论文标题
局部迭代的障碍汉密尔顿人：一种新工具的单数扰动理论
Local iterative block-diagonalization of gapped Hamiltonians: a new tool in singular perturbation theory
论文作者
论文摘要
In this paper the local iterative Lie-Schwinger block-diagonalization method, introduced in [FP], [DFPR1], and [DFPR2] for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation consisting of bounded interaction potentials of short range由真实的耦合常数t进行了mutltigl。我们的目标是证明此类汉密尔顿人的基础能量上方的光谱差距持续存在，因为| t |的值与晶格的大小无关。新思想和概念对于将我们的方法扩展到维度d> 1的系统是必要的：与我们较早的工作一样，基于明智选择的原始汉密尔顿的统一结合，介绍了一系列局部块 - 二进制步骤。可以通过我们称之为晶格中包含的最小矩形来确定在这些区块 - 划分之间产生的有效相互作用电位的支持，该概念有助于解决在迭代块对基础化步骤过程中出现的组合问题。对于给定的最小矩形，通过利用各种相当微妙的机制来实现对给定矩形中每个块对基于的有效相互作用电位的控制，并控制由大型矩形和大型分类构成的构成型号的构成构成的构成构成的矩形和概述，这些机制包括在内在局部区块 - 二角化步骤中涉及的相互作用潜在术语的单一结合产生的分子中。
In this paper the local iterative Lie-Schwinger block-diagonalization method, introduced in [FP], [DFPR1], and [DFPR2] for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation consisting of bounded interaction potentials of short range mutltiplied by a real coupling constant t. Our goal is to prove that the spectral gap above the ground-state energy of such Hamiltonians persists for sufficiently small values of |t|, independently of the size of the lattice. New ideas and concepts are necessary to extend our method to systems in dimension d > 1: As in our earlier work, a sequence of local block-diagonalization steps based on judiciously chosen unitary conjugations of the original Hamiltonian is introduced. The supports of effective interaction potentials generated in the course of these block-diagonalization steps can be identified with what we call minimal rectangles contained in the lattice, a concept that serves to tackle combinatorial problems that arise in the course of iterating the block-diagonalization steps. For a given minimal rectangle, control of the effective interaction potentials generated in each block-diagonalization step with support in the given rectangle is achieved by exploiting a variety of rather subtle mechanisms which include, for example, the use of weighted sums of paths consisting of overlapping rectangles and of large denominators, expressed in terms of sums of orthogonal projections, that serve to control analogous sums of projections in the numerators resulting from the unitary conjugations of the interaction potential terms involved in the local block-diagonalization step.
