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$_{7}$.