Connecting global and local energy distributions in quantum spin models on a lattice
Connecting global and local energy distributions in quantum spin models on a lattice
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DOI：
10.1088/1742-5468/2016/03/033301
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发表时间：
2016-03-01
影响因子：
2.4
通讯作者：
Landau, Zeph
中科院分区：
文献类型：
--
作者：
Arad, Itai;Kuwahara, Tomotaka;Landau, Zeph
多体量子系统中的局域相互作用通常是非对易的，因此局域的哈密顿量不能与全局的哈密顿量同时测量。局部和全局哈密顿量的测量结果的概率分布之间的联系将取决于这两个哈密顿量的对角化基之间的角度。在本文中，我们描述了这两个分布之间的关系。一方面，如果全局系统处于能量τ <τ的本征态的叠加中，则我们在局部区域中测量能量τ的概率上界。另一方面，我们在一个二分系统中，是在它的两个子系统的本征态的张量积测量的整体能量的概率绑定。非常粗略地说，我们表明，由于当地的相互作用的管理性质，这些分布是相同的，一个人遇到的通勤情况下，指数小的修正。最后，我们使用这些边界来研究局部截断哈密顿量的谱，其中连续区域的能量被截断超过一些阈值能量。我们表明，该哈密顿量的频谱的下半部分是指数接近的原始哈密顿量。这一结果在一维中的限制版本是最近改进一维面积定律的核心组成部分。
Local interactions in many-body quantum systems are generally non-commuting and consequently the Hamiltonian of a local region cannot be measured simultaneously with the global Hamiltonian. The connection between the probability distributions of measurement outcomes of the local and global Hamiltonians will depend on the angles between the diagonalizing bases of these two Hamiltonians. In this paper we characterize the relation between these two distributions. On one hand, we upperbound the probability of measuring an energy tau in a local region, if the global system is in a superposition of eigenstates with energies epsilon < tau. On the other hand, we bound the probability of measuring a global energy epsilon in a bipartite system that is in a tensor product of eigenstates of its two subsystems. Very roughly, we show that due to the local nature of the governing interactions, these distributions are identical to what one encounters in the commuting cases, up to exponentially small corrections. Finally, we use these bounds to study the spectrum of a locally truncated Hamiltonian, in which the energies of a contiguous region have been truncated above some threshold energy. We show that the lower part of the spectrum of this Hamiltonian is exponentially close to that of the original Hamiltonian. A restricted version of this result in 1D was a central building block in a recent improvement of the 1D area-law.