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arXiv:1104.4096 · Physical Review Letters

3D Loop Models and the CPn−1 Sigma Model

Adam Nahum, J. T. Chalker, Pablo Serna, M. Ortuño, A. M. Somoza

Published Sep 8, 2011• 70 citations
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Abstract

Many statistical mechanics problems can be framed in terms of random curves; we consider a class of three-dimensional loop models that are prototypes for such ensembles. The models show transitions between phases with infinite loops and short-loop phases. We map them to CP(n-1) sigma models, where n is the loop fugacity. Using Monte Carlo simulations, we find continuous transitions for n=1, 2, 3, and first order transitions for n≥5. The results are relevant to line defects in random media, as well as to Anderson localization and (2+1)-dimensional quantum magnets.

DOI
10.1103/physrevlett.107.110601

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Many statistical mechanics problems can be framed in terms of random curves; we consider a class of three-dimensional loop models that are prototypes for such ensembles. The models show transitions between phases with infinite loops and short-loop phases. We map them to CP(n-1) sigma models, where n is the loop fugacity. Using Monte Carlo simulations, we find continuous transitions for n=1, 2, 3, and first order transitions for n≥5. The results are relevant to line defects in random media, as well as to Anderson localization and (2+1)-dimensional quantum magnets.
We consider the statistical mechanics of a class of models involving close-packed loops with fugacity nnn on three-dimensional lattices. The models exhibit phases of two types as a coupling constant is varied: in one, all loops are finite, and in the other, some loops are infinitely extended. We show that the loop models are discretizations of $C{P}^{n\ensuremath{-}1}$ $\ensuremath{\sigma}$ models. The finite and infinite loop phases represent, respectively, disordered and ordered phases of the $\ensuremath{\sigma}$ model, and we discuss the relationship between loop properties and $\ensuremath{\sigma}$ model correlators. On large scales, loops are Brownian in an ordered phase and have a nontrivial fractal dimension at a critical point. We simulate the models, finding continuous transitions between the two phases for n=1,2,3n=1,2,3n=1,2,3 and first order transitions for $n\ensuremath{\ge}4$. We also give a renormalization-group treatment of the $C{P}^{n\ensuremath{-}1}$ model that shows how a continuous transition can survive for values of nnn larger than (but close to) 2, despite the presence of a cubic invariant in the Landau-Ginzburg description. The results we obtain are of broader relevance to a variety of problems, including SU(n)\mathrm{SU}(n)SU(n) quantum magnets in (2+1) dimensions, Anderson localization in symmetry class CCC, and the statistics of random curves in three dimensions.
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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The loop model is an important model of statistical mechanics and has been extensively studied in two-dimensional lattices. However, it is still difficult to simulate the loop model directly in three-dimensional lattices, especially in lattices with coordination numbers larger than 3. In this paper, a cluster weight Ising model is proposed by introducing an additional cluster weight n in the partition function of the traditional Ising model. This model is equivalent to the loop model on the two-dimensional lattice, but on the three-dimensional lattice, it is still not very clear whether or not these models have the same universality. By using a Monte Carlo method with cluster updates and color assignment, we obtain the global phase diagram containing the paramagnetic and ferromagnetic phases. The phase transition between the two phases is second order at 1≤n<n_{cri} and first order at n≥n_{cri}, where n_{cri}≈2. The thermal exponent y_{t} is equal to the system dimension d when the first-order transition occurs. For the second-order transitions, the numerical estimation of y_{t} and the magnetic exponent y_{m}, shows that the universalities of the two models on the three-dimensional lattice are different. Our results are helpful in the understanding of some traditional statistical mechanics models.
3D Unoriented loop models and the RPn−1\mathrm{RP}^{n-1}RPn−1 sigma model
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. Estimates of critical exponents include ν=0.918(5)ν=0.918(5)ν=0.918(5) and η=−0.091(9)η=-0.091(9)η=−0.091(9). We also study the scaling dimension of the 4-leg 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).
We study a lattice model of interacting loops in three dimensions with a 1/r21/{r}^{2}1/r2 interaction. Using Monte Carlo methods, we have found that the phase diagram contains a line of second-order phase transitions between a phase where the loops are gapped and a phase where they proliferate. The correlation length exponent and critical conductivity vary continuously along this line. Our model is exactly self-dual at a special point on the critical line, which allows us to calculate the critical conductivity exactly at this point.