  Nematic order is an exotic property observed in several strongly correlated
systems, such as the iron-based superconductors. Using large-scale density
matrix renormalization group (DMRG) techniques, we study at zero-temperature
the nematic spin liquid that competes with spin dipolar and quadrupolar orders.
We use these nematic orders to characterize different quantum phases and
quantum phase transitions. More specifically, we study a spin-$1$
bilinear-biquadratic Heisenberg model on the square lattice with couplings
beyond nearest neighbors. We focus on parameter regions around the highly
symmetric $SU(3)$ point where the bilinear and biquadratic interactions are
equal. With growing further-neighbor biquadratic interactions, we identify
different spin dipolar and quadrupolar orders. We find that the DMRG results on
cylindrical geometries correctly detect nematicity in different quantum states
and accurately characterize the quantum phase transitions among them.
Therefore, spin-driven nematicity -- here defined as the spontaneous breaking
of the lattice invariance under a 90$^o$ rotation -- is an order parameter
which can be studied directly in DMRG calculations in two dimensions in
different quantum states.
