Spontaneous onset of a low temperature topologically ordered phase in a 2-dimensional (2D) lattice model of uniaxial liquid crystal (LC) was debated extensively pointing to a suspected underlying mechanism affecting the RG flow near the topological fixed point. A recent MC study clarified that a prior crossover leads to a transition to nematic phase. The crossover was interpreted as due to the onset of a perturbing relevant scaling field originating from the extra spin degree of freedom. As a counter example and in support of this hypothesis, we now consider V-shaped bent-core molecules with rigid rod-like segments connected at an assigned angle. The two segments of the molecule interact with the segments of all the nearest neighbours on a square lattice, prescribed by a biquadratic interaction. We compute equilibrium averages of different observables with Monte Carlo techniques as a function of temperature and sample size. For the chosen molecular bend angle and symmetric inter-segment interaction between neighbouirng molecules, the 2D system shows two transitions as a function of T: the higher one at T1 leads to a topological ordering of defects associated with the major molecular axis without a crossover, imparting uniaxial symmetry to the medium described by the first fundamental group of the order parameter space $pi_{1}$= $Z_{2}$ (inversion symmetry). The second at T2 leads to a medium displaying biaxial symmetry with $pi_{1}$ = Q (quaternion group). The biaxial phase shows a self-similar microscopic structure with the three axes showing power law correlations with vanishing exponents as the temperature decreases.
Download