分形随机电路中的局域化
Localization in fractonic random circuits
摘要：
在一维随机量子线路模型中研究了初始局域算符在幺正时间演化下的扩展，该模型受分形子相的量子动力学的激励，约束条件为$U(1)$电荷及其偶极矩守恒.我们发现，电荷仍然局限于其初始位置，提供了一个清晰的例子，随机电路动态的非遍历动力学阶段。这种局部化可以理解为低维随机游动的返回特性的结果，通过一种机制让人想起弱局部化，但对失相不敏感。电荷动力学很好地描述了一个系统的耦合流体动力学方程，这使得几个非平凡的预测与数值吻合得很好。重要的是，这些方程还预测了2d分形电路中的局部化。不动分形电荷发射非保守算子，其扩展由不同于非分形电路的指数决定。分形算符的纠缠呈现出短时间的线性增长，并满足面积定律和亚热体积定律。纠缠谱遵循半泊松统计，类似于MBL系统的本征态。非遍历现象持续到包含非零密度的偶极或分形电荷的初始条件。我们的工作意味着，低维分形系统永远保持其初始条件的记忆在本地观测噪声量子动力学，从而构成理想的记忆。这也意味着，1d和2d分形子系统应该实现真正的MBL下的哈密顿动力学，即使在没有无序，与MBL的障碍，在平移不变的系统和d>1被逃避的性质的机制负责本地化。我们还提出了一个可能的路线，新的非遍历阶段在高维。
英文摘要：
We study the spreading of initially-local operators under unitary time evolution in a 1d random quantum circuit model which is constrained to conserve a $U(1)$ charge and its dipole moment, motivated by the quantum dynamics of fracton phases. We discover that charge remains localized at its initial position, providing a crisp example of a non-ergodic dynamical phase of random circuit dynamics. This localization can be understood as a consequence of the return properties of low dimensional random walks, through a mechanism reminiscent of weak localization, but insensitive to dephasing. The charge dynamics is well-described by a system of coupled hydrodynamic equations, which makes several nontrivial predictions in good agreement with numerics. Importantly, these equations also predict localization in 2d fractonic circuits. Immobile fractonic charge emits non-conserved operators, whose spreading is governed by exponents distinct to non-fractonic circuits. Fractonic operators exhibit a short time linear growth of observable entanglement with saturation to an area law, as well as a subthermal volume law for operator entanglement. The entanglement spectrum follows semi-Poisson statistics, similar to eigenstates of MBL systems. The non-ergodic phenomenology persists to initial conditions containing non-zero density of dipolar or fractonic charge. Our work implies that low-dimensional fracton systems preserve forever a memory of their initial conditions in local observables under noisy quantum dynamics, thereby constituting ideal memories. It also implies that 1d and 2d fracton systems should realize true MBL under Hamiltonian dynamics, even in the absence of disorder, with the obstructions to MBL in translation invariant systems and in d>1 being evaded by the nature of the mechanism responsible for localization. We also suggest a possible route to new non-ergodic phases in high dimensions.
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