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

A system of $2\times d$ hard rectangles on square lattice is known to showfour different phases for $d \geq 14$. As the covered area fraction $\rho$ isincreased from $0$ to $1$, the system goes from low-density disordered phase,to orientationally-ordered nematic phase, to a columnar phase withorientational order and also broken translational invariance, to a high densityphase in which orientational order is lost. For large d, the threshold densityfor the first transition $\rho_1^*$ tends to $0$, and the critical density forthe third transition $\rho_3^*$ tends to $1$. Interestingly, simulations haveshown that the critical density for the second transition $\rho_2^*$ tends to anon-trivial finite value $\approx 0.73$, as $d \rightarrow \infty$, and$\rho_2^* \approx 0.93$ for $d=2$. We provide a theoretical explanation of thisinteresting result. We develop an approximation scheme to calculate the surfacetension between two differently ordered columnar phases. The density at whichthe surface tension vanishes gives an estimate $\rho_2^* = 0.746$, for $d\to\infty$, and $\rho_2^*=0.923$ for $d=2$. For all values of $d$, these estimatesare in good agreement with Monte Carlo data.

Stability of columnar order in assemblies of hard rectangles or squares