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Absolute Poisson's ratio and the bending rigidity exponent of a crystalline two-dimensional membrane

We compute the absolute Poisson's ratio $ν$ and the bending rigidity exponent $η$ of a free-standing two-dimensional crystalline membrane embedded into a space of large dimensionality $d = 2 + d_c$, $d_c \gg 1$. We demonstrate that, in the regime of anomalous Hooke's law, the absolute Poisson's ratio approaches material independent value determined solely by the spatial dimensionality $d_c$: $ν= -1 +2/d_c-a/d_c^2+\dots$ where $a\approx 1.76\pm 0.02$. Also, we find the following expression for the exponent of the bending rigidity: $η= 2/d_c+(73-68ζ(3))/(27 d_c^2)+\dots$. These results cannot be captured by self-consistent screening approximation.