Spectral origin of conformal invariance in active nematic turbulence

TL;DR

This paper explains why zero-vorticity contours in active nematic turbulence exhibit conformal invariance and belong to the critical percolation universality class, based on spectral properties and long-range correlations.

Contribution

It provides a spectral explanation for the conformal invariance in active nematic turbulence, linking energy spectrum and correlation decay to universality class.

Findings

Zero-vorticity contours obey SLE_6 with diffusivity κ ≈ 6.

Energy spectrum E(q) ∼ q^{-1} implies correlation decay exponent a = 3/2.

Gaussian surrogate fields confirm the spectral and conformal invariance results.

Abstract

Zero-vorticity contours in the collective flows of living cells obey Schramm-Loewner evolution with diffusivity $\kappa = 6$ and thus fall in the universality class of critical percolation. This observation is surprising because the underlying vorticity field has long-range correlations that, according to the Weinrib-Halperin criterion, should alter the universality class. Here we propose a spectral explanation for this apparent paradox in two-dimensional active nematic turbulence. The universal energy spectrum $E(q) \sim q^{-1}$ implies sign-field correlations whose decay exponent $a = 3/2$ matches the Weinrib-Halperin marginal threshold $2/\nu_0 = 3/2$ for two-dimensional percolation. At this marginal point the long-range correlations are irrelevant under renormalization, so the system flows to the uncorrelated percolation fixed point. Gaussian surrogate fields with the same spectrum confirm $a = 3/2$ to three significant figures, and left-passage analysis of their zero-vorticity interfaces yields $\kappa = 5.98 \pm 0.08$, consistent with SLE_6.