  The spin-1/2 chain with isotropic exchange J1, J2 > 0 between first and
second neighbors is frustrated for either sign of J1 and has a singlet ground
state (GS) for J1/J2 $\ge - 4$. Its rich quantum phase diagram supports
gapless, gapped, commensurate (C), incommensurate (IC) and other phases.
Critical points J1/J2 are evaluated using exact diagonalization (ED) and
density matrix renormalization group (DMRG) calculations. The wave vector $q_G$
of spin correlations is related to GS degeneracy and obtained as the peak of
the spin structure factor $S(q)$. Variable $q_G$ indicates IC phases in two
$J1/J2$ intervals, [ -4, -1.24] and [0.44, 2], and a C-IC point at J1/J2 = 2.
The decoupled C phase in [-1.24, 0.44] has constant $q_G = {\pi}/2$,
nondegenerate GS, and a lowest triplet state with broken spin density on
sublattices of odd and even numbered sites. The lowest triplet and singlet
excitations, $E_m$ and $E_{\sigma}$, are degenerate in finite systems at
specific frustration $J1/J2$. Level crossing extrapolates in the thermodynamic
limit to the same critical points as $q_G$. The $S(q)$ peak diverges at $q_G =
{\pi}$ in the gapless phase with $J1/J2 > 4.148$ and quasi-long-range order
(QLRO({$\pi$})). $S(q)$ diverges at $\pm {\pi}/2$ in the decoupled phase with
QLRO({$\pi$}/2), but is finite in gapped phases with finite range correlations.
Numerical results and field theory agree at small $J2/J1$ but disagree for the
decoupled phase with weak exchange $J1$ between sublattices. Two related models
are summarized: one has an exact gapless decoupled phase with QLRO({$\pi$}/2)
and no IC phases; the other has a single IC phase without a decoupled phase in
between.
