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arXiv:1503.02498 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Universal self-similar dynamics of relativistic and nonrelativistic field theories near nonthermal fixed points

Asier Piñeiro Orioli, Kirill Boguslavski, Jürgen Berges

Published Jul 27, 2015• 130 citations
arXivHTMLPDF

Abstract

We investigate universal behavior of isolated many-body systems far from equilibrium, which is relevant for a wide range of applications from ultracold quantum gases to high-energy particle physics. The universality is based on the existence of nonthermal fixed points, which represent nonequilibrium attractor solutions with self-similar scaling behavior. The corresponding dynamic universality classes turn out to be remarkably large, encompassing both relativistic as well as nonrelativistic quantum and classical systems. For the examples of nonrelativistic (Gross-Pitaevskii) and relativistic scalar field theory with quartic self-interactions, we demonstrate that infrared scaling exponents as well as scaling functions agree. We perform two independent nonperturbative calculations, first by using classical-statistical lattice simulation techniques and second by applying a vertex-resummed kinetic theory. The latter extends kinetic descriptions to the nonperturbative regime of overoccupied modes. Our results open new perspectives to learn from experiments with cold atoms aspects about the dynamics during the early stages of our universe.

DOI
10.1103/physrevd.92.025041

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We investigate universal behavior of isolated many-body systems far from equilibrium, which is relevant for a wide range of applications from ultracold quantum gases to high-energy particle physics. The universality is based on the existence of nonthermal fixed points, which represent nonequilibrium attractor solutions with self-similar scaling behavior. The corresponding dynamic universality classes turn out to be remarkably large, encompassing both relativistic as well as nonrelativistic quantum and classical systems. For the examples of nonrelativistic (Gross-Pitaevskii) and relativistic scalar field theory with quartic self-interactions, we demonstrate that infrared scaling exponents as well as scaling functions agree. We perform two independent nonperturbative calculations, first by using classical-statistical lattice simulation techniques and second by applying a vertex-resummed kinetic theory. The latter extends kinetic descriptions to the nonperturbative regime of overoccupied modes. Our results open new perspectives to learn from experiments with cold atoms aspects about the dynamics during the early stages of our universe.
Quantum many-body systems far from equilibrium can approach a nonthermal fixed point during their real-time evolution. One example is scalar field theory, which occurs in models \nof cosmological inflation, and similar examples are found for non-Abelian plasmas relevant for heavy-ion collisions and for ultracold Bose gases. Investigating nonthermal fixed points of different microscopic theories, we present two novel universality classes that provide links between these systems. \n \nOne of them involves nonrelativistic, N-component relativistic and expanding scalar systems. It occurs in the deep infrared regime of very high occupancies and is governed by a self-similar evolution. Its nonequilibrium dynamics leads to the formation of a Bose-Einstein condensate. The scaling properties of this region can be described by a vertex-resummed kinetic theory that is based on a systematic large-N expansion at next-to-leading order. \n \nThe other novel universality class encompasses scalar field theories and non-Abelian plasmas in a longitudinally expanding background and corresponds to an early dynamical stage of heavy-ion collisions in the high-energy limit. We show that these systems share the same self-similar scaling properties for a wide range of momenta in a limit where particles are weakly coupled but their occupancy is high. \n \nBoth universality classes are found in separate momentum regions in a longitudinally expanding N-component scalar field theory. We argue that the important role of the infrared dynamics ensures that key features of our results for scalar and gauge theories cannot be reproduced consistently in conventional kinetic theory frameworks. Moreover, the observed universality connects different physics disciplines from heavy-ion collisions to ultracold atoms, making a remarkable link between the world's hottest and coldest matter.
We compute universal scaling exponents and scaling functions for massless scalar field theories far from equilibrium. Using a large-NNN expansion to next-to-leading order, we investigate the nonperturbative infrared behaviour associated with the transport of a) energy or b) effective particle number towards low frequencies ωωω and momenta pˉ\bar ppˉ​. For massless dispersion ω=pˉω= \bar{p}ω=pˉ​, we determine the scaling form of the distribution function whose universal power-law tail fS(pˉ)∼1/pˉκf_S(\bar p) \sim 1/\bar{p}^κfS​(pˉ​)∼1/pˉ​κ is found to be a) κ=d+1κ= d+1κ=d+1 and b) κ=dκ= dκ=d in d=3d=3d=3 spatial dimensions. Our results establish that relativistic and nonrelativistic theories do not belong to the same universality class in this case, which opens up new applications to far-from-equilibrium phenomena in high-energy and (quantum) many-body physics with linear dispersions.
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.
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.
We study the dynamics of perturbations around nonthermal fixed points associated with universal scaling phenomena in quantum many-body systems far from equilibrium. For an N-component scalar quantum field theory in 3+1 space-time dimensions, we determine the stability scaling exponents using a self-consistent large-N expansion to next-to-leading order. Our analysis reveals the presence of both stable and unstable perturbations, the latter leading to quasiexponential deviations from the fixed point in the infrared. We identify a tower of far-from-equilibrium quasiparticle states and their dispersion relations by computing the spectral function. With the help of linear response theory, we demonstrate that unstable dynamics arises from a competition between elastic scattering processes among the quasiparticle states. What ultimately renders the fixed point dynamically attractive is the phenomenon of a "scaling instability," which is the universal scaling of the unstable regime toward the infrared due to a self-similar quasiparticle cascade. Our results provide ab initio understanding of emergent stability properties in self-organized scaling phenomena.
We map the infrared dynamics of a relativistic single-component (N=1N=1N=1) interacting scalar field theory to that of nonrelativistic complex scalar fields. The Gross-Pitaevskii (GP) equation, describing the real-time dynamics of single-component ultracold Bose gases, is obtained at first nontrivial order in an expansion proportional to the powers of $\ensuremath{\lambda}{\ensuremath{\phi}}^{2}/{m}^{2}$ where $\ensuremath{\lambda}$, $\ensuremath{\phi}$, and mmm are the coupling constant, the scalar field, and the particle mass respectively. Our analytical studies are corroborated by numerical simulations of the spatial and momentum structure of overoccupied scalar fields in (2+1)-dimensions. Universal scaling of infrared modes, vortex-antivortex superfluid dynamics, and the off-equilibrium formation of a Bose-Einstein condensate are observed. Our results for the universal scaling exponents are in agreement with those extracted in the numerical simulations of the GP equation. As in these simulations, we observe coarsening phase kinetics in the Bose superfluid with strongly anomalous scaling exponents relative to that of vertex resummed kinetic theory. Our relativistic field theory framework further allows one to study more closely the coupling between superfluid and normal fluid modes, specifically the turbulent momentum and spatial structure of the coupling between a quasiparticle cascade to the infrared and an energy cascade to the ultraviolet. We outline possible applications of the formalism to the dynamics of vortex-antivortex formation and to the off-equilibrium dynamics of the strongly interacting matter formed in heavy-ion collisions.