Spectral gaps of frustration-free spin systems with boundary
Spectral gaps of frustration-free spin systems with boundary
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DOI：
10.1063/1.5089773
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发表时间：
2018-01
影响因子：
1.3
通讯作者：
M. Lemm;E. Mozgunov
中科院分区：
文献类型：
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作者：
M. Lemm;E. Mozgunov
在量子多体系统中，基态之上光谱间隙的存在具有深远的影响。在本文中，我们讨论了无挫折自旋系统及其应用中具有光谱间隙的“有限尺寸”标准。我们将最初由 Knabe 和 Gosset-Mozgunov 为周期系统开发的准则扩展到有边界的系统。我们的有限尺寸标准表明，如果线性系统尺寸 $n$ 的光谱间隙超过阶 $n^{-3/2}$ 的显式阈值，则整个系统将出现间隙。该准则以精确的方式考虑了有限系统的“体间隙”和“边缘间隙”。 $n^{-3/2}$ 缩放是稳健的：它适用于 1D 和 2D 系统、任意晶格和任意有限范围相互作用。我们的结果的一个应用是为二维无挫败模型不能容纳手性边缘模式（其有限大小的光谱间隙将像 $n^{-1}$ 一样缩放）的民间传说提供严格的基础。
In quantum many-body systems, the existence of a spectral gap above the ground state has far-reaching consequences. In this paper, we discuss "finite-size" criteria for having a spectral gap in frustration-free spin systems and their applications. We extend a criterion that was originally developed for periodic systems by Knabe and Gosset-Mozgunov to systems with a boundary. Our finite-size criterion says that if the spectral gaps at linear system size $n$ exceed an explicit threshold of order $n^{-3/2}$, then the whole system is gapped. The criterion takes into account both "bulk gaps" and "edge gaps" of the finite system in a precise way. The $n^{-3/2}$ scaling is robust: it holds in 1D and 2D systems, on arbitrary lattices and with arbitrary finite-range interactions. One application of our results is to give a rigorous foundation to the folklore that 2D frustration-free models cannot host chiral edge modes (whose finite-size spectral gap would scale like $n^{-1}$).