Critical theory of the topological quantum phase transition in the AKLT--SZH chain

We systematically study the phase diagram of $S=2$ spin chain by means of density-matrix renormalization group and exact diagonalization. We find two gapped phases which are topologically distinct: the AKLT phase is characterized by $S=1$ edge spins while the SZH phase by $S=3/2$ edge spins. Except from a multicritical point where a direct topological quantum phase transition occurs, we identify a dimer phase between both gapped phases. The whole phase boundary between dimer and SZH phases, including the multicritical point, is a critical line with central charge $c=5/2$. Finally, we propose and confirm that this line corresponds to $\rm SO(5)_1$ Wess--Zumino--Witten conformal field theory.