The unprecedentedly bright optical afterglow of GRB 130606A located by Swift at a redshift close to the reionization era (z = 5.913) provides a new opportunity to probe the ionization status of intergalactic medium (IGM). Here we present an analysis of the red Ly alpha damping wing of the afterglow spectrum taken by Subaru/FOCAS during 10.4-13.2 hr after the burst. We find that the minimal model including only the baseline power-law and HI absorption in the host galaxy does not give a good fit, leaving residuals showing concave curvature in 8400-8900 A with an amplitude of about 0.6% of the flux. Such a curvature in the short wavelength range cannot be explained either by extinction at the host with standard extinction curves, intrinsic curvature of afterglow spectra, or by the known systematic uncertainties in the observed spectrum. The red damping wing by intervening HI gas outside the host can reduce the residual by about 3 sigma statistical significance. We find that a damped Ly alpha system is not favored as the origin of this intervening HI absorption, from the observed Ly beta and metal absorption features. Therefore absorption by diffuse IGM remains as a plausible explanation. A fit by a simple uniform IGM model requires HI neutral fraction of f_HI ~ 0.1-0.5 depending on the distance to the GRB host, implying high f_HI IGM associated with the observed dark Gunn-Peterson (GP) troughs. This gives a new evidence that the reionization is not yet complete at z ~ 6.
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