Many important concepts used in the study of complex systems have their origin in the lattice statistical mechanics
of cooperative phase transitions. The classic order-disorder transitions of Potts/Ising models illustrate
emergent phenomena such as spontaneous symmetry breaking, scale-free critical behaviour and power-law singularities.
However, a class of lattice statistics models show that rather more complex behaviour can be obtained,
under conditions which are suffciently general that they might also occur in less idealised models.
A 5-state lattice statistics model is analysed to explore behaviour in and around a so-called "massless phase".
The massless phase, characterised by power law decay of correlations, appears as an intermediate regime between
high temperature disorder and low-temperature ordering. The critical exponents (and fractal spatial distributions)
occur over a finite range of temperatures. The state of the massless phase is characterised by a topological
ordering with analogues of spin-waves being the dominant perturbations.
The transition from the ordered to massless phases is analysed using new exact series expansions obtained by
the finite lattice method. The results are also compared to Monte Carlo simulations. The results are compared
to other studies in the small number of cases where such series expansions exist for comparable models, including
the 6-state "clock model".
One of the properties that makes the massless phase dificult to study is the weak nature of the associated
Kosterlitz-Thouless transition with exponentially-weak behaviour rather than power-law behaviour. As a further
diffculty, the limiting behaviour is confined to a narrow regime, outside which one sees an apparent "cross-over"
to power law behaviour. This suggests that behaviour analogous to the "massless phases" will be diffcult to
characterise as one moves beyond idealised lattice systems.