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Numerical study of incommensurate and decoupled phases of spin-1/2 chains with isotropic exchange J1, J2 between first and second neighbors

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 $q_G = π/2$, 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 $E_σ$, 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 $q_G = π$ in the gapless phase with $J1/J2 > 4.148$ and quasi-long-range order (QLRO({$π$})). $S(q)$ diverges at $\pm π/2$ in the decoupled phase with QLRO({$π$}/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({$π$}/2) and no IC phases; the other has a single IC phase without a decoupled phase in between.