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: it is provably impossible to determine 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 DMRG—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 neighbour interactions with undecidable spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable as in higher dimensions, but also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with constant spectral gap and unique classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behaviour 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。近来，这个重要的问题对于量子自旋系统在两个（或多个）空间维度上是无法确定的：证明一般无法确定系统是有间隙的还是无间隙的，这一结果对物理学的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，一维自旋系统比高维自旋系统更简单：例如，它们不具有热相变或拓扑顺序，并且存在高效的数值算法，例如DMRG，甚至可证明是多项式，时间—对于空白的一维系统，利用这样的系统遵守熵面积定律的事实。此外，光谱间隙不确定性的构建主要依赖于非周期性的平铺，这在1D中是不可能的，那么光谱间隙问题在1D中是否可以判定？在本文中，我们通过构造一族一维自旋链来证明事实并非如此，该一族自旋链具有平移不变的最近邻域相互作用，且谱隙不确定。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。特别是，这意味着存在一维系统，该系统具有恒定的光谱间隙和独特的经典基态，适用于所有尺寸达到无可争议的大尺寸的系统，随后它们切换为具有密集光谱的无间隙行为。