English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We consider the statistical mechanics of a class of models involvingclose-packed loops with fugacity $n$ on three-dimensional lattices. The modelsexhibit phases of two types as a coupling constant is varied: in one, all loopsare finite, and in the other, some loops are infinitely extended. We show thatthe loop models are discretisations of $CP^{n-1}$ $\sigma$ models. The finiteand infinite loop phases represent, respectively, disordered and ordered phasesof the $\sigma$ model, and we discuss the relationship between loop propertiesand $\sigma$ model correlators. On large scales, loops are Brownian in anordered phase and have a non-trivial fractal dimension at a critical point. Wesimulate the models, finding continuous transitions between the two phases for$n=1,2,3$ and first order transitions for $n\geq 4$. We also give arenormalisation group treatment of the $CP^{n-1}$ model that shows how acontinuous transition can survive for values of $n$ larger than (but close to)two, despite the presence of a cubic invariant in the Landau-Ginzburgdescription. The results we obtain are of broader relevance to a variety ofproblems, including SU(n) quantum magnets in (2+1) dimensions, Andersonlocalisation in symmetry class C, and the statistics of random curves in threedimensions.

Phase transitions in 3D loop models and the $CP^{n-1}$ $\sigma$ model