标题： 具有指数衰减关联的态的无间隙性强制 显示英文标题

标题： Enforced Gaplessness from States with Exponentially Decaying Correlations

摘要： 众所周知，如果其基态具有代数关联，则指数局域化的哈密顿量必须是无隙的。我们表明，甚至某些指数衰减的关联也可以暗示无隙性。这由变形的拓扑码$\propto \exp(\beta \sum_{\ell} Z_{\ell}) |\mathsf{TC}\rangle$举例说明，其中$|\mathsf{TC}\rangle$是一个固定点拓扑码波函数。尽管它在$\beta > \beta_c$有一个约束区域，但近期的研究引起了对其边界定律环关联的关注。在这里，我们证明这些不寻常的环关联——即边界定律与一种 1 形对称性共存，其无序算符具有长程序——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而变化。此外，我们为任意局部无冲突哈密顿量构造了一个变分低能态，在周期性边界条件下，该态将有限尺寸间隙上界为$O(1/L^3)$。引人注目的是，这些变分态看起来像环波——自旋波的非准粒子类比——由非局部环算符从基态生成。我们的发现对于识别无隙基态所属的希尔伯特空间子集具有重要意义，且这些技术具有广泛的应用性。例如，我们第一个结果的一个推论是，在环面二维经典伊辛模型有序相的 Glauber 动力学中，马尔可夫转移矩阵必须是无隙的，而我们的第二个结果对其间隙进行了限制。 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.