AI 中文总结
该研究针对量子多体物理中找哈密顿量谱隙下界难的问题，重新审视鞅方法并改进，设计新技术精确高效计算关键量，改进了方法，优于现有技术，还简化了MPS母哈密顿量有能隙的证明。
AI 中文摘要
证明基态空间之上的谱隙是量子多体物理中的核心问题。然而，找到谱隙的下界非常困难。我们重新审视了最初由法内斯、纳赫特盖尔和维尔纳开发的鞅方法，以证明矩阵乘积态（MPS）的母哈密顿量谱隙的存在，并对该方法的不同步骤进行了改进。最重要的是，我们设计了一种新技术，能够精确且高效地计算鞅方法中的关键量——局部基态空间的重叠。这使得该方法得到明显改进，在多个模型上的基准测试表明它优于其他现有技术来下界谱隙。值得注意的是，我们基于数值的方法同时也显著简化了任何（行为良好的）MPS的母哈密顿量总是有能隙这一事实的证明。
英文摘要
Proving spectral gaps above the ground space is a central problem in quantum many-body physics. Yet, finding lower bounds on the gap is notoriously difficult. We revisit the martingale method, originally developed by Fannes, Nachtergaele, and Werner [Fannes '92, Nachtergaele '96] to prove the existence of spectral gaps of parent Hamiltonians of Matrix Product States (MPS), and provide improvements to the different steps of the method. Most importantly, we devise a new technique which allows to compute the key quantity in the martingale method -- the overlap of local ground spaces -- exactly and efficiently. This enables a clear improvement of the method, allowing it to outperform other existing techniques to lower bound gaps, which we demonstrate by benchmarking on several models. Remarkably, our -- numerically motivated -- approach at the same time also yields a significantly simplified proof of the fact that parent Hamiltonians of any (well-behaved) MPS are always gapped.
