Loop models with crossings

doi:10.1103/PhysRevB.87.184204

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Loop models with crossings

2013-05-28

DOI：10.1103/PhysRevB.87.184204

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Abstract

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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 Z(2) lattice gauge theory, we show that the continuum description of the model is a replica limit of the sigma model on real projective space (RPn-1). This field theory sustains Z(2) 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 = 10(6) at loop fugacity n = 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 Theta point; we use the RG treatment of a perturbed 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.

Keywords:

KINETIC GROWTH SIMULATIONS

QUANTUM HALL TRANSITION

2 DIMENSIONS

SQUARE LATTICE

TRICRITICAL EXPONENTS

SPIN CHAINS

EPSILON-DIMENSIONS

MAGNETIC-FIELD

SIGMA-MODELS

LOCALIZATION