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

TL;DR

This paper investigates a three-dimensional loop model with crossings, confirming its description by $ ext{RP}^{n-1}$ sigma models, and provides new critical exponent estimates for the $n ightarrow1$ universality class using large-scale Monte Carlo simulations.

Contribution

It offers the first critical exponent estimates for the $n ightarrow1$ limit of $ ext{RP}^{n-1}$ sigma models through extensive Monte Carlo simulations.

Findings

Critical exponent $ u=0.918(5)$

Critical exponent $ ext{eta}=-0.091(9)$

Scaling dimension $x_4=1.292(8)$ for $n=1$

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 $1800\times1800\times1800$, to obtain the critical exponents for a universality class with no previously reported estimates in the literature, namely the replica-like limit $n\rightarrow1$ 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 $x_4=1.292(8)$.