The s=1/2 quasi-one-dimensional Heisenberg antiferromagnetic chain with asymmetrical sublattices possesses three branches of spin-wave spectra (one in acoustic mode and the other two in optical mode). The frustration relieves the gap-degeneracy of two optical branches and softens the spectrum in acoustic mode, but doesn't soften the spectra in optical mode. Also, the frustration moves down obviously one of the two spectra in optical mode. It means that the frustration makes the spin-wave excitation described by spectrum in optical mode easier and more important. The status of spin on the chain changes more easily than that of side spin under the effect of frustration. The weakening of magnetic order in ground state becomes strong with the strengthening of frustration. By comparison with numerical results of exact diagonalization and DMRG methods, we discuss the merit and drawbacks of the spin-wave approximation.
The quantum phases of one-dimensional spin
s=1/2 chains are discussed for models with two parameters, frustrating exchange
g=J2>0 between second neighbors and normalized nonfrustrating power-law exchange with exponent $\ensuremath{\alpha}$ and distance dependence ${r}^{\ensuremath{-}\ensuremath{\alpha}}$. The ground state (GS) at
g=0 has a long-range order (LRO) for $\ensuremath{\alpha}<2$ and long-range spin fluctuations for $\ensuremath{\alpha}>2$. The models conserve total spin
S=SA+SB, have singlet GS for any
g, $\ensuremath{\alpha}\ensuremath{\ge}0$ 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 $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$, the magnetic gap
Em to the lowest triplet state, and the excitation ${E}_{\ensuremath{\sigma}}$ 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
gc=1/4ln2 from a GS with perfect LRO to a decoupled phase with
SA=SB=0 for $g\ensuremath{\ge}4/{\ensuremath{\pi}}^{2}$ and no correlation between spins in different sublattices. The model with $\ensuremath{\alpha}=1$ 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 $\ensuremath{\alpha}\ensuremath{\ge}2$ are discussed in terms of GS degeneracy where ${E}_{\ensuremath{\sigma}}(g)=0$, excited state degeneracy where ${E}_{\ensuremath{\sigma}}(g)={E}_{m}(g)$, and $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$. The decoupled phase at large frustration has nondegenerate GS for any exponent $\ensuremath{\alpha}$ and excited states related to sublattice excitations.
By using density-matrix renormalization group(DMRG) method,the authors investigate a system of quasi-one-dimensional Heisenberg antiferromagnetic chain with symmetry breaking of sublattice.The ground state energy per unit cell,spin correlation function and staggered susceptibility are calculated.The calculations show that the system exhibits ferrimagnetic properties,although the spin-spin interactions are antiferromagnetic.The results are consistent qualitatively with the conclusion predicted by LIEB-MATTIS theorem and the results calculated by spin-wave theory.