Anomalous scaling at non-thermal fixed points of the sine-Gordon model

TL;DR

This work extends the theory of non-thermal fixed points to the sine-Gordon model by deriving a non-perturbative wave-Boltzmann kinetic equation for the momentum distribution $f(t,\mathbf{p})$ using a 2PI effective action with $s$-channel resummation of the cosine potential. It reveals two scaling regimes in the infrared: an anomalous fixed point with $(\beta,\alpha,\kappa)=(1/(z+d),\;d/(z+d),\;2d+z)$ (with $z=2$), and a Gaussian fixed point with $(\beta_G,\alpha_G,\kappa_G)=(1/z,\;d/z,\;d+z/2)$, where large occupancies select high-order multi-particle scattering that is non-local in momentum but local in position space. The analysis shows that anomalous transport arises from many-quasiparticle interactions, leading to slow coarsening-like evolution, with numerical simulations in accompanying work supporting the predicted exponents (e.g., in $d=2$, $\beta=1/4$, $\alpha=1/2$, $\kappa=6$). The framework provides a principled way to classify coarsening dynamics for transcendental potentials and points toward possible RG formulations for non-thermal fixed points. Overall, the results illuminate how non-polynomial interactions shape universal far-from-equilibrium scaling and how non-local momentum transport coexists with spatially localized correlations.

Abstract

We extend the theory of non-thermal 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 non-perturbative two-particle irreducible effective action, which re-sums a series of closed loop chains akin to a large-$N$ 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 non-local 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.

Figures (2)

• Figure 1: (a) A single $m$-vertex at space-time point $x$ is connected, by $l_{1}$ of its legs, via full propagators $G(x,x_{1})$, to a neighboring vertex $\sim\varphi(x_{1})^{m_{1}}$, which we label by $1$, by $l_{2}$ legs to a different vertex $2$, and so on to $l_{L}$ legs linked to vertex $L$. In addition to this, $l_{0}=2l$ legs of each vertex are connected in a pair-wise manner to each other by local propagators $G(x,x)$. The multiplicities sum up to $\sum_{\nu=0}^{L}l_{\nu}=m$. (b) Re-summation of the diagrammatic series over powers $[G(x,x_{\nu})]^{l_{\nu}}$, within the otherwise identical 2PI diagrams. Such a re-summed link $\overline{G}_\mathrm{nl}(x,x_{\nu})$ is denoted by a thick line. The combinatorial factors indicate the non-local line (\ref{['paper:eq:evenoddResummedLinesSG']}), which consists of even and odd contributions of opposite sign. There are no vertex couplings included in this re-summation.
• Figure 2: Exemplary diagrams contributing to the $s$-channel bubble-re-summed two-particle irreducible effective action $\Gamma_{2}[G]$, Eqs. (\ref{['paper:eq:Gamma2engeq3']}) and (\ref{['paper:eq:Gamma2ongeq3']}), of the sine-Gordon model. Lines represent fully dressed Green's functions $\overline{G}(x,y)$, dots represent $m$-point vertices ${v}_{m}\varphi^{m}$, Eq. \ref{['paper:eq:Vexpansion']}. Diagrams with an even (odd) number of lines linking neighboring vertices contribute to $\Gamma_{2,\mathrm{e}}[G]$ ($\Gamma_{2,\mathrm{e}}[G]$), cf. Eqs. (\ref{['paper:eq:Gamma2engeq3']}) and (\ref{['paper:eq:Gamma2ongeq3']}), respectively.