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Spin parity and broken symmetry in finite spin-1/2 chains with frustrated exchange: quantum transition from high to low spin

Exact diagonalization of finite spin-1/2 chains with periodic boundary conditions is applied to the ground state (gs) of chains with ferromagnetic (F) exchange $J_1 < 0$between first neighbors, antiferromagnetic (AF) exchange $J_2 = αJ_1 > 0$between second neighbors, and axial anisotropy $0 \le Δ\le 1$. In zero field, the gs is in the $S_z = 0$ sector for the relevant parameters and is doubly degenerate at multiple points $γ_m = (α_m, Δ_m)$ in the $α$, $Δ$ plane. Degeneracy under inversion at sites or spin parity or both leads, respectively, to a bond order wave (BOW), to staggered magnetization or to vector chiral (VC) order. Exact results up to $N = 28$ spins directly yield order parameters and spin correlation functions whose weak N dependencies allow inferences about infinite chains. The high-spin gs at $J_2 = 0$ changes discontinuously at $γ_1 = (-1/4, 1)$ to a singlet in the isotropic ($Δ= 1$) chain. The transition from high to low spin $S(α, Δ)$ is continuous for $ Δ< Δ_B = 0.95 \pm 0.01$ on the degeneracy line $α_1(Δ)$. The gs has staggered magnetization between $Δ_A = 0.72$ and $Δ_B$, and a BOW for $Δ< Δ_A$. When both inversion and spin parity are reversed at $γ_m$, the correlation functions $C(p)$ for spins separated by $p$ sites are identical. $C(p)$ minima are shifted by $π/2$ from the minima of VC order parameters at separation $p$, consistent with right and left-handed helices along the z axis and spins in the xy plane. Degenerate gs of finite chains are related to quantum phase diagrams of extended $α$, $Δ$ chains, with good agreement for order parameters along the line $α_1(Δ)$.