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arXiv:2607.26620

Critical non-thermal fixed point and the dynamical condensation phase transition

Anne-Solène Bornens, Elisabeth Gliott, Nicolas Cherroret

cond-mat.quant-gasquant-phSubmitted Jul 29, 2026• Announced Jul 30, 2026• 0 citations
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Abstract

Using a non-perturbative quantum kinetic framework, we develop a unified description of the far-from-equilibrium dynamics of three-dimensional Bose gases following cooling quenches across the Bose-Einstein condensation transition. By tracking the spatio-temporal evolution of the momentum distribution, we show that the equilibrium condensation threshold simultaneously acts as a dynamical critical point, separating distinct far-from-equilibrium universality classes governed by different non-equilibrium attractors. While quenches above the transition exhibit a single-timescale relaxation toward a thermal fixed point, quenches below the transition display a crossover from a transient weak-turbulence regime to a coarsening fixed point governed by the diffusive recombination of vortex lines. Quenches directly to the condensation threshold, finally, are controlled by a previously unidentified critical fixed point characterized by the superdiffusive spreading of critical fluctuations and a distinct set of dynamical exponents. Together, these dynamical scaling laws establish a far-from-equilibrium counterpart of the condensation phase transition, in which the equilibrium critical point also organizes the long-time non-equilibrium dynamics.

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Using a non-perturbative quantum kinetic framework, we develop a unified description of the far-from-equilibrium dynamics of three-dimensional Bose gases following cooling quenches across the Bose-Einstein condensation transition. By tracking the spatio-temporal evolution of the momentum distribution, we show that the equilibrium condensation threshold simultaneously acts as a dynamical critical point, separating distinct far-from-equilibrium universality classes governed by different non-equilibrium attractors. While quenches above the transition exhibit a single-timescale relaxation toward a thermal fixed point, quenches below the transition display a crossover from a transient weak-turbulence regime to a coarsening fixed point governed by the diffusive recombination of vortex lines. Quenches directly to the condensation threshold, finally, are controlled by a previously unidentified critical fixed point characterized by the superdiffusive spreading of critical fluctuations and a distinct set of dynamical exponents. Together, these dynamical scaling laws establish a far-from-equilibrium counterpart of the condensation phase transition, in which the equilibrium critical point also organizes the long-time non-equilibrium dynamics.
Abstract Using a non-perturbative quantum kinetic framework, we develop a unified description of the far-from-equilibrium dynamics of three-dimensional Bose gases following cooling quenches across the Bose-Einstein condensation transition. By tracking the spatio-temporal evolution of the momentum distribution, we show that the equilibrium condensation threshold simultaneously acts as a dynamical critical point, separating distinct far-from-equilibrium universality classes governed by different non-equilibrium attractors. While quenches above the transition exhibit a single-timescale relaxation toward a thermal fixed point, quenches below the transition display a crossover from a transient weak-turbulence regime to a coarsening fixed point governed by the diffusive recombination of vortex lines. Quenches directly to the condensation threshold, finally, are controlled by a previously unidentified critical fixed point characterized by the superdiffusive spreading of critical fluctuations and a distinct set of dynamical exponents. Together, these dynamical scaling laws establish a far-from-equilibrium counterpart of the condensation phase transition, in which the equilibrium critical point also organizes the long-time non-equilibrium dynamics.
Critical dynamics of an ultracold Bose gas far from equilibrium is studied in two spatial dimensions. Superfluid turbulence is created by quenching the equilibrium state close to zero temperature. Instead of immediately rethermalizing, the system approaches a meta-stable transient state, characterized as a nonthermal fixed point. A focus is set on the vortex density and vortex-antivortex correlations which characterize the evolution towards the nonthermal fixed point and the departure to final (quasi-)condensation. Two distinct power-law regimes in the vortex-density decay are found and discussed in terms of a vortex unbinding process and a kinetic description of vortex scattering. A possible relation to decaying turbulence in classical fluids is pointed out. By comparing the results to equilibrium studies of a two-dimensional Bose gas, an intuitive understanding of the location of the nonthermal fixed point in a reduced phase space is developed.
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.
Spatio-temporal scaling dynamics connected to non-thermal fixed points has been suggested as a universal framework to describe the relaxation of isolated far-from-equilibrium systems. Experimental studies in weakly-interacting cold atom systems have found scaling dynamics connected to specific attractors. In our experiments, we study a quantum gas of strongly interacting 6^66Li2_22​ Feshbach molecules, brought far out of equilibrium by imprinting a white-noise phase profile. The observed relaxation follows the same universal dynamics as for the previously observed formation of the order parameter in a shock-cooled gas of weakly interacting 87^{87}87Rb atoms. Our results point to a single universal fixed point with a large basin of attraction governing the relaxation of quasi-1D bosonic systems, independent of their specific initial conditions and microscopic details.
Universal scaling behavior in the relaxation dynamics of an isolated two-dimensional Bose gas is studied by means of semi-classical stochastic simulations of the Gross–Pitaevskii model. The system is quenched far out of equilibrium by imprinting vortex defects into an otherwise phase-coherent condensate. A strongly anomalous non-thermal fixed point is identified, associated with a slowed decay of the defects in the case that the dissipative coupling to the thermal background noise is suppressed. At this fixed point, a large anomalous exponent and, related to this, a large dynamical exponent are identified. The corresponding power-law decay is found to be consistent with three-vortex-collision induced loss. The article discusses these aspects of non-thermal fixed points in the context of phase-ordering kinetics and coarsening dynamics, thus relating phenomenological and analytical approaches to classifying far-from-equilibrium scaling dynamics with each other. In particular, a close connection between the anomalous scaling exponent η , introduced in a quantum-field theoretic approach, and conservation-law induced scaling in classical phase-ordering kinetics is revealed. Moreover, the relation to superfluid turbulence as well as to driven stationary systems is discussed.
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.