标题： 强制无能隙性来自指数衰减关联的状态 显示英文标题

标题： Enforced Gaplessness from States with Exponentially Decaying Correlations

摘要： 众所周知，若一个指数局域化的哈密顿量的基态具有代数关联，则它必须是无隙的。 我们证明了甚至某些指数衰减关联也能导致无隙性。 这通过变形 toric 代码 $\propto \exp(\beta \sum_{\ell} Z_{\ell}) |\mathsf{TC}\rangle$ 得到了说明，其中 $|\mathsf{TC}\rangle$ 是一个固定的 toric 代码波函数。 尽管它在 $\beta > \beta_c$ 的情况下存在局限区域，但最近的工作已经引起了对其周长法则环路关联的关注。 在这里，我们展示了这些不寻常的环路关联——即周长法则与具有长程有序离序算符的 1-形式对称性的共存——意味着任何局部母哈密顿量要么是无隙的，要么其简并度随系统大小按比例增长。 此外，我们为任意局部无阻尼哈密顿量构造了一个变分低能态，在周期边界条件下将有限尺寸间隙上限为 $O(1/L^3)$。 令人震惊的是，这些变分态看起来像环波——自旋波的非准粒子类似物——由非局域环算符从基态生成。 我们的发现对识别带隙基态属于希尔伯特空间的子集有影响，并且这些技术具有广泛适用性。 例如，我们的第一个结果的一个推论是，二维经典伊辛模型在环面上有序相的 Glabuer 动力学必须具有无隙的马尔可夫转移矩阵，而我们的第二个结果则限定了其间隙。 显示英文摘要

摘要： It is well known that an exponentially localized Hamiltonian must be gapless if its ground state has algebraic correlations. We show that even certain exponentially decaying correlations can imply gaplessness. This is exemplified by the deformed toric code $\propto \exp(\beta \sum_{\ell} Z_{\ell}) |\mathsf{TC}\rangle$, where $|\mathsf{TC}\rangle$ is a fixed-point toric code wavefunction. Although it has a confined regime for $\beta > \beta_c$, recent work has drawn attention to its perimeter law loop correlations. Here, we show that these unusual loop correlations -- namely, perimeter law coexisting with a 1-form symmetry whose disorder operator has long-range order -- imply that any local parent Hamiltonian must either be gapless or have a degeneracy scaling with system size. Moreover, we construct a variational low-energy state for arbitrary local frustration-free Hamiltonians, upper bounding the finite-size gap by $O(1/L^3)$ on periodic boundary conditions. Strikingly, these variational states look like loop waves -- non-quasiparticle analogs of spin waves -- generated from the ground state by non-local loop operators. Our findings have implications for identifying the subset of Hilbert space to which gapped ground states belong, and the techniques have wide applicability. For instance, a corollary of our first result is that Glauber dynamics for the ordered phase of the two-dimensional classical Ising model on the torus must have a gapless Markov transition matrix, with our second result bounding its gap.