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

We compute the absolute Poisson’s ratio $\nu$ and the bending rigidity exponent $\eta$ 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 $d_c$: $\nu =-1+2\mathrm{/}{d}_c-a\mathrm{/}{d}_c^2+\dots$ where $a\approx 1.76\pm 0.02$. Also, we find the following expression for the exponent of the bending rigidity: $\eta =2\mathrm{/}{d}_c+(73-68\zeta (3))\mathrm{/}(27d_c^2)+\dots$. These results cannot be captured by self-consistent screening approximation.