Dimensional change of quadrupole orders in pseudospin-\(\frac{1}{2}\) pyrochlore magnets under [111] field

We have studied long range orders of electric quadrupole moments described by an effective pseudospin-\(\frac{1}{2}\) Hamiltonian representing pyrochlore magnets with non-Kramers ions under [111] magnetic field, in relevance to Tb\(_{2}\)Ti\(_{2}\)O\(_{7}\). Order parameters and phase transitions of this frustrated system are investigated using classical Monte-Carlo simulations. In zero field, the model undergoes a first-order phase transition from a paramagnetic state to an ordered state with an antiparallel arrangement of pseudospins. This pseudospin order is characterized by the wavevector \(\boldsymbol{k}=0\) and is selected by an energetic or an order-by-disorder mechanism from degenerate \(\boldsymbol{k}=(h,h,h)\) mean-field orders. Under [111] magnetic field this three-dimensional quadrupole order is transformed to a quasi two-dimensional quadrupole order on each kagom\'{e} lattice separated by field-induced ferromagnetic triangular lattices. We discuss implication of the simulation results with respect to experimental data of Tb\(_{2}\)Ti\(_{2}\)O\(_{7}\).