Dimensional change of the quadrupole order in pseudospin-1/2 pyrochlore magnets under magnetic field in the [111] direction

We have studied long-range orders of electric quadrupole moments described by an effective pseudospin-1/2 Hamiltonian representing pyrochlore magnets with non-Kramers ions under [111] magnetic field, in relevance to Tb2Ti2O7. 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 wave vector k = 0 and is selected by an energetic or an order-by-disorder mechanism from degenerate 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 kagome lattice separated by field-induced ferromagnetic triangular lattices. We discuss the implication of the simulation results with respect to experimental data of Tb2Ti2O7.