The simplest topologically ordered phase in $2+1\mathrm{D}$ is the deconfined phase of ${\mathbb{Z}}_2$ gauge theory (realized in the toric code, for example). This phase permits a duality that exchanges electric and magnetic excitations (“$e$” and “$m$” particles). The phase transition where one of these particles condenses, while the other remains gapped, has 3D Ising exponents. But the transition out of the deconfined phase when self-duality symmetry is preserved is more mysterious. It has so far been unclear whether this transition is continuous, but if continuous, it may be the simplest critical point for which a useful continuum Lagrangian is still lacking. These questions are relevant to soft matter, too, since the gauge theory also describes classical membranes in 3D. Here, we study the self-dual transition with Monte Carlo simulations of the ${\mathbb{Z}}_2$ gauge-Higgs model on cubic lattices of linear size $L\le 96$. Our results indicate a continuous transition, for example via a striking parameter-free scaling collapse. We use duality symmetry to distinguish the leading duality-odd and duality-even scaling operators $A$ and $S$. We explain why standard techniques for locating the critical point are ineffective, and we develop an alternative using “renormalization group trajectories” of cumulants. We check that two- and three-point functions are scale invariant, with scaling dimensions $x_A$ and $x_S$ (autocorrelations in the Monte Carlo dynamics also yield a dynamical exponent $z$). Separately, we propose a general picture for emergent 1-form symmetries, in terms of “patching” of membranes or world surfaces. We relate this to the percolation of anyon worldlines in spacetime. The latter yields a fourth exponent for the self-dual transition. We propose variations of the model for further investigation.

中文翻译：

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三维Z2规范中物质的自对偶临界性

最简单的拓扑有序相 $2+1\mathrm{D}$ 是解除限制的阶段 ${\mathbb{Z}}_2$规范理论（例如在复曲面代码中实现）。这个阶段允许交换电和磁激励的二元性（“$\mathrm{电子}$“ 和 ”$米$“ 粒子）。其中一个粒子凝聚而另一个粒子保持间隙的相变具有 3D Ising 指数。但是，当自对偶对称出deconfined相位的转变保存是更加神秘。到目前为止，尚不清楚这种转变是否是连续的，但如果是连续的，它可能是最简单的临界点，但仍然缺乏有用的连续统拉格朗日量。这些问题也与软物质有关，因为规范理论还描述了 3D 中的经典膜。在这里，我们用蒙特卡罗模拟研究了自对偶转换${\mathbb{Z}}_2$ 线性尺寸立方晶格上的规范-希格斯模型 $升\le 96$. 我们的结果表明一个连续的转变，例如通过一个引人注目的无参数缩放崩溃。我们使用对偶对称来区分领先的奇偶对偶缩放算子$\mathrm{一种}$ 和 $秒$. 我们解释了为什么用于定位临界点的标准技术无效，并且我们开发了使用累积量的“重整化群轨迹”的替代方法。我们检查两点和三点函数是尺度不变的，具有尺度维度$X_{\mathrm{一种}}$ 和 $X_{秒}$ （蒙特卡洛动力学中的自相关也产生一个动态指数 $z$）。另外，我们根据膜或世界表面的“修补”提出了涌现的 1 型对称性的一般图景。我们将这与时空中任何一个世界线的渗透联系起来。后者产生自对偶转换的第四个指数。我们提出了模型的变体以供进一步研究。