Spectral functions and dynamic critical behavior of relativistic $Z_2$ theories (2007.03374v2)

Abstract: We investigate the dynamic critical behaviour 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.

• The paper computes spectral functions using lattice simulations, revealing quasi-particle peaks and critical scaling behavior near the critical temperature.
• The paper demonstrates dynamic scaling by extracting universal scaling functions and critical exponents for both conservative and dissipative dynamics.
• The paper identifies a soft mode in the ordered phase and shows that finite momentum acts as an effective infrared cutoff for the critical contributions.

Summary of "Spectral Functions and Dynamic Critical Behavior of Relativistic $Z_2$ Theories"

The paper presents a detailed investigation into the dynamic critical phenomena of relativistic scalar field theories with $Z_2$ symmetry, focusing on the computation of spectral functions for the order parameter using classical-statistical lattice simulations. This analysis is performed above, near, and below the critical temperature ($T_c$), providing insights into the behavior of the spectral functions across different phases.

Key Findings

1. Spectral Functions and Quasi-Particle Peaks:
   • In the symmetric phase ($T > T_c$), the spectral functions are predominantly characterized by relativistic quasi-particle peaks, conforming to a relativistic dispersion relation.
   • As the system approaches criticality ($T \approx T_c$), significant infrared contributions emerge, illustrating critical scaling behavior.
2. Dynamic Critical Scaling:
   • Spectral functions near the critical point exhibit universal scaling behavior. The paper successfully extracts corresponding dynamic scaling functions and critical exponents.
   • The dynamic critical exponent $z$ is computed for both conservative (Model C) and dissipative (Model A) dynamics, revealing expected alterations in the critical dynamics due to changes in conservation laws.
3. Order Parameter and Soft Mode:
   • In the ordered phase ($T < T_c$), besides the quasi-particle peak, a soft mode emerges, suggestive of collective excitations with a distinct dispersion relation.
   • This collective mode correlates with the presence of capillary waves at low temperatures, particularly in 3+1 dimensions.
4. Finite Spatial Momentum and Critically Slowing Modes:
   • Analysis at finite spatial momenta shows that momentum provides an effective infrared cutoff, which modifies the critical contributions to the spectral functions.

Methodology

The researchers employ a lattice simulation approach in real time, using both Hamiltonian and Langevin dynamics to simulate Models C and A within the dynamic universality classification. By leveraging the fluctuation-dissipation theorem, they extract the spectral functions from classical-statistical correlations.

Implications

This paper provides a comprehensive framework for understanding how dynamic critical phenomena manifest in spectral properties, offering a quantitative tool to probe similar transitions in complex systems, including condensed matter and high-energy physics. The findings are crucial for developing a quantitative understanding of processes near the critical point in systems with $Z_2$ symmetry, with potential implications for future theoretical and experimental investigations, particularly in the context of the QCD critical endpoint.

Future Directions

The paper suggests the potential extension of this research to more complex systems, including those with continuous symmetries (e.g., O(4) models), and non-equilibrium dynamics simulations. Such explorations could provide further insight into the universality and robustness of critical phenomena across different theoretical frameworks and physical systems.

This work lays a strong foundation for the systematic exploration of critical dynamics in theoretical physics, emphasizing the crucial role of spectral analysis in unveiling the characteristics of phase transitions in classical-statistical systems.