Nonthermal Fixed-Point Realizations in Quantum Matter

A nonthermal fixed point is inferred when far-from-equilibrium dynamics becomes self-similar over a growing range of scales and different preparations approach the same scaling function and exponent relations. A transient power law at one time is not enough: the claim requires temporal collapse, conservation-compatible transport, robustness to initial conditions, and controlled finite-time boundaries.

Required background. Quenches and relaxation fixes the preparation and limiting procedures; nonthermal fixed points and wave turbulence develops the general continuum formalism. Helpful background. Phase–density hydrodynamics supplies the infrared variables for dilute-gas realizations.

Self-similar evolution

For an isotropic occupation distribution in $d$ dimensions, a common scaling ansatz is

$$n(k,t)=\left(\frac{t}{t_r}\right)^\alpha f\!\left[\left(\frac{t}{t_r}\right)^\beta\frac{k}{k_r}\right],$$

within a stated momentum interval $k_{\min}(t)<k<k_{\max}(t)$. Here $t_r$ and $k_r$ are fixed reference scales, so the argument of the dimensionless shape $f$ is dimensionless. Positive $\beta$ shifts structure toward smaller momenta and gives $\ell(t)/\ell_r\sim(t/t_r)^\beta$, with $\ell_r=k_r^{-1}$. The reference scales are conventional; $\alpha$, $\beta$, and the shape $f$ carry the scaling claim.

Suppose the particle content of that scaling interval is conserved and flux through its boundaries is negligible. Changing variables $q=(t/t_r)^\beta k/k_r$ gives

$$N_{\mathrm{sc}}(t) \propto \int_{\mathrm{sc}}\!\mathrm d^dk\,n(k,t) \propto \left(\frac{t}{t_r}\right)^{\alpha-d\beta},$$

so particle transport requires $\alpha=d\beta$. If instead an energy integral with dispersion $\omega_k\sim k^z$ is conserved in the relevant range, then $\alpha=(d+z)\beta$. One scaling range cannot generally conserve both with nonzero $\beta$; direct and inverse cascades can occupy different momentum intervals and carry different invariants.

The nonthermal-fixed-point interpretation of such self-similar transport was formulated for strongly occupied quantum fields by Berges, Rothkopf, and Schmidt 2008.

The conservation relation is an independent check, not a fit constraint to be imposed silently. Loss, a trap, and a moving boundary of the analysis window contribute surface or source terms. If $N_{\mathrm{sc}}$ drifts, the exponent relation must be modified or the fixed-point interpretation weakened.

From images to a collapse test

A reproducible analysis begins with calibrated $n(k,t)$ rather than normalized curves whose changing total weight has been hidden. For trial exponents, rescale

$$\widetilde n(q;t)= \left(\frac{t}{t_r}\right)^{-\alpha} n\!\left(k_r\left(\frac{t}{t_r}\right)^{-\beta}q,t\right),$$

and minimize deviations among times over a common physical momentum interval. The fit must propagate covariance between $\alpha$ and $\beta$, resolution convolution, atom-number uncertainty, and correlations between bins derived from the same image.

Three tests make the result materially stronger:

1. fit exponents on part of the time series and predict held-out times;
2. vary the initial state while keeping conserved density and low-energy theory fixed; and
3. verify the transported invariant directly, including flux through the fitted window.

The 2018 spinor-gas and one-dimensional Bose-gas experiments observed temporal scaling collapses, approximate invariant transport, and robustness across preparations in distinct platforms Prüfer et al. 2018, Erne et al. 2018. They established finite-window realizations of nonthermal universality, not proof that the measured gases remain at an asymptotic fixed point for arbitrarily long times. The later evolution in the one-dimensional experiment proceeded toward a thermal quasicondensate. A 2024 driven two-dimensional gas instead resolved an inverse turbulent-wave cascade with both a weak-wave-turbulent range and a lower-momentum spectrum reminiscent of coarsening Karailiev et al. 2024; because continuous driving maintains a steady flux, it should not be conflated with an isolated post-quench fixed-point trajectory.

Coarsening, turbulence, and alternatives

Self-similar coarsening and wave turbulence can produce similar plots but different physics. Coarsening ties $\ell(t)$ to growing domains or defect separation. Weak wave turbulence ties a stationary or slowly changing spectrum to a flux through momentum space. A nonthermal fixed point describes attraction in the space of distributions under time evolution. These descriptions can overlap in a regime, yet none follows from a power-law $n(k)\propto k^{-\kappa}$ alone.

Plausible alternatives include ordinary phase ordering, ballistic redistribution from the initial state, a finite-time bottleneck, imaging saturation at low $k$, and loss-driven normalization drift. Useful negative controls scramble the initial phases, change the interaction or density while holding the imaging pipeline fixed, vary the fitted momentum bounds, and compare with synthetic spectra passed through the measured point-spread function.

Finite volume imposes $k_{\min}\sim2\pi/L$; once the moving infrared scale reaches it, condensation or saturation replaces the scaling flow. The ultraviolet cutoff must remain below the lattice, healing, or range scale at which the assumed continuum dispersion and interaction cease to apply. A reported exponent without both boundaries is not portable between platforms.

Evidence status

The experimental and theoretical evidence summarized here was checked through 10 August 2026. The robust conclusion is that several quantum-gas platforms exhibit controlled, finite-window self-similar dynamics consistent with nonthermal-fixed-point descriptions. Universality classes, asymptotic basins of attraction, and the relation among coarsening, turbulence, and driven steady cascades remain active research questions. Dated updates and contrary evidence belong in the Quantum Matter and Emergence Research dossier.

Exercises

1. Conserved moment. For $I_s=\int\mathrm d^dk\,k^s n(k,t)$, derive the exponent relation required for $I_s$ to be time independent inside the scaling window.

Solution

With $q=t^\beta k$, the measure contributes $t^{-d\beta}$, $k^s$ contributes $t^{-s\beta}$, and $n$ contributes $t^\alpha$. Therefore $I_s\propto t^{\alpha-(d+s)\beta}$ and conservation requires $\alpha=(d+s)\beta$. Particle number has $s=0$; for a dispersion proportional to $k^z$, energy has $s=z$.

2. Resolution false positive. Suppose the true scaling length grows until $1/\ell(t)$ falls below a fixed momentum resolution $\Delta k$. What trend should appear in a fitted $\beta$ if unresolved late times are included?

Solution

Once the feature is narrower than $\Delta k$, the measured distribution stops shifting appreciably even though the underlying length may continue to grow. A single fit then tends to reduce the inferred $\beta$ and can produce an apparent late-time saturation. Convolving candidate scaling functions with the measured resolution before fitting, or excluding times without resolved motion, exposes the effect.

References

• Berges, Jürgen, Alexander Rothkopf, and Jonas Schmidt. “Nonthermal Fixed Points: Effective Weak Coupling for Strongly Correlated Systems Far from Equilibrium.” Physical Review Letters 101, 041603 (2008). DOI.
• Erne, Sebastian, Robert Bücker, Thomas Gasenzer, Jürgen Berges, and Jörg Schmiedmayer. “Universal Dynamics in an Isolated One-Dimensional Bose Gas Far from Equilibrium.” Nature 563, 225–229 (2018). DOI.
• Karailiev, Andrey, Martin Gazo, Maciej Gałka, Christoph Eigen, Tanish Satoor, and Zoran Hadzibabic. “Observation of an Inverse Turbulent-Wave Cascade in a Driven Quantum Gas.” Physical Review Letters 133, 243402 (2024). DOI.
• Prüfer, Maximilian, Philipp Kunkel, Helmut Strobel, Stefan Lannig, Daniel Linnemann, Christian-Marcel Schmied, Jürgen Berges, Thomas Gasenzer, and Markus K. Oberthaler. “Observation of Universal Dynamics in a Spinor Bose Gas Far from Equilibrium.” Nature 563, 217–220 (2018). DOI.