Title: Low energy spectrum of the XXZ model coupled to a magnetic field Show Chinese title

Title: XXZ 模型在磁场中的低能谱

Abstract: For a class of Hamiltonians of $XXZ$ spin chains in a uniform external magnetic field that are small quantum perturbations of an Ising Hamiltonian, it is shown that the spectral gap above the ground-state energy remains strictly positive when the perturbation is turned on, uniformly in the length of the chain. This result is proven for perturbations of both the ferromagnetic and the antiferromagnetic Ising Hamiltonian. In the antiferromagnetic case, the external magnetic field is required to be small. For a chain of an even number of sites, the two-fold degenerate ground-state energy of the unperturbed antiferromagnetic Hamiltonian may split into two energy levels separated by a very small gap. These results are proven by using a new, quite subtle refinement of a method developed in earlier work and used to iteratively block-diagonalize Hamiltonians of systems confined to ever larger subsets of a lattice by using strictly local unitary conjugations. The new method developed in this paper provides complete control of boundary effects on the low-energy spectrum of perturbed Ising chains uniformly in their length. Show Chinese abstract

Abstract: 对于均匀外磁场中的 $XXZ$ 自旋链的一类哈密顿量，它们是伊辛哈密顿量的小量子扰动，证明了当扰动打开时，基态能量以上的能隙仍然严格为正，并且该性质在链的长度上是一致的。 此结果对于铁磁和反铁磁伊辛哈密顿量的扰动都得到了证明。 在反铁磁情况下，需要外部磁场足够小。 对于偶数个位点的链，未扰动的反铁磁哈密顿量的两重简并基态能量可能会分裂成两个能量级，其间由一个非常小的间隙分开。 这些结果通过使用一种新的、相当微妙的方法来证明，该方法是对早期工作中开发的一种方法的改进，并用于通过严格局部幺正共轭逐步对格子的更大子集中的系统哈密顿量进行块对角化。 本文中提出的新方法提供了对扰动伊辛链低能谱边界效应的完全控制，并且这种控制在链的长度上一致有效。