We investigate the dynamic critical behavior of a relativistic scalar field theory with $Z_2$ symmetry by calculating spectral functions of the order parameter at zero and non-vanishing momenta from first-principles classical-statistical lattice simulations in real-time. We find that at temperatures above the critical point ($T>T_c$), the spectral functions are well described by relativistic quasi-particle peaks. Close to the transition temperature ($T\sim T_c$), we observe strong infrared contributions building up. In the ordered phase at low temperatures ($T<T_c$), in addition to the quasi-particle peak, we observe a soft mode with a dispersion relation indicative of collective excitations. Investigating the spectral functions close to $T_c$, we demonstrate that the behavior in the vicinity of the critical point is controlled by dynamic scaling functions and the dynamic critical exponent z, which we determine from our simulations. By considering the equations of motion for a closed system and a system coupled to a heat bath, we extract the dynamic critical behavior for two different dynamic universality classes (Models A & C) in two and three spatial dimensions.

我们研究相对论标量场理论的动态临界行为 ${ž}_2$通过实时从第一原理经典统计格子模拟中计算零阶和零阶矩处阶参数的谱函数来实现对称性。我们发现在高于临界点的温度下（$Ť>{Ť}_C$），光谱函数很好地用相对论准粒子峰描述。接近转变温度（$Ť〜{Ť}_C$），我们会观察到强烈的红外辐射积聚。在低温下的有序相中（$Ť<{Ť}_C$），除了准粒子峰外，我们还观察到一种软模式，其色散关系表示集体激发。研究接近${Ť}_C$，我们证明了临界点附近的行为是由动态缩放函数和动态临界指数z控制的，我们可以从仿真中确定它们。通过考虑封闭系统和与热浴耦合的系统的运动方程，我们在两个和三个空间维度上提取了两个不同的动态通用性类（模型A和C）的动态临界行为。