Dynamical phase transitions are defined through non-analyticities of the survival probability of an out-of-equilibrium time-evolving state at certain critical times. They ensue from zeros of the corresponding survival amplitude. By extending the time variable onto the complex domain, we formulate the complex-time survival amplitude. The complex zeros of this quantity near the time axis correspond, in the infinite-size limit, to non-analytical points where the survival probability abruptly vanishes. Our results are numerically exemplified in the fully-connected transverse-field Ising model, which displays a symmetry-broken phase delimited by an excited-state quantum phase transition. A detailed study of the behavior of the complex-time survival amplitude when the characteristics of the out-of-equilibrium protocol changes is presented. The influence of the excited-state quantum phase transition is also put into context.
Distribution of dynamical quantities in the contact process, random walks, and quantum spin chains in random environments
我们研究了各种一维无序模型中动力学量的分布，这些模型的临界行为由无限随机不动点描述。在{ \it无序接触过程}中，在固定随机环境中定义的猝灭生存概率$\mathcal{P}(t)$在临界点处表现出多标度性，这意味着当$t\to\infty$时，$\mathcal{P}(t)=t^{-δ}$，其中（环境和时间相关的）指数$δ$具有普适极限分布.利用强无序重整化群方法，在半无限格点的端点处解析地确定了极限分布，发现极限分布是指数分布，而在无限格点处，则基于数值结果，给出了小、大$δ$的极限行为的表达式.用同样的方法，我们还证明了随机环境中的随机游动问题{ \it random walks in random environments}中的猝灭生存概率也是指数极限分布的多尺度问题.此外，还发现{ \it随机横场伊辛链}的自旋-自旋自相关函数在重正化水平上具有与接触过程的生存概率相似的形式.因此，可以建立两个问题中相应的极限分布之间的关系。最后讨论了该模型中自发磁化强度的分布。
Phase field crystal dynamics for binary systems: Derivation from dynamical density functional theory, amplitude equation formalism, and applications to alloy heterostructures
