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arXiv:1801.09490 · Physical Review A

Kinetic theory of nonthermal fixed points in a Bose gas

Isara Chantesana, Asier Piñeiro Orioli, Thomas Gasenzer

Published Apr 19, 2019• 73 citations
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Abstract

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.

DOI
10.1103/physreva.99.043620

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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 \ensuremathη\ensuremath{\eta}\ensuremathη. The approach of the nonthermal fixed point is, in particular, found to involve a rescaling of momenta in time ttt by t\ensuremathβ{t}^{\phantom{\rule{0.16em}{0ex}}\ensuremath{\beta}}t\ensuremathβ, with \ensuremathβ=1/z\ensuremath{\beta}=1/z\ensuremathβ=1/z, within our kinetic approach independent of \ensuremathη\ensuremath{\eta}\ensuremathη. 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.
Universal dynamics of a dilute Bose gas is studied within the kinetic regime where the time evolution of the mode occupation number is governed by a wave-Boltzmann equation. The universality manifests itself in the form of the dynamical evolution which can be a cascade in the context of Kolmogorov-Zakharov wave turbulence or a self-similar shift in time and space. Which is the case depends on the relevant global conservation laws and on the particular range of scales in which the respective dynamics takes place. A nonperturbative kinetic equation is derived by applying the Schwinger-Keldysh closed-time functional integral and the 2-particle-irreducible formalism to an action of complex scalar Bose fields with quartic interaction. The resulting dynamic equation for two-point correlations are reduced to a wave-Boltzmann-type kinetic equation with an effective TTT-matrix which depends explicitly on the mode occupation numbers. This nonperturbative dependence on the solution occurs in the infrared regime where wave numbers are large and collective scattering of many particles prevails. Thus, the scaling analysis of wave turbulence theory can be applied. We explicitly calculate the effective TTT-matrix analytically taking into account an infrared cutoff for dealing with the infrared divergences. Our results show that the scaling behaviour of the TTT-matrix departs from the one predicted by naive dimensional counting due to the presence of the infrared cutoff. The Kolmogorov-Zakharov and the self-similar exponents are evaluated by power counting using our TTT-matrix and the results are confirmed by numerical integration of the wave-Boltzmann scattering integral. The scaling exponents governing the time evolution are determined by means of the global conservation laws and the kinetic equation. Depending on the scaling properties of the quasiparticle spectrum, the momentum cutoff scale in the infrared evolves critically slowed in time. The respective scaling exponent is universal in the nonperturbative regime regardless of whether the process is an inverse cascade or a self-similar shift towards the infrared. Thus, our results provide a general framework for classifying nonthermal fixed points in dilute ultracold Bose gases. They pave the way to a straightforward generalisation and application to trapped systems in different dimensionalities and to systems with more than one internal degree of freedom. Our results furthermore provide a possible interpretation of recent experimental results on wave turbulence in an ultracold Bose gas. %We studied the universal dynamics of diluted Bose gas within the kinetic regime where time evolution of occupation number is governed by the Boltzmann's scattering integral. %There are three possible scenarios, depending on the scaling behaviour of occupation number. They are the particle and energy cascade which are the steady state of turbulence transport and self-similar evolution where local occupation evolves in time. %We evaluated the scaling exponents in both momentum and time scales in each case under the constraints of global number and energy conservation. Furthermore, we can go beyond the perturbative regime by employing the closed-time-path contour and the 2-particle-irreducible formalism to the Gross-Pitaevskii action. The kinetic equation is modified by an effective T-matrix as a result of the resummation scheme. However, the structure of kinetic equation remains the same as in the perturbative case. Thus, all exponents in the nonperturbative regime depart from those in the perturbative regime because of the scaling of the effective T-matrix.
Nonthermal fixed points in the evolution of a quantum many-body system quenched far out of equilibrium manifest themselves in a scaling evolution of correlations in space and time. We develop a low-energy effective theory of nonthermal fixed points in a bosonic quantum many-body system by integrating out long-wavelength density fluctuations. The system consists of NNN distinguishable spatially uniform Bose gases with U(N)\text{U}(N)U(N)-symmetric interactions. The effective theory describes interacting Goldstone modes of the total and relative-phase excitations. It is similar in character to the nonlinear Luttinger-liquid description of low-energy phonons in a single dilute Bose gas, with the markable difference of a universal nonlocal coupling function depending, in the large-NNN limit, only on momentum, single-particle mass, and density of the gas. Our theory provides a perturbative description of the nonthermal fixed point, technically easy to apply to experimentally relevant cases with a small number of fields NNN. Numerical results for N=3N=3N=3 allow us to characterize the analytical form of the scaling function and confirm the analytically predicted scaling exponents. The predicted and observed exponentially suppressed coherence at short distances takes the form of that of a quasicondensate in low-dimensional equilibrium systems. The fixed point which is dominated by the relative phases is found to be Gaussian, while a non-Gaussian fixed point is anticipated to require scaling evolution with a distinctly lower power of time.
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.
It is demonstrated how a many-body system far from thermal equilibrium can exhibit universal dynamics in passing a nonthermal fixed point. As an example, the process of Bose–Einstein (BE) condensation of a dilute cold gas is considered. If the particle flux into the low-energy modes, induced, for example by a cooling quench, is sufficiently strong, the Bose gas develops a characteristic power-law single-particle spectrum , and critical slowing down in time occurs. The fixed point is shown to be marked by the creation and dilution of tangled vortex lines. Alternatively, for a weak cooling quench and particle flux, the condensation process runs quasi-adiabatically, passing by the fixed point in the far distance, and the signatures of critical scaling remain absent.
We extend the theory of nonthermal fixed points to the case of anomalously slow universal scaling dynamics according to the sine-Gordon model. This entails the derivation of a kinetic equation for the momentum occupancy of the scalar field from a nonperturbative two-particle irreducible effective action, which resums a series of closed loop chains akin to a large-NNN expansion at next-to-leading order. The resulting kinetic equation is analyzed for possible scaling solutions in space and time that are characterized by a set of universal scaling exponents and encode self-similar transport to low momenta. Assuming the momentum occupancy distribution to exhibit a scaling form we can determine the exponents by identifying the dominating contributions to the scattering integral and power counting. If the field exhibits strong variations across many wells of the cosine potential, the scattering integral is dominated by the scattering of many quasiparticles such that the momentum of each single participating mode is only weakly constrained. Remarkably, in this case, in contrast to wave turbulent cascades, which correspond to local transport in momentum space, our results suggest that kinetic scattering here is dominated by rather nonlocal processes corresponding to a spatial containment in position space. The corresponding universal correlation functions in momentum and position space corroborate this conclusion. Numerical simulations performed in accompanying work yield scaling properties close to the ones predicted here.
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.