Quantum phase transition in a dimerized chain with hexamer distortion

We consider the dimerized spin-1/2 Heisenberg chain with spin hexameric distortion of the exchange pattern and study the zero-temperature phase diagram in the parameter space $(J_1,J_2,J_3)$ by continuum-limit bosonization approach and the exact diagonalization method. The phase diagram is rich and has two gaped dimer phases. We obtain an estimate of the critical line separating the different gapped dimer phases by the bosonization approach. The existence of the transition line and the difference between dimer phases is checked numerically. The behavior of the energy gap and the dimer order parameter supports the exact location of the gapless line.