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We use a tensor network renormalization group method to study random $S=2$antiferromagnetic Heisenberg chains with alternating bond strengthdistributions. In the absence of randomness, bond alternation induces twoquantum critical points between the $S=2$ Haldane phase, a partially dimerizedphase 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 bondsforming across even links and $4-\sigma$ valence bonds on odd links. Here westudy the effects of bond randomness on the ground states of the dimerized spinchain, calculating disorder-averaged twist order parameters and spincorrelations. We classify the types of random VBS phases depending on thestrength of bond randomness $R$ and dimerization $D$ using the twist orderparameter, which has a negative/positive sign for a VBS phase with odd/even$\sigma$. Our results demonstrate the existence of a multicritical point in theintermediate disorder regime with finite dimerization, where (2,2), (3,1) and(4,0) phases meet. This multicritical point is at the junction of three phaseboundaries in the $R$-$D$ plane: the (2,2)-(3,1) and (3,1)-(4,0) boundariesthat extend to zero randomness, and the (2,2)-(4,0) phase boundary thatconnects another multicritical point in the undimerized limit. The undimerizedmulticritical point separates a gapless Haldane phase and aninfinite-randomness critical line with the diverging dynamic critical exponentin the large $R$ limit at $D=0$. Furthermore, we identify the (3,1)-(4,0) phaseboundary as an infinite-randomness critical line even at small $R$, and findthe signature of infinite randomness at the (2,2)-(3,1) phase boundary only inthe vicinity of the multicritical point.

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