We study the energy and entanglement dynamics of $(1+1)\mathrm{D}$ conformal field theories (CFTs) under a Floquet drive with the sine-square deformed (SSD) Hamiltonian. Previous work has shown that this model supports both a nonheating and a heating phase. Here, we analytically establish several robust and “superuniversal” features of the heating phase which rely on conformal invariance but not on the details of the CFT involved. First, we show the energy density is concentrated in two peaks in real space, a chiral and an antichiral peak, which leads to an exponential growth in the total energy. The peak locations are set by fixed points of the Möbius transformation. Second, all of the quantum entanglement is shared between these two peaks. In each driving period, a number of Bell pairs are generated, with one member pumped to the chiral peak and the other member pumped to the antichiral peak. These Bell pairs are localized, accumulate at these two peaks, and can serve as a source of quantum entanglement. Third, in both the heating and nonheating phases, we find that the total energy is related to the half system entanglement entropy by a simple relation $E(t)\propto c\exp [(6/c)S(t)]$ with $c$ being the central charge. In addition, we show that the nonheating phase, in which the energy and entanglement oscillate in time, is unstable to small fluctuations of the driving frequency in contrast to the heating phase. Finally, we point out an analogy to the periodically driven harmonic oscillator which allows us to understand global features of the phases and introduce a quasiparticle picture to explain the spatial structure, which can be generalized to setups beyond the SSD construction.

中文翻译：

---

球形共形场理论中的新兴空间结构与纠缠局部化

我们研究了能量和纠缠动力学 $（\mathrm{1个}+\mathrm{1个}）\mathrm{d}$正弦方变形（SSD）哈密顿量的Floquet驱动下的共形场理论（CFT）。先前的工作表明，该模型支持非加热阶段和加热阶段。在这里，我们分析性地建立了加热阶段的几个鲁棒性和“超通用性”特征，这些特征依赖于保形不变性，而不依赖于所涉及的CFT的细节。首先，我们表明能量密度集中在现实空间的两个峰中，即手性峰和反手性峰，这导致总能量呈指数增长。峰值位置由Möbius变换的固定点设置。其次，所有量子纠缠在这两个峰之间共享。在每个行驶周期中，都会生成许多贝尔对，其中一个成员被泵送到手性峰，另一成员被泵送到反手性峰。这些贝尔对是局部的，在这两个峰处积累，可以用作量子纠缠的来源。第三，在加热和非加热阶段，我们发现总能量与半系统纠缠熵之间的关系很简单$Ë（Ť）\propto C\mathrm{经验值}[（6/C）\mathrm{小号}（Ť）]$ 与 $C$作为中央负责人。另外，我们表明，与加热阶段相比，能量和纠缠随时间振荡的非加热阶段对驱动频率的小波动不稳定。最后，我们指出了与周期性驱动的谐波振荡器的类比，该类比使我们能够了解相的全局特征，并引入准粒子图片来解释空间结构，可以将其推广到SSD结构以外的设置。