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Spontaneous onset of a low temperature topologically ordered phase in a2-dimensional (2D) lattice model of uniaxial liquid crystal (LC) was debatedextensively pointing to a suspected underlying mechanism affecting the RG flownear the topological fixed point. A recent MC study clarified that a priorcrossover leads to a transition to nematic phase. The crossover was interpretedas due to the onset of a perturbing relevant scaling field originating from theextra spin degree of freedom. As a counter example and in support of thishypothesis, we now consider V-shaped bent-core molecules with rigid rod-likesegments connected at an assigned angle. The two segments of the moleculeinteract with the segments of all the nearest neighbours on a square lattice,prescribed by a biquadratic interaction. We compute equilibrium averages ofdifferent observables with Monte Carlo techniques as a function of temperatureand sample size. For the chosen molecular bend angle and symmetricinter-segment interaction between neighbouirng molecules, the 2D system showstwo transitions as a function of T: the higher one at T1 leads to a topologicalordering of defects associated with the major molecular axis without acrossover, imparting uniaxial symmetry to the medium described by the firstfundamental group of the order parameter space $\pi_{1}$= $Z_{2}$ (inversionsymmetry). The second at T2 leads to a medium displaying biaxial symmetry with$\pi_{1}$ = Q (quaternion group). The biaxial phase shows a self-similarmicroscopic structure with the three axes showing power law correlations withvanishing exponents as the temperature decreases.

Topological phase transitions in 2-dimensional bent-core liquid crystal models