The spectral gap problem—determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations—pervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum-spin systems in two (or more) spatial dimensions: There exists no algorithm that determines in general whether a system is gapped or gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one-dimensional spin systems are simpler than their higher-dimensional counterparts: For example, they cannot have thermal phase transitions or topological order, and there exist highly effective numerical algorithms such as the density matrix renormalization group—and even provably polynomial-time ones—for gapped 1D systems, exploiting the fact that such systems obey an entropy area law. Furthermore, the spectral gap undecidability construction crucially relied on aperiodic tilings, which are not possible in 1D. So does the spectral gap problem become decidable in 1D? In this paper, we prove this is not the case by constructing a family of 1D spin chains with translationally invariant nearest-neighbor interactions for which no algorithm can determine the presence of a spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable as in higher dimensions, but it also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with a constant spectral gap and nondegenerate classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behavior with dense spectrum.

中文翻译：

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一维光谱间隙的不确定性

谱隙问题（确定系统的能谱是否具有高于基态的能隙，或者是否存在连续范围的低能激发）遍布整个量子多体物理学。最近，对于在两个（或多个）空间维度上的量子自旋系统，这个重要问题是无法确定的：没有一种算法可以确定系统是有间隙的还是无间隙的，其结果对物理学产生了许多意想不到的后果这样的系统。但是，有许多迹象表明，一维自旋系统比其高维自旋系统更简单：例如，它们不能具有热相变或拓扑顺序，并利用空白的一维系统遵循熵面积定律这一事实，存在一些高效的数值算法，例如密度矩阵重归一化组，甚至可证明的多项式时间组。此外，光谱间隙不确定性的构建至关重要地依赖于非周期性平铺，这在一维中是不可能的。那么在一维中光谱间隙问题是否可以确定？在本文中，我们通过构造一族具有平移不变的最近邻相互作用的一维自旋链来证明不是这种情况，对于这些相互作用，没有任何算法可以确定光谱间隙的存在。这不仅证明一维系统的光谱缺口与高维一样难以处理，而且还预示了一维自旋链中定性新型复杂物理的存在。尤其是，