标题： 拓扑序在$U(1)$棋盘状 toric 码中的缺失 显示英文标题

标题： Absence of topological order in the $U(1)$ checkerboard toric code

摘要： 我们研究了对应于具有两个不同星型子晶格的$U(1)$-对称性扩展 toric 代码的$U(1)$棋盘式 toric 代码。因此，可以从孤立星的极限调整到均匀系统。基于量子蒙特卡洛模拟表明有限团簇的紧致化可能导致非平凡基态简并度，均匀系统被认为可能具有非阿贝尔拓扑序。在这里，我们证明这些非平凡性质可以在孤立星的微扰极限下自然解释。实际上，基态简并度的紧致化依赖性可以追溯到来自面算符的几何约束。此外，在四阶退化微扰理论中，基态简并度被完全消除，导致出现一种具有束缚分数量子激发的非拓扑相。这些分数量子激发对于小扰动是束缚的，因此它们不能作为热力学极限中的单一低能激发存在，而只能作为拓扑平凡的复合粒子存在。然而，束缚尺度被证明出乎意料的大，所以在紧致极限之前有限尺寸的能隙在有限尺寸的簇上非常小，通过高阶级数展开显式计算得出。我们的发现表明，最近在均匀极限下的量子蒙特卡洛模拟未能区分这些能隙和有限尺寸效应。因此，所有结果都指向$U(1)$棋盘式 toric 代码在整个参数轴上不存在拓扑序。 显示英文摘要

摘要： We investigate the $U(1)$ checkerboard toric code which corresponds to the $U(1)$-symmetry enriched toric code with two distinct star sublattices. One can therefore tune from the limit of isolated stars to the uniform system. The uniform system has been conjectured to possess non-Abelian topological order based on quantum Monte Carlo simulations suggesting a non-trivial ground-state degeneracy depending on the compactification of the finite clusters. Here we show that these non-trivial properties can be naturally explained in the perturbative limit of isolated stars. Indeed, the compactification dependence of the ground-state degeneracy can be traced back to geometric constraints stemming from the plaquette operators. Further, the ground-state degeneracy is fully lifted in fourth-order degenerate perturbation theory giving rise to a non-topological phase with confined fracton excitations. These fractons are confined for small perturbations so that they cannot exist as single low-energy excitation in the thermodynamic limit but only as topologically trivially composite particles. However, the confinement scale is shown to be surprisingly large so that finite-size gaps are extremely small on finite clusters up to the uniform limit which is calculated explicitly by high-order series expansions. Our findings suggest that these gaps were not distinguished from finite-size effects by the recent quantum Monte Carlo simulation in the uniform limit. All our results therefore point towards the absence of topological order in the $U(1)$ checkerboard toric code along the whole parameter axis.