The critical behaviour of a relativistic scalar field theory with Z2 symmetry is investigated near a second-order phase transition. Two sets of dynamic equations are employed, allowing to control conservation laws of both order parameter and energy density. We calculate spectral functions of the order parameter as well as unequal-time correlation functions of the energy momentum tensor at zero and non-vanishing spatial momenta from first-principles classical-statistical lattice simulations in real-time. For both, we investigate general properties and relevant degrees of freedom in distinct regions of the phase diagram. Close to the critical point, we find signatures of dynamic scaling behaviour and calculate the dynamic critical exponent z controlling the divergence of the critical time scale. For both the order parameter spectral function as well as the energy density autocorrelation, we extract universal dynamic scaling functions. Modifying the simulation framework to include dynamically changing temperature and external field allows us to study non-equilibrium phenomena. For the special case of instant quenches to the critical point, we identify universal scaling behaviour controlled by the initial magnetization, and calculate the related additional dynamic critical exponent. We extract the universal non-equilibrium scaling functions for the evolution of both the order parameter as well as the correlation length.
The critical behaviour of a relativistic scalar field theory with Z2 symmetry is investigated near a second-order phase transition. Two sets of dynamic equations are employed, allowing to control conservation laws of both order parameter and energy density. We calculate spectral functions of the order parameter as well as unequal-time correlation functions of the energy momentum tensor at zero and non-vanishing spatial momenta from first-principles classical-statistical lattice simulations in real-time. For both, we investigate general properties and relevant degrees of freedom in distinct regions of the phase diagram. Close to the critical point, we find signatures of dynamic scaling behaviour and calculate the dynamic critical exponent z controlling the divergence of the critical time scale. For both the order parameter spectral function as well as the energy density autocorrelation, we extract universal dynamic scaling functions. Modifying the simulation framework to include dynamically changing temperature and external field allows us to study non-equilibrium phenomena. For the special case of instant quenches to the critical point, we identify universal scaling behaviour controlled by the initial magnetization, and calculate the related additional dynamic critical exponent. We extract the universal non-equilibrium scaling functions for the evolution of both the order parameter as well as the correlation length.
We study the dynamics of self-interacting scalar fields with Z2 symmetry governed by a relativistic Israel-Stuart type diffusion equation in the vicinity of a critical point. We calculate spectral functions of the order parameter in mean-field approximation as well as using first-principles classical-statistical lattice simulations in real-time. We observe that the spectral functions are well-described by single Breit-Wigner shapes. Away from criticality, the dispersion matches the expectations from the mean-field approach. At the critical point, the spectral functions largely keep their Breit-Wigner shape, albeit with non-trivial power-law dispersion relations. We extract the characteristic time-scales as well as the dynamic critical exponent z, verifying the existence of a dynamic scaling regime. In addition, we derive the universal scaling functions implied by the Breit-Wigner shape with critical power-law dispersion and show that they match the data. Considering equations of motion for a system coupled to a heat bath as well as an isolated system, we perform this study for two different dynamic universality classes, both in two and three spatial dimensions.
We investigate the out-of-equilibrium dynamics of a relativistic Z2-symmetric scalar field theory with Langevin dynamics in two and three spatial dimensions under linear driving across magnetic first-order phase transitions, close to and far below the critical temperature Tc. Using classical-statistical lattice simulations, we find that if the driving timescale is sufficiently fast, the system exhibits finite-time scaling behavior independent of temperature and dimensionality, identical to that observed in mean-field simulations. In slow quenches near Tc this mean-field behavior crosses over to critical Kibble-Zurek scaling behavior, while for temperatures T≪Tc nucleation and growth dominate the transition dynamics, resulting in corrections to scaling. Near the transition point where the order parameter changes sign, the crossover between mean-field and critical out-of-equilibrium dynamics is found to be well described by the leading algebraic correction to Kibble-Zurek scaling. We find that universal non-equilibrium scaling behavior can be observed for T≲Tc, provided the driving is fast enough to avoid nucleation but slow enough for correlations to form, and compute the associated universal scaling functions for the order parameter.
We study the critical dynamics of a scalar field theory with Z 2 symmetry in the dynamic universality class of Model A in two and three spatial dimensions with classical-statistical lattice simulations. In particular, we measure the non-equilibrium behavior of the system under a quench protocol in which the symmetry-breaking external field is changed at a constant rate through the critical point. Using the well-established Kibble-Zurek scaling theory we compute non-equilibrium scaling functions of cumulants of the order parameter up to fourth order. Together with the static critical exponents and the dynamic critical exponent, these fully describe the universal non-equilibrium evolution of the system near the critical point. We further extend the analysis to include finite-size effects and observe good collapse of our data onto two-dimensional universal non-equilibrium and finite-size scaling functions.
In context of the search for the QCD critical endpoint in heavy-ion collisions, a deep understanding of the out-of-equilibrium dynamics of the system is necessary to make well-grounded predictions for signatures in final states. To this end, we investigate the dynamic critical behavior of a classical scalar field theory with Z2 symmetry in the dynamic universality class of Model A in two spatial dimensions. The critical dynamics of the system is studied under a linear quench protocol, in which the external symmetry breaking field is changed at a constant rate through the critical point. We discuss the connection to the Kibble-Zurek mechanism and determine universal scaling functions, which fully describe the non-equilibrium evolution of the system near the critical point for all quench rates under consideration. We find that, while the scaling functions are non-trivial, the corresponding scaling exponents are fully determined by static critical exponents and the dynamic critical exponent.
We now begin our exposition of universal dynamic scaling behavior that emerges under non-equilibrium conditions. We are mainly concerned with the relaxational models A and B here, but also touch on other dynamic universality classes. First, we consider non-equilibrium critical relaxation from a disordered initial state, and compute the associated universal scaling exponents within the dynamic renormalization group framework. Related phenomena in the early-time regime, prior to reaching temporally translationally invariant asymptotics, are ‘aging’ and a non-equilibrium fluctuation-dissipation ratio, both of which have enjoyed prominence in the literature on ‘glassy’ kinetics. In this context, we derive the critical initial slip exponent for the order parameter, and briefly discuss interesting persistence properties. Second, we explore the long-time scaling laws in phase ordering kinetics and coarsening for the relaxational models, following a fast temperature ‘quench’ into the ordered phase, in systems with either non-conserved or conserved order parameters. Aside from a few explicit computations in the spherical model limit (wherein the number of order parameter components n → ∞), we largely employ phenomenological considerations and scaling theory, stressing the importance of topological defects for energy dissipation during coarsening. Next we address the question how violations of Einstein's relation that links relaxation coefficients and noise correlations might affect the asymptotic dynamic critical scaling behavior. Whereas in systems with non-conserved order parameter typically the equilibrium scaling laws are recovered in the vicinity of the critical point, in contrast genuinely novel universality classes may emerge in the case of a conserved parameter, provided it is driven out of equilibrium in a spatially anisotropic fashion.
The collective behaviour of statistical systems close to critical points is characterized by an extremely slow dynamics which, in the thermodynamic limit, eventually prevents them from relaxing to an equilibrium state after a change in the thermodynamic control parameters. The non-equilibrium evolution following this change displays some of the features typically observed in glassy materials, such as ageing, and it can be monitored via dynamic susceptibilities and correlation functions of the order parameter, the scaling behaviour of which is characterized by universal exponents, scaling functions, and amplitude ratios. This universality allows one to calculate these quantities in suitable simplified models and field-theoretical methods are a natural and viable approach for this analysis. In addition, if a statistical system is spatially confined, universal Casimir-like forces acting on the confining surfaces emerge and they build up in time when the temperature of the system is tuned to its critical value. We review here some of the theoretical results that have been obtained in recent years for universal quantities, such as the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics, with particular focus on the Ising model with Glauber dynamics in the bulk. The non-equilibrium dynamics of the Casimir force acting in a film is discussed within the Gaussian model.