Finite-size localization scenarios in condensation transitions

Abstract

We consider the phenomenon of condensation of a globally conserved quantity $H={\ensuremath{\sum}}_{i=1}^{N}{\ensuremath{\epsilon}}_{i}$ distributed on $N$ sites, occurring when the density $h=H/N$ exceeds a critical density ${h}_{c}$. We numerically study the dependence of the participation ratio ${Y}_{2}=\ensuremath{\langle}{\ensuremath{\epsilon}}_{i}^{2}\ensuremath{\rangle}/(N{h}^{2})$ on the size $N$ of the system and on the control parameter $\ensuremath{\delta}=(h\ensuremath{-}{h}_{c})$, for various models: (i) a model with two conservation laws, derived from the discrete nonlinear Schr\"odinger equation; (ii) the continuous version of the zero-range process class, for different forms of the function $f(\ensuremath{\epsilon})$ defining the factorized steady state. Our results show that various localization scenarios may appear for finite $N$ and close to the transition point. These scenarios are characterized by the presence or the absence of a minimum of ${Y}_{2}$ when plotted against $N$ and by an exponent $\ensuremath{\gamma}\ensuremath{\ge}2$ defined through the relation ${N}^{*}\ensuremath{\simeq}{\ensuremath{\delta}}^{\ensuremath{-}\ensuremath{\gamma}}$, where ${N}^{*}$ separates the delocalized region ($N\ensuremath{\ll}{N}^{*}$, ${Y}_{2}$ vanishes with increasing $N$) from the localized region ($N\ensuremath{\gg}{N}^{*}$, ${Y}_{2}$ is approximately constant). We finally compare our results with the structure of the condensate obtained through the single-site marginal distribution.