Analysis of Quantum Spin Systems by a Nonlinear Sigma Model Method

在具有反铁磁相互作用的量子自旋系统中，强的量子涨落会破坏自旋有序，形成无序的基态。那么从基态的自旋激发具有有限的激发能或自旋隙。我们提出了一种新的非线性σ模型方法，并利用它研究了量子自旋系统的无序态，特别是对正方晶格上具有第一和第二近邻交换作用的自旋系统（称为J1-J2海森堡模型）建立了一种非线性σ模型方法.在此基础上，结合已有的结果，我们发现该模型的基态是无序相的元格态，如果该模型的经典形式具有反铁磁自旋序，则该模型原则上可以推广到另一个二维自旋系统.我们研究的情况下，三角形晶格与正方形晶格样的各向异性和蜂窝晶格。无序基态的性质是一个值得研究的问题。在正方形晶格附近的各向异性三角形晶格的基态与正方形晶格的基态是连续的，这意味着它们的无序状态基本相同。另一方面，强阻挫系统的非线性σ模型方法尚未建立。例如，在强阻挫的情况下，量子自旋系统的经典版本具有120度结构的序，而不是反铁磁序。非线性σ模型方法推广到这样的系统是不容易的。对一维量子自旋系统，我们将非线性σ模型方法推广到带侧链自旋的情形.具有无带隙自旋激发的条件从没有侧链的情况改变。

In a quantum spin system with antiferromagnetic interactions, strong quantum fluctuation may destroy the spin order so as to form a disordered ground state. Then a spin excitation from the ground state has a finite excitation energy or spin-gap. We formulated a novel nonlinear σ model method and, by using it, we examined disordered states for quantum spin systems.Especially, we established a nonlinear σ model method for a spin system with first- and second-neighbor exchange interactions on a square lattice, which is called the J1-J2 Heisenberg model. Based on the formulation and assisted by some known results, we found that the ground state of this model is the plaquette state in the disordered phase.The formulation can be extended, in priciple, to another two-dimensional spin system, if its classical version has an antiferromagnetic spin order. We examined the cases of a triangular lattice with square-lattice-like anisotropy and a honycomb lattice. The characters of the disorderd ground states are issues. The ground state for an anisotropic triangular lattice near a square lattice continues to that for a sqare lattice, meaning that their disordered state is basically the same.On the other hand, the nonlinear σ model method for a system with strong frustration is not established yet. In the case of strong frustration, the classical version of a quantum spin system has, for example, an order of 120-degree structure and not an antiferromagnetic order. The extention of the nonlinear σ model method to such a system is not easy. We have done some trials for a general formulation.For one-dimensional quantum spin systems, we extend the nonlinear σ model method to the case of spins with side chains. The condition for having gapless spin excitations is changed from the case of no side chains.