Topological Transition to a Critical Phase in a Two-dimensional 3-Vector Model with non-Abelian Fundamental Group: A Simulational Study

Two-dimensional 3-vector (\textit{d}=2, \textit{n}=3) lattice model with inversion site symmetry and fundamental group of its order-parameter space $\Pi_1 (\mathcal{R})= Z_{2}$, did not exhibit the expected topological transition despite stable defects associated with its uniaxial orientational order. This model is investigated specifically requiring the medium to host distinct classes of defects associated with the three ordering directions, facilitating their simultaneous interactions. The necessary non-Abelian isotropy subgroup of $\mathcal{R}$ is realized by assigning $D_{2}$ site symmetry, resulting in $\Pi_1 (\mathcal{R})= \mathbb{Q }$ (the group of quaternions). With liquid crystals serving as prototype model, a general biquadratic Hamiltonian is chosen to incorporate equally attractive interactions among the three local directors resulting in an orientational order with the desired topology. A Monte Carlo investigation based on the density of states shows that this model exhibits a transition, simultaneously mediated by the three distinct defects with topological charge $1/2$ (disclinations), to a low-temperature critical state characterized by a line of critical points with quasi-long range order of its directors, their power-law exponents vanishing as temperature tends to zero. It is argued that with \textit{n}=3, simultaneous participation of all spin degrees through their homotopically inequivalent defects is necessary to mediate a transition in the two-dimensional system to a topologically ordered state.