Polydispersed rods on the square lattice

We study the grand-canonical solution of a system of hard polydispersed
rods placed on the square lattice using transfer matrix and finite size
scaling calculations. Only excluded volume interactions are considered.
In order to treat both directions on the square lattice
in a simmetric way, the transfer matrix is defined along the diagonal direction
of the lattice. The polydispersity of the rods is determined by distinct
activities for internal and endpoint monomers of a rod, as is done in
a lattice model for equilibrium polymerization. We determine the critical line
separating an isotropic from a nematic phase, extrapolating data for the
correlation length of the model defined on strips of finite width with periodic
boundary conditions. In the full packing limit it was possible to handle
strips of larger widths, and therefore more precise estimates could be
obtained. No second transition to a disordered phase is
found at high density, contrary to what is observed in
the monodispersed case, therefore the critical line extends up to the full packing limit. The
estimates of critical exponents and the central charge, on the whole critical
line, and also in the full packing limit, are consistent with the Ising universality class.
The extrapolated phase diagram is compared with Bethe lattice results for the same model and with
simulational results for monodispersed rods on the square lattice.