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arXiv:2107.13366

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

Pablo Serna

Published Jul 28, 2021• 0 citations
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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 $ν=0.918(5)$ and $η=-0.091(9)$. We also study the scaling dimension of the 4-leg watermelon correlators, particularly for $n=1$ we obtain the value $x_4=1.292(8)$.

DOI
10.48550/arxiv.2107.13366

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3D Unoriented loop models and the RPn−1\mathrm{RP}^{n-1}RPn−1 sigma models
We study a completely-packed loop model with crossings in a three-dimensional lattice and confirm it is described by RPn−1\mathrm{RP}^{n-1}RPn−1 sigma field theories. We use Monte Carlo simulations, with systems sizes up to 1800×1800×18001800\times1800\times18001800×1800×1800, to obtain the critical exponents for a universality class with no previously reported estimates in the literature, namely the replica-like limit n→1n\rightarrow1n→1 of RPn−1\mathrm{RP}^{n-1}RPn−1 sigma field theory. Classic critical exponents include ν=0.918(5)\nu=0.918(5)ν=0.918(5) and η=−0.091(9)\eta=-0.091(9)η=−0.091(9). We also provide estimates of the scaling dimension of the 4-legs watermelon correlators, particularly for n=1n=1n=1 we obtain the value x4=1.292(8)x_4=1.292(8)x4​=1.292(8).
The universal behavior of two-dimensional loop models can change dramatically when loops are allowed to cross. We study models with crossings both analytically and with extensive Monte Carlo simulations. Our main focus (the ``completely packed loop model with crossings'') is a simple generalization of well-known models that shows an interesting phase diagram with continuous phase transitions of a new kind. These separate the unusual ``Goldstone'' phase observed previously from phases with short loops. Using mappings to Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theory, we show that the continuum description of the model is a replica limit of the $\ensuremath{\sigma}$ model on real projective space (${\mathbb{RP}}^{n\ensuremath{-}1}$). This field theory sustains Z2{\mathbb{Z}}_{2}Z2​ point defects, which proliferate at the transition. In addition to studying the new critical points, we characterize the universal properties of the Goldstone phase in detail, comparing renormalization group (RG) calculations with numerical data on systems of linear size up to L=106L={10}^{6}L=106 at loop fugacity n=1n=1n=1. (Very large sizes are necessary because of the logarithmic form of correlation functions and other observables.) The model is relevant to polymers on the verge of collapse, and a particular point in parameter space maps to self-avoiding trails at their $\ensuremath{\Theta}$ point; we use the RG treatment of a perturbed $\ensuremath{\sigma}$ model to resolve some perplexing features in the previous literature on trails. Finally, one of the phase transitions considered here is a close analog of those in disordered electronic systems---specifically, Anderson metal-insulator transitions---and provides a simpler context in which to study the properties of these poorly understood (central-charge-zero) critical points.
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We study a generalized clock model on the simple cubic lattice. The parameter of the model can be tuned such that the amplitude of the leading correction to scaling vanishes. In the main part of the study, we simulate the model with Z8{Z}_{8}Z8​ symmetry. At the transition, with increasing length scale, O(2)O(2)O(2) symmetry emerges. We perform Monte Carlo simulations using a hybrid of local Metropolis and cluster algorithms of lattices with a linear size up to L=512L=512L=512. The field variable requires less memory and the updates are faster than for a model with O(2)O(2)O(2) symmetry at the microscopic level. Our finite-size scaling analysis yields accurate estimates for the critical exponents of the three-dimensional XYXYXY-universality class. In particular, we get $\ensuremath{\eta}=0.03810(8),\ensuremath{\nu}=0.67169(7)$, and $\ensuremath{\omega}=0.789(4)$. Furthermore, we obtain estimates for fixed point values of phenomenological couplings and critical temperatures.
We perform high-statistics Monte Carlo simulations of three three-dimensional Ising spin glass models: the ± J Ising model for two values of the disorder parameter p , p = 1/2 and 0.7, and the bond-diluted ± J model for bond-occupation probability p b = 0.45. A finite-size scaling analysis of the quartic cumulants at the critical point shows conclusively that these models belong to the same universality class and allows us to estimate the scaling-correction exponent ω related to the leading irrelevant operator, ω = 1.0(1). We also determine the critical exponents ν and η. Taking into account the scaling corrections, we obtain ν = 2.53(8) and η = −0.384(9).