3D Unoriented loop models and the ${\mathrm{RP}}^{n-1}$ sigma model

Abstract

We study a completely-packed loop model with crossings in a three-dimensional lattice and confirm it is described by ${\mathrm{RP}}^{n-1}$ sigma field theories. We use Monte Carlo simulations, with systems sizes up to $1800\times 1800\times 1800$, to obtain the critical exponents for a universality class with no previously reported estimates in the literature, namely the replica-like limit $n\to 1$ of ${\mathrm{RP}}^{n-1}$ sigma field theory. Estimates of critical exponents include $\nu =0.918(5)$ and $\eta =-0.091(9)$. We also study the scaling dimension of the 4-leg watermelon correlators, particularly for $n=1$ we obtain the value $x_4=1.292(8)$.