In this article we give an overview of the concept of universal dynamics near non-thermal fixed points in isolated quantum many-body systems. We outline a non-perturbative kinetic theory derived within a Schwinger–Keldysh closed-time path-integral approach, as well as a low-energy effective field theory which enable us to predict the universal scaling exponents characterizing the time evolution at the fixed point. We discuss the role of wave-turbulent transport in the context of such fixed points and discuss universal scaling evolution of systems bearing ensembles of (quasi) topological defects. This is rounded off by the recently introduced concept of prescaling as a generic feature of the evolution towards a non-thermal fixed point.
In this article we give an overview of the concept of universal dynamics near non-thermal fixed points in isolated quantum many-body systems. We outline a non-perturbative kinetic theory derived within a Schwinger–Keldysh closed-time path-integral approach, as well as a low-energy effective field theory which enable us to predict the universal scaling exponents characterizing the time evolution at the fixed point. We discuss the role of wave-turbulent transport in the context of such fixed points and discuss universal scaling evolution of systems bearing ensembles of (quasi) topological defects. This is rounded off by the recently introduced concept of prescaling as a generic feature of the evolution towards a non-thermal fixed point.
Abstract Closed quantum systems far from thermal equilibrium can show universal dynamics near attractor solutions, known as non-thermal fixed points, generically in the form of scaling behaviour in space and time. A systematic classification and comprehensive understanding of such scaling solutions are tasks of future developments in non-equilibrium quantum many-body theory. In this tutorial review, we outline several analytical approaches to non-thermal fixed points and summarise corresponding numerical and experimental results. The analytic methods include a non-perturbative kinetic theory derived within the two-particle irreducible effective action formalism, as well as a low-energy effective field theory framework. As one of the driving forces of this research field are numerical simulations, we summarise the main results of exemplary cases of universal dynamics in ultracold Bose gases. This encompasses quantum vortex ensembles in turbulent superfluids as well as recently observed real-time instanton solutions in one-dimensional spinor condensates.
In these notes we discuss recent developments in the field of non-equilibrium quantum dynamics. Specifically, we consider nearly coherent Bose gases brought far out of equilibrium and study their behaviour in view of connections between universal properties, (quasi-)topological field configurations and turbulent dynamics. We demonstrate that the isolated Bose gas, on its way back to thermal equilibrium, can approach metastable non-equilibrium configurations and spend a long time in their vicinity. In such configurations, which have been termed non-thermal fixed points, the system shows universal long-range properties manifest through scaling, i.e., self-similar correlations. The spatial field pattern, at the same time, is characterized by the appearance of defects and domain formation whose geometry gives rise to the particular scaling laws seen in the correlation functions. We obtain an overall picture which connects well-known concepts for describing universal dynamics such as wave-turbulence, superfluid turbulence, and (quasi-)topological excitations. This allows to bring together a wide range of concepts and methods with a large spectrum of applications.
Abstract This chapter considers nearly coherent Bose gases brought far out of equilibrium and their behavior in the context of connections between universal properties, (quasi-)topological field configurations, and turbulent dynamics. The isolated Bose gas, on its way back to thermal equilibrium, can approach metastable nonequilibrium configurations and spend a long time in their vicinity. In such configurations, termed nonthermal fixed points, the system shows universal long-range properties manifest through scaling, i.e. self-similar correlations. The time evolution near such fixed points undergoes critical slowing down. The spatial field pattern, at the same time, is characterized by the appearance of defects and domain formation whose geometry gives rise to the particular scaling laws seen in the correlation functions. An overall picture is obtained that connects well-known concepts for describing universal dynamics such as wave turbulence, superfluid turbulence, and (quasi-)topological excitations. The extension to fermionic and gauge fields should reveal many interesting new aspects.
Recent progress in out-of-equilibrium closed quantum systems has significantly advanced the understanding of mechanisms behind their evolution toward thermalization. Notably, the concept of nonthermal fixed points (NTFPs)-responsible for the emergence of spatiotemporal universal scaling in far-from-equilibrium systems-has played a crucial role in both theoretical and experimental investigations. In this work, we introduce a differential equation that has the universal scaling associated with NTFPs as a solution. The advantage of working with a differential equation, rather than only with its solution, is that we can extract several insightful properties not necessarily present in the solution alone. How the differential equation is derived allows physical interpretation of the universal exponents in terms of the time dependence of the amplitude of the distributions and their momentum scaling. Employing two limiting cases of the equation, we determined the universal exponents related to the scaling using the distributions near just two momentum values. We established a solid agreement with previous investigations by validating this approach with three distinct physical systems. This consistency highlights the universal nature of scaling due to NTFPs and emphasizes the predictive capabilities of the proposed differential equation. Moreover, under specific conditions, the equation predicts a power-law related to the ratio of the two universal exponents, leading to implications concerning particle and energy transport. This suggests that the observed power-laws in far-from-equilibrium turbulent fluids could be related to the universal scaling due to NTFPs, potentially offering insights into the study of turbulence.
We outline a kinetic theory of nonthermal fixed points for the example of a dilute Bose gas, partially reviewing results obtained earlier, thereby extending, complementing, generalizing, and straightening them out. We study universal dynamics after a cooling quench, focusing on situations where the time evolution represents a pure rescaling of spatial correlations, with time defining the scale parameter. The nonequilibrium initial condition set by the quench induces a redistribution of particles in momentum space. Depending on conservation laws, this can take the form of a wave-turbulent flux or of a more general self-similar evolution, signaling the critically slowed approach to a nonthermal fixed point. We identify such fixed points using a nonperturbative kinetic theory of collective scattering between highly occupied long-wavelength modes. In contrast, a wave-turbulent flux, possible in the perturbative Boltzmann regime, builds up in a critically accelerated self-similar manner. A key result is the simple analytical universal scaling form of the nonperturbative many-body scattering matrix, for which we lay out the concrete conditions under which it applies. We derive the scaling exponents for the time evolution as well as for the power-law tail of the momentum distribution function, for a general dynamical critical exponent z and an anomalous scaling dimension $\ensuremath{\eta}$. The approach of the nonthermal fixed point is, in particular, found to involve a rescaling of momenta in time t by ${t}^{\phantom{\rule{0.16em}{0ex}}\ensuremath{\beta}}$, with $\ensuremath{\beta}=1/z$, within our kinetic approach independent of $\ensuremath{\eta}$. We confirm our analytical predictions by numerically evaluating the kinetic scattering integral as well as the nonperturbative many-body coupling function. As a side result, we obtain a possible finite-size interpretation of wave-turbulent scaling recently measured by Navon et al.
Abstract We review the recent progress in the understanding of the relaxation of isolated near-integrable quantum many-body systems. Focusing on prethermalization and universal dynamics following a quench, we describe the experiments with ultracold atomic gases that illustrate these phenomena and summarize the essential theoretical concepts employed to interpret them. Our discussion highlights the key topics that link the different approaches to this interdisciplinary field, including the generalized Gibbs ensemble, non-thermal fixed points, critical slowing and universal scaling. Finally, we point to new experimental challenges demonstrating these fundamental features of many-body quantum systems out of equilibrium.