  The phase diagram of the quantum spin-1/2 antiferromagnetic
$J^{\,}_{1}$-$J^{\,}_{2}$ XXZ chain was obtained by Haldane using bosonization
techniques. It supports three distinct phases for $0\leq
J^{\,}_{2}/J^{\,}_{1}<1/2$, i.e., a gapless algebraic spin liquid phase, a
gapped long-range ordered Neel phase, and a gapped long-range ordered dimer
phase. Even though the Neel and dimer phases are not related hierarchically by
a pattern of symmetry breaking, it was shown that they meet along a line of
quantum critical points with a U(1) symmetry and central charge $c=1$. Here, we
extend the analysis made by Haldane on the quantum spin-1/2 antiferromagnetic
$J^{\,}_{1}$-$J^{\,}_{2}$ XYZ chain using both bosonization and numerical
techniques. We show that there are three Neel phases and the dimer phase that
are separated from each other by six planes of phase boundaries realizing U(1)
criticality when $0\leq J^{\,}_{2}/J^{\,}_{1}<1/2$. We also show that each
long-range ordered phase harbors topological point defects (domain walls) that
are dual to those across the phase boundary in that a defect in one ordered
phase locally binds the other type of order around its core. By using the
bosonization approach, we identify the critical theory that describes
simultaneous proliferation of these dual point defects, and show that it
supports an emergent U(1) symmetry that originates from the discrete symmetries
of the XYZ model. To confirm this numerically, we perform DMRG calculation and
show that the critical theory is characterized by the central charge $c=1$ with
critical exponents that are consistent with those obtained from the
bosonization approach. Furthermore, we generalize the field theoretic
description of direct continuous phase transition to higher dimensions,
especially in $d=3$, by using a non-linear sigma model (NLSM) with a
topological term.
