Size-dependent low-energy excitations in an alternating spin-1/spin-$\frac{1}{2}$ antiferromagnetic chain: Spin-wave theory and density-matrix renormalization-group studies

We study the properties of the ground state and the low-lying excited states of an alternating spin-1 and spin-$\frac{1}{2}$ antiferromagnetic chain with ferromagnetic next-nearest-neighbor coupling with a variety of methods, namely, spin-wave analysis, density-matrix renormalization-group (DMRG) method, and exact-diagonalization method. The ground state of this model with $2N$ spins is ferrimagnetic with the total spin $S_G=N∕2$ for all parameter values. In the long chain limit, a gapless excitation is found with characteristics of goldstone mode of the ferromagnetic order with total spin $S=S_G-1$. The lowest gapped excitation, however, exists in the total spin $S=S_G+1$ sector. Interestingly, we find that with the increase in ferromagnetic coupling, spin-wave gapped excitation becomes lower than the massless mode for the large wave numbers. Correspondingly, from DMRG analysis we find that the excitation in the total spin $S=S_G+1$ is lower than that in the total spin $S=S_G-1$ when the size of the chain is short and the ferromagnetic interaction is large. Characteristics of these excitations are also reflected in the low temperature thermodynamic quantities, where the low-energy properties are strongly affected by the system size and the coupling strength.

Figure 1

The two branches of the spin-wave dispersion curves, $E_{1k}$ and $E_{2k}$, in units of $J^{\prime }$ as a function of $k$. The solid line is for $E_{1k}$ mode and the dashed line is for $E_{2k}$ mode for $J∕J^{\prime }=0.2$, 0.3, 0.7, and 1.0.

Figure 2

Schematic spin configurations of (a) the classical ground state, $\left({\mathrm{a}}^{\prime }\right)$ the classical ground state with a pairwise flip, (b) the spin configuration with the magnetization $S_G-1$, $\left({\mathrm{b}}^{\prime }\right)$ the spin configuration as in (b) with a pairwise flip, (c) the spin configuration with the magnetization $S_G+1$, and $\left({\mathrm{c}}^{\prime }\right)$ the spin configuration as in (c) with a pairwise flip.

Figure 3

The ground state energy per site $\left(E_0\right)$ in units of $J^{\prime }$, as a function of $J∕J^{\prime }$, for alternating spin-1/spin-$\frac{1}{2}$ chain.

Figure 4

Excitation gaps as a function of the size of the system $\left(2N\right)$ for $J∕J^{\prime }=0.2$, 0.3, 0.7, and 1.0. The solid line with circles represents the gap to the state $S=S_G-1$. The dashed line and the dotted line represent the gap to the excitations $S=S_G+1$ and $S=S_G+2$, respectively.

Figure 5

Magnetic susceptibility multiplied by $T\left(\chi T\right)$ as a function of temperature $\left(T\right)$. The solid line represents $\chi T$ for $J∕J^{\prime }=0.2$, the dotted line for $J∕J^{\prime }=0.3$, the dashed line for $J∕J^{\prime }=0.7$, the solid line with circles for $J∕J^{\prime }=1.0$, and the solid line with diamonds for $J∕J^{\prime }=2.0$.