Random singlets and permutation symmetry in the disordered spin-2 Heisenberg chain: A tensor network renormalization group study

We use a tensor network renormalization group method to study random $S=2$ antiferromagnetic Heisenberg chains with alternating bond strength distributions. In the absence of randomness, bond alternation induces two quantum critical points between the $S=2$ Haldane phase, a partially dimerized phase and a fully dimerized phase, depending on the strength of dimerization. These three phases, called ($\sigma$,$4-\sigma$)=(2,2), (3,1) and (4,0) phases, are valence-bond solid (VBS) states characterized by $\sigma$ valence bonds forming across even links and $4-\sigma$ valence bonds on odd links. Here we study the effects of bond randomness on the ground states of the dimerized spin chain, calculating disorder-averaged twist order parameters and spin correlations. We classify the types of random VBS phases depending on the strength of bond randomness $R$ and dimerization $D$ using the twist order parameter, which has a negative/positive sign for a VBS phase with odd/even $\sigma$. Our results demonstrate the existence of a multicritical point in the intermediate disorder regime with finite dimerization, where (2,2), (3,1) and (4,0) phases meet. This multicritical point is at the junction of three phase boundaries in the $R$-$D$ plane: the (2,2)-(3,1) and (3,1)-(4,0) boundaries that extend to zero randomness, and the (2,2)-(4,0) phase boundary that connects another multicritical point in the undimerized limit. The undimerized multicritical point separates a gapless Haldane phase and an infinite-randomness critical line with the diverging dynamic critical exponent in the large $R$ limit at $D=0$. Furthermore, we identify the (3,1)-(4,0) phase boundary as an infinite-randomness critical line even at small $R$, and find the signature of infinite randomness at the (2,2)-(3,1) phase boundary only in the vicinity of the multicritical point.