Probing the existence of phase transitions in one-dimensional fluids of penetrable particles

Phase transitions in one-dimensional classical fluids are usually ruled out by making appeal to van Hove's theorem. A way to circumvent the conclusions of the theorem is to consider an interparticle potential that is everywhere bounded. Such is the case of, {\it e.g.}, the generalized exponential model of index 4 (GEM-4 potential), which in three dimensions gives a reasonable description of the effective repulsion between flexible dendrimers in a solution. An extensive Monte Carlo simulation of the one-dimensional GEM-4 model [S. Prestipino, {\em Phys. Rev. E} {\bf 90}, 042306 (2014)] has recently provided evidence of an infinite sequence of low-temperature cluster phases, however also suggesting that upon pushing the simulation forward what seemed a true transition may eventually prove to be only a sharp crossover. We hereby investigate this problem theoretically, by three different and increasingly sophisticated approaches ({\it i.e.}, a mean-field theory, the transfer matrix of a lattice model of clusters, and the exact treatment of a system of point clusters in the continuum), to conclude that the alleged transitions of the one-dimensional GEM4 system are likely just crossovers.