Incommensurate state in a quasi-one-dimensional $S=1∕2$ bond-alternating antiferromagnet with frustration in magnetic fields

We investigate the critical properties of the $S=1∕2$ bond-alternating spin chain with a next-nearest-neighbor interaction in magnetic fields. By the numerical calculation and the exact solution based on the effective Hamiltonian, we show that there is a parameter region where the longitudinal incommensurate spin correlation becomes dominant around the half-magnetization of the saturation. Possible interpretations of our results are discussed. We next investigate the effects of the interchain interaction $\left(J^{\prime }\right)$. The staggered susceptibility and the uniform magnetization are calculated by combining the density-matrix renormalization group method with the interchain mean-field theory. For the parameters where the dominant longitudinal incommensurate spin correlation appears in the case $J^{\prime }=0$, the staggered long-range order does not emerge up to a certain critical value of $J^{\prime }$ around the half-magnetization of the saturation. We calculate the static structure factor in such a parameter region. The size dependence of the static structure factor at $k=2k_F$ implies that the system has a tendency to form an incommensurate long-range order around the half-magnetization of the saturation. We discuss the recent experimental results for the NMR relaxation rate in magnetic fields performed for pentafluorophenyl nitronyl nitroxide.

Figure 1

The extrapolated ${\eta }^x$ and ${\eta }^z$ for $\alpha =0.10$, 0.15, 0.18, and 0.20 in $\delta =0.15$ as function of $m$.

Figure 2

In the darkly shaded area, the relation ${\eta }^z<1<{\eta }^x$ is satisfied around the half-magnetization of the saturation. In the lightly shaded area, the relation ${\eta }^x<1<{\eta }^z$ is satisfied in $0<m<1∕2$. The half-magnetization plateau appears in the “plateau” area. Two broken lines are the boundaries for the half-magnetization plateau obtained by the level spectroscopy analysis (Ref. 19). The open circle corresponds to the parameters for ${\mathrm{F}}_5\mathrm{PNN}$ ($\alpha =0.15$ and $\delta =0.43$). The four open triangles correspond to the parameters used in Fig. 1 ($\alpha =0.10$, 0.15, 0.18, and 0.20 in $\delta =0.15$).

Figure 3

$H_{\mathrm{eff}}$ dependence of ${\eta }^z$ and $m_{\mathrm{eff}}$ for several $\Delta$.

Figure 4

The divergence exponents $\gamma$ for $\alpha =0.10$, 0.15, 0.18, and 0.20 in $\delta =0.15$ as function of $m$, where $1∕T_1\sim T^{-\gamma }$. The lines are to guide the eyes.

Figure 5

The staggered magnetization $m_s$ for several values of $H_u$ as function of $h_s$. Parameters used are (a) $\alpha =0.20$ and $\delta =0.15$ and (b) $\alpha =0.05$ and $\delta =0.40$.

Figure 6

The uniform magnetization $m_u$ for $(\alpha ,\delta )=(0.20,0.15)$, (0.05, 0.40), and (0.15, 0.43) as function of $H_u$. The interchain couplings are set to be $z{J}^{\prime }=8.95\times {10}^{-3}$, $2.50\times {10}^{-2}$, and $6.85\times {10}^{-3}$, respectively.

Figure 7

(a) Static structure factor $S^z\left(k\right)$ in $\alpha =0.20$ and $\delta =0.15$ for $m_u=1∕3$, 1/4, and 1/6. (b) $N$ dependence of $S^z\left(2k_F\right)$. The solid lines are obtained by the least-squares method. The divergence powers are evaluated as 0.196 for $m_u=1∕6$, 0.237 for $m_u=1∕4$, and 0.170 for $m_u=1∕3$.

Figure 8

The divergence exponent $\gamma$ for ${\mathrm{F}}_5\mathrm{PNN}$ ($\alpha =0.15$ and $\delta =0.43$) as function of $H$, where $1∕T_1\sim T^{-\gamma }$. Using the experimental data, $H_{p1}$ and $H_{p2}$ are evaluated as $H_{p1}\sim 3.77\mathrm{T}$ and $H_{p2}\sim 5.79\mathrm{T}$. The line is to guide the eyes.

Figure 9

(a) The staggered magnetization $m_s$ for several values of $H_u$ as function of $h_s$. Parameters are $\alpha =0.15$ and $\delta =0.43$. (b) Static structure factor $S^z\left(k\right)$ for $m_u=1∕4$. Inset: $L$ dependence of $S^z\left(2k_F\right)$. From the solid lines obtained by the least-squares method, the divergence power is evaluated as 0.346.