Finite-size localization scenarios in condensation transitions

We consider the phenomenon of condensation of a globally conserved quantity H = Sigma(N)(i=1) 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 = /(Nh(2)) on the size N of the system and on the control parameter delta = (h - h(c)), for various models: (i) a model with two conservation laws, derived from the discrete nonlinear Schrodinger equation; (ii) the continuous version of the zero-range process class, for different forms of the function f (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 gamma >= 2 defined through the relation N* similar or equal to delta(-gamma), where N* separates the delocalized region (N << N*, Y-2 vanishes with increasing N) from the localized region (N >> N*, Y-2 is approximately constant). We finally compare our results with the structure of the condensate obtained through the single-site marginal distribution.

Automated summary

The numerical study explores the phenomenon of condensation of a globally conserved quantity on N sites when the density exceeds a critical value, showing various localization scenarios characterized by the presence or absence of a minimum of participation ratio Y-2 and an exponent gamma defined through a relation. The comparison with the condensate structure obtained through single-site marginal distribution is also conducted.