以利用玻戈留波夫变换的变换系数求解子系统的关联函数。根据子系统关联函数与约化密度矩阵之间的关系，我们得到了约化密度矩阵的计算方法。最后我们可以根据约化密度矩阵来计算纠缠熵。这种方法的优点在于它的复杂性是随着系统尺寸的增加而线性增长的，而其它方法的复杂性是以指数形式增长的。

我们使用这种数值方法研究了伊辛模型的浸润相变中的纠缠熵。首先，我们固定右边界场 h_R ，横场 g ，改变左边界场强度 h_L 的数值，使之经过浸润相变点 h_w 。通过这种方法，我们研究了在浸润相变点左邻域内的半链纠缠熵的行为，发现在相变点左邻域内半链纠缠熵 S_[1,N/2] 与左边界场强度 h_L 满足线性关系，并且发现在相变点处，半链纠缠熵的偏导数 ∂_[1,/2]/ ∂ℎ 与系统尺寸 N 成正比。然后我们通过改变自旋链尺寸 N 的大小，来探究半链纠缠熵与系统尺寸N的依赖关系。我们通过 Origin 来处理计算得到的数据，并且得到了拟合函数。最后，我们研究了纠缠熵与子系统尺寸的关系，并且验证了当子系统尺寸非常小时，子系统纠缠熵 S_[1,n] 与子系统尺寸 n 存在对数关系。

接着，我们研究了XY模型的浸润相变中的纠缠熵，在XY模型中存在两个二级的浸润相变以及一个四级的浸润相变。我们对这三个相变重复了上面对伊辛模型的讨论中做过的工作。我们发现：对于XY模型中的两个二级相变，纠缠熵的临界行为与伊辛模型中的十分相似，这也验证了这三个二级的浸润相变确实属于同一个普适类。而对于四级的浸润相变，纠缠熵的临界行为与二级相变中的存在明显的差异：当我们改变左边界场强度 h_L 时，半链纠缠熵 S_[1,N/2] 在线性变化之前会有一次小振荡以及非线性的快速增长部分。此外，子系统纠缠熵 S_[1,n] 与子系统尺寸 n 的关系也与二级相变中的不同，即使在子系统尺寸非常小时，它们也并不满足对数关系。

In this paper, we mainly study the entanglement entropy in the one-dimensional transvers-field Ising model and XY model with boundary-fields.

The wetting transitions of the two models have been studied in depth by Kun Hu, and the critical behavior of the energy gap, the thickness of the wetting layer, the boundary susceptibility and other phtsical quantities have also been clarified. However, the nature of entanglement entropy at the critical point of the wetting transition has not been studied, while entanglement entropy is an important tool for studying quantum many-body systems and quantum phase transitions. In order to further study the wetting transitions of these two models, we used numerical methods to investigate the critical behavior of entanglement entropy, in order to obtain a more in-depth understanding of the properties of wetting transitions.

The correlation matrix method is used to calculate the entanglement entropy of subsystem. First, we add two ghost sites, which only interact with the spins at sites 1 and N, to the boundary of spin chain to get effective Hamiltonian. So that the spin operators at ghost sites are commute with the effective Hamiltonian. Then, by using the Jordan-Wigner transformation and Bogoliubov transformation, we change the spin system into spinless fermionic system and then diagonalize the effective Hamiltonian. According to the anticommutation relation between fermionic operators, we can obtain the correlation functions in subsystem by using the Bogoliubov transformation coefficients. According to the relation between subsystem correlation functions and reduced density matrix of subsystem, we get the calculation method of reduced density matrix. Finally, we can calculate the entanglement entropy by using the reduced density matrix. The advantage of this method is that its complexity increases linearly with the system size, while the complexity of other method increases exponentially.

We use this method to study the entanglement entropy in the wetting transition of Ising model. First, we let the right boundary field h_R and transvers-field g fixed, changing the left boundary field h_L to pass through the transition point h_w. By this way, we study the behavior of the half-chain entanglement entropy in the neighbourhood to the left of wetting transition point, and find that there is a linear relationship between half-chain entropy S_[1,N/2] and left boundary field h_L, and find that partial derivative of half-chain entanglement entropy ∂_[1,/2] /∂ℎ is proportional to the size of system N at the critical point. Then, we change the spin chain size N to study the relationship between half-chain entropy S_[1,N/2] and system size N. We use Origin to deal with the data of calculation, and obtain the fitting function. Finally, we study the relationship between entanglement entropy S_[1,n] and subsystem size n, and verify that there is a logarithmic relationship between entropy S_[1,n] and subsystem size n as n is very small.

Then, we study the entanglement entropy in the wetting transitions of XY model, in which there are two second-order wetting transitions and a fourth-order wetting transition. we repeat the work that has been done above. We find that, for the two second-order transitions, the critical behavior of entanglement entropy is similiar with that in the Ising model, which verify that this three transitions belong to one universality. For the fourth-order wetting transition, the critical behavior of entanglement entropy is different with that in the second-order transitions. When we change the left boundary field h_L, there is a small oscillation and a nonlinear rapid increasing in the curve of half-chain entropy S_[1,N/2] before the linear part. Furthermore, it is also different that the relationship between entropy S_[1,n] and subsystem size n. There is not a logarithmic relationship even though the subsystem size n is very small.