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Symmetry-protected topological phases and transition in a frustrated spin-$\frac{1}{2}$ XXZ chain

Frustrated spin-$1/2$ XXZ zigzag chains relevant to Rb$_2$Cu$_2$Mo$_3$O$_{12}$ are revisited in the light of symmetry-protected topological (SPT) phases. Using a density-matrix renormalization group method for infinite systems, we identify projective representations for four distinct time-reversal invariant SPT phases; two parity-symmetric dimer phases near the Heisenberg and XX limits and two parity-broken vector-chiral (VC) dimer phases in between. A small bond alternation in the nearest-neighbor ferromagnetic exchange coupling induces a direct SPT transition between the two distinct VC dimer phases. It is also found numerically that two Berezinskii-Kosterlitz-Thouless transitions from the gapless to the two distinct gapped VC phases meet each other at a Gaussian criticality of the same Tomonaga-Luttinger parameter value as in the SU(2)-symmetric case.