Lie-Schwinger块对角化与带隙量子链：基态能量的解析性
Lie-Schwinger block-diagonalization and gapped quantum chains: analyticity of the ground-state energy
摘要：
我们考虑的量子链的哈密顿量是由一个哈密顿量，不耦合的自由度位于链的不同位置的相互作用，并有一个严格的正的基态能量以上的能隙的短程微扰。对于形式受限的相互作用。现场哈密顿条款，我们已经证明，频谱间隙的扰动哈密顿以上的基态能量是有界的从下面由一个正常数均匀的长度链，为小值的耦合常数，见[DFPR]。本文的主要结果是，在相同的假设条件下，当耦合常数在一定区间内，且在链长上均匀分布时，基态能量是解析的。此外，假设相互作用势在平移下不变，我们证明了，在热力学极限下，每个站点的能量是解析的耦合常数的值在相同的固定间隔。在我们的证明中，我们使用了一个新的方法在[FP]中介绍，这是基于本地Lie-Schwinger共轭的哈密尔顿与连接子集的链。我们证明了一个相当强的结果，复哈密顿对应的复值的耦合常数。
英文摘要：
We consider quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we have proven that the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain, for small values of a coupling constant; see [DFPR]. The main result of this paper is that, under the same hypotheses, the ground-state energy is analytic for values of the coupling constant belonging to a fixed interval, uniformly in the length of the chain. Furthermore, assuming that the interaction potentials are invariant under translations, we prove that, in the thermodynamic limit, the energy per site is analytic for values of the coupling constant in the same fixed interval. In our proof we use a new method introduced in [FP], which is based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain. We prove a rather strong result concerning complex Hamiltonians corresponding to complex values of the coupling constant.
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