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Oxford University Press eBooks

Nonthermal fixed points: universality, topology, and turbulence in Bose gases

Boris Nowak, Sebastian Erne, Markus Karl, Jan Schole, Dénes Sexty, Thomas Gasenzer

Published Jun 23, 2016• 4 citations
Publisher

Abstract

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.

DOI
10.1093/acprof:oso/9780198768166.003.0007

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•Jan 1, 2026•0 citations
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.
Quasistationary far-from-equilibrium critical states of a two-component Bose gas are studied in two spatial dimensions. After the system has undergone an initial dynamical instability it approaches a nonthermal fixed point. At this critical point the structure of the gas is characterized by ensembles of (quasi)topological defects such as vortices, skyrmions, and solitons which give rise to universal power-law behavior of momentum correlation functions. The resulting power-law spectra can be interpreted in terms of strong-wave-turbulence cascades driven by particle transport into long-wavelength excitations. Scaling exponents are determined on both sides of the miscible-immiscible transition controlled by the ratio of the intraspecies to interspecies couplings. Making use of quantum turbulence methods, we explain the specific values of the exponents from the presence of transient (quasi)topological defects.
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 zzz 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 ttt 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.
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 Understanding the relaxation dynamics of isolated quantum systems remains a central challenge in nonequilibrium physics. In this short review, we discuss recent experimental and theoretical advances in universal dynamics of turbulent Bose gases, with particular emphasis on nonthermal fixed points and wave turbulence in our experimental system at the São Carlos Institute of Physics of the University of São Paulo. We highlight the emergence of self-similar scaling behavior and its connection to particle and energy transport across momentum scales. In addition, we discuss an approach in which a differential equation has a universal scaling solution, providing a practical framework for extracting universal exponents from limited regions of the momentum distribution. Furthermore, we recap experimental observations of direct and inverse cascades in a three–dimensional trapped Bose–Einstein condensate, revealing distinct relaxation stages governed by universal scaling laws. These results demonstrate that turbulence plays a key role in far–from–equilibrium dynamics, offering a unified perspective on transport processes and thermalization in quantum many–body systems.
We study the emergence of universal scaling in the time-evolving momentum distribution of a harmonically trapped three-dimensional Bose-Einstein condensate, parametrically driven to a turbulent state. We demonstrate that the out-of-equilibrium dynamics post excitation is described by a single function due to nearby nonthermal fixed points. The observed behavior connects the dynamics of a quantum turbulent state to several far-from-equilibrium phenomena. We present a controllable protocol to explore universality in such systems, obtaining the associated scaling exponents. Our experimental results thus offer a promising route to investigate the complex dynamics of the quantum turbulent regime.
We study the emergence of distinctly subdiffusive scaling in the vicinity of an anomalous nonthermal fixed point, which reflects spatio-temporally self-similar scaling dynamics, in a quasi-two-dimensional Bose gas in the superfluid phase. In this regime, the far-from-equilibrium dynamics is governed by ensembles of quantum vortices. Mutual annihilation of vortices and antivortices drives coarsening, leading to the observed scaling behavior and an algebraically growing mean inter-defect distance over time. Introducing dipolar interactions provides additional control over the anisotropy and long-range character of the interactions. This control, combined with various parameter regimes and initial conditions, allows us to explore the universality of the subdiffusive scaling near the anomalous nonthermal fixed point. Scaling exponents are extracted from occupation-number spectra, the temporal evolution of inter-defect distances, and the decay of the total vortex number. At late times, compressible excitations of the background condensate grow due to ongoing vortex-antivortex annihilation, resulting in a crossover toward diffusion-type scaling near the Gaussian nonthermal fixed point. The observed temporal scaling depends only weakly on the strength and anisotropy of the dipolar interactions, demonstrating the universality of both anomalous and diffusion-type nonthermal fixed points. Nevertheless, the dipolar case exhibits qualitatively distinct vortex-clustering behavior. In contrast to non-dipolar gases, roton excitations in anisotropic, dipolar interactions appear to suppress same-sign vortex clustering, favoring the formation of vortex dipoles and maximizing the separation between vortices of equal sign. For pure contact interaction, we additionally observe spatial scaling characteristics consistent with Kraichnan--Kolmogorov turbulence. During a universal interval, in which the characteristic length scale associated with the inter-defect distance exhibits subdiffusive power-law growth, the moments of the superfluid velocity circulation around a square contour scale as predicted from classical turbulence theory. Higher-order moments of the velocity circulation reveal intermittent deviations, which are found to be consistent with values measured in fully developed classical turbulence. Notably, the subdiffusive coarsening clearly deviates from values known in classical systems. Collectively, these results provide strong evidence for the universality of the subdiffusive scaling near the anomalous nonthermal fixed point and establish a direct connection to decaying quantum turbulence cascades, highlighting the relevance of nonthermal fixed points in turbulence theory.