Decoupled phase of frustrated spin-1/2 antiferromagnetic chains with and without long range order in the ground state

The quantum phases of one-dimensional spin $s= 1/2$ chains are discussed for models with two parameters, frustrating exchange $g = J_2 > 0$ between second neighbors and normalized nonfrustrating power-law exchange with exponent $\alpha$ and distance dependence $r^{-\alpha}$. The ground state (GS) at $g = 0$ has long-range order (LRO) for $\alpha < 2$, long-range spin fluctuations for $\alpha > 2$. The models conserve total spin $S = S_A + S_B$, have singlet GS for any $g$, $\alpha \ge 0$ and decouple at $1/g = 0$ to linear Heisenberg antiferromagnets on sublattices $A$ and $B$ of odd and even-numbered sites. Exact diagonalization of finite chains gives the sublattice spin $\ < S^2_A \ >$, the magnetic gap $E_m$ to the lowest triplet state and the excitation $E_{\sigma}$ to the lowest singlet with opposite inversion symmetry to the GS. An analytical model that conserves sublattice spin has a first order quantum transition at $g_c = 1/4{\rm ln2}$ from a GS with perfect LRO to a decoupled phase with $S_A = S_B = 0$ for $g \ge 4/\pi^2$ and no correlation between spins in different sublattices. The model with $\alpha = 1$ has a first order transition to a decoupled phase that closely resembles the analytical model. The bond order wave (BOW) phase and continuous quantum phase transitions of finite models with $\alpha \ge 2$ are discussed in terms of GS degeneracy where $E_{\sigma}(g) = 0$, excited state degeneracy where $E_{\sigma}(g) = E_m(g)$, and $\ < S^2_A \ >$. The decoupled phase at large frustration has nondegenerate GS for any exponent $\alpha$ and excited states related to sublattice excitations.