在体临界点，表面可能会有更丰富更新奇的临界行为，称为表面临界行为。一般来说，表面与体的临界性存在着多对一的关系。本文主要研究二维量子系统的 (2+1) 维 O(3) 临界点以及去禁闭量子临界点附近的表面临界行为。我们通过耦合一维的对角梯子形成二维的量子系统，在体 O(3) 临界点上研究其表面临界行为。由于数据的不稳定以及缺乏理论的支持，在去禁闭临界点处，对于其表面临界行为我们一时还不能给出明确的结论。所以在本文中，我们研究了其对应的二维系统剥离出的一维表面，即，两条腿梯子晶格上的 J­-Q₂模型和三条腿梯子晶格上的 J­-Q₃ 模型。本文主要分为以下三个部分。第一部分，我们利用量子蒙特卡罗模拟，深入地研究了二维反铁磁耦合自旋 1/2 海森堡对角梯子模型的体和表面临界行为。我们发现系统存在受对称性保护的拓扑 (symmetry protected topological, SPT) 相—二维 SPT Haldane 相，反铁磁有序相和拓扑平庸的 RS (rung singlet) 相。调节梯子的耦合强度，该模型会出现两个相变点：从 Haldane 相到反铁磁相以及从反铁磁相到 RS 相。我们数值地证明，二维的拓扑序并不改变体相变的临界指数，两个相变点都属于三维 O(3) 的普适类。在体临界点，通过不同的表面构造，我们实现了由SPT 相无能隙表面态和悬挂自旋链两种不同的机制所诱导的特殊表面相变。所得到的结果强有力地支持了量子特殊表面临界现象的量子起源。第二部分，我们利用量子蒙特卡罗模拟，深入地研究了二维铁磁耦合自旋 1/2 反铁磁海森堡对角梯子模型的体和表面临界行为。这个模型还可以被视为，反铁磁对角耦合的一般自旋 1/2 海森堡梯子 (usual ladder) 模型。我们发现系统存在磁有序的条纹 (striped) 相和两种二维 SPT Haldane 相。我们数值地证明，两个相变点都属于三维 O(3) 的普适类。从而进一步说明，体的拓扑序的确不改变体相变的临界性质。我们进而发现：两种 SPT 相的无能隙表面态，在体临界点都会直接诱导磁有序的非凡表面相变，而不需要增强表面的耦合。这种表面有序行为，在三维经典模型的 O(3) 临界点上是不会发生的，所以其起因应该归于 Haldane 相的拓扑性质。第三部分，我们研究了两条腿梯子晶格上的 J­-Q₂ 模型和三条腿梯子晶格上的 J­-Q₃ 模型。我们发现在一定的耦合强度下，两种模型都会自发性的破缺晶格的 Z2 对称性，形成价键固体态 (valence­ bond solids， VBS)，而在无序相却有着完全不同的性质，其中两条腿梯子处在 Haldane 相，而三条腿梯子处在共振价键态 (resonating valence­ bond states， RVB)。我们数值地证明，一维的拓扑序也不改变相变的临界指数，两条腿梯子中的量子相变属于二维伊辛的普适类。而对于三条腿梯子，其发生 BKT (Berezinskii–Kosterlitz–Thouless) 类的相变。通过对比 Haldane 相和 RVB 相中的性质，我们也验证了 Haldane 猜想。
本文的结构如下：第一章，我们主要来介绍相变和表面临界现象研究中的一些基本概念和研究进展。第二章，我们主要介绍量子蒙特卡罗模拟中一种常用的方法，随机序列展开方法(Stochastic Series Expansion， SSE)。并通过海森堡模型，介绍其在实际问题中的应用。第三章，通过以上方法来研究，反铁磁耦合对角梯子模型中的体和表面临界行为。第四章，通过随机序列展开方法来研究，铁磁耦合对角梯子模型中的体和表面临界行为。第五章，应用同样的方法来研究，两条腿梯子晶格上的 J-­Q₂ 和三条腿梯子晶格上的J-­Q₃ 模型中的相与相变。第六章，对本文进行总结，并对未来的工作进行展望。

At a bulk critical point, the surface may show rich and novel critical behaviors, called the surface critical behavior (SCB). In general, there is a many­ to­ one correspondence between the surface and the bulk criticality. In this work, we want to study the SCBs of the two-­dimensional (2d) quantum system, including O(3) critical points and deconfined quantum critical points. By coupling one­-dimensional (1d) diagonal ladders to form a 2d quantum system, we study SCBs at its O(3) bulk critical points. Due to the instability of the data and the lack of theoretical support, we cannot draw definitive conclusions about the SCBs at the deconfined quantum critical point. Therefore, in this work, we study the 1d surface peeled off from its corresponding 2d quantum system. We studied the J­Q₂ model on a two-­leg ladder lattice and the J­-Q₃ model on a three-­leg ladder lattice, respectively. In the first work, using Quantum Monte Carlo simulations, we study the bulk and surface critical behaviors of the 2d antiferromagnetic coupled diagonal ladders model. With the spin­1/2 Heisenberg model on this lattice, we find that the system has a symmetry protected topological (SPT) phase, 2d Haldane phase, antiferromagnetic phase and a rung singlet product phase. Adjusting the coupling strength ratio of the system, two phase transition points will appear in the model. We show numerically show that the 2d topological order does not change the critical exponents of the bulk phase transition, and both quantum transition points belong to the three-dimensional O(3) universality class. At the bulk critical points, we realize the special SBCs by two different mechanisms: the gapless surface states of the SPT phase and the dangling spin chain. Our work strongly supports the quantum origin of the nonordinary surface critical behaviors found in various quantum models. In the second work, using Quantum Monte Carlo simulations, we study the bulk and surface critical behaviors of the 2d ferromagnetic coupled diagonal ladders model. The model can also be viewed as usual ladders with ferromagnetic rung couplings coupled by antiferromagnetic diagonal couplings. With the spin­1/2 Heisenberg model on this lattice, we find that the model hosts a striped magnetic ordered phase and two topological nontrivial Haldane phases. We show numerically that the two quantum critical points are all in the three­-dimensional O(3) universality class irrelevant to the topological properties of the Haldane phases. We further demonstrate that extraordinary surface critical behaviors are realized at both critical points on such gapless surfaces originating from Haldane phases without enhancing the surface coupling. Notably, the surface is not expected to be ordered in the three­-dimensional classical O(3) critical point, suggesting that the topological properties of the Haldane phases are responsible for such surface critical behavior. In the third work, we studied the J­-Q₂ model on a two­-leg ladder lattice and the -J­Q₃ model on a three­-leg ladder lattice, respectively. We found that the two models will spontaneously break the Z2 translational symmetry of the lattice and form valence bond solids (VBS) state, under certain coupling strength. However, there are completely different properties in the disordered phase. The two-­leg ladder is in the Haldane phase, while the three­-leg ladder is in the resonating valence bond (RVB) state. We show numerically that the 1d topological order does not change the critical exponents of the phase transition and the quantum phase transitions in the two-leg ladder belong to 2d Ising universality. For the three­-leg ladder, a BKT (Berezinskii–Kosterlitz–Thouless) type phase transition occurs. By contrasting the properties in the Haldane and RVB phases, we also potentially prove the Haldane’s conjecture. The paper is organized as follows: In the first chapter, we mainly introduce some basic concepts and research progress in the study of phase transitions and SCB. In the second chapter, we mainly introduce a quantum Monte Carlo simulation method, Stochastic Series Expansion (SSE). And through the Heisenberg model, its application in practical problems is introduced. In the third chapter, we study the bulk and SCB in the antiferromagnetic coupled diagonal ladders model using above method. In the fourth chapter, we study the bulk and SCB in the ferromagnetic coupled diagonal ladders model using the SSE method. In the fifth chapter, the same method is used to study the phase and phase transition in the 1d J­-Q models. In the sixth chapter, the paper is summarized and the future work is prospected.