Correlated spin and orbital dimerizations in spin-orbital models

We study a one-dimensional spin-orbital model with Heisenberg symmetry for spins and $\mathrm{XY}$ symmetry for orbital degrees of freedom. We establish the ground-state phase diagram of this model. The gapless phases of the model consist of fully polarized ferromagnetic spin states together with the filled Jordan-Wigner Fermi sea for the orbitals. Two such phases, with ferromagnetic and antiferromagnetic $\mathrm{XY}$ correlations for the orbitals, are found. All other phases are found to have a spin gap with degenerate ground states. Two types of correlated spin and orbital dimerizations are observed: (i) spin singlet with orbital singlet in alternating rungs and (ii) spin singlet with orbital triplet in the same rung. The phase boundaries, energy gaps, and order parameters are evaluated using the density matrix renormalization group method.