  Spin systems with honeycomb structures have recently attracted a great deal
of attention in connection with the Kitaev quantum spin liquid state (QSL)
predicted theoretically. One possible Kitaev QSL candidate is
Na$_2$Co$_2$TeO$_6$ realizing a honeycomb lattice of pseudo-spin-1/2.
Field-dependent single-crystal neutron diffraction technique allows us to
determine the microscopic spin-spin correlations across the field induced phase
transitions for H//a and H//a* in plane field directions. Our results reveal
phase transitions, initially to a canted zigzag antiferromagnetic state at
approximately 60 kOe, followed by a possible transition to a partially
polarized state over the range of 90-120 kOe, and finally to a field-induced
fully polarized state above 120 kOe. We observe distinct field dependencies of
the magnetic peak intensities for H//a and H//a*. In addition, low-temperature
electron spin resonance in magnetic fields H//c yields a complete softening for
one of the antiferromagnetic resonances at ~ 40 kOe, revealing a field-induced
phase transition. The present work, thus, provides new insights into the field
evolution of the important Kitaev-Heisenberg spin system Na$_2$Co$_2$TeO$_6$.
