Antiferromagnetic order in weakly coupled random spin chains

The ordering of weakly coupled random antiferromagnetic $S=1/2$ chains, as relevant for the recent experimentally investigated spin chain materials, is considered theoretically. The one-dimensional isotropic Heisenberg model with random exchange interactions is treated numerically on finite chains with the density-matrix renormalization-group approach as well as with the standard renormalization analysis, both within the mean-field approximation for interchain coupling $J_{\perp }$. Results for the ordering temperature $T_N$ and for the ordered moment $m_0$ are presented and are both reduced with increasing disorder, agreeing with the experimental observations. The most pronounced effect of the random singlet concept appears to be a very large span of local ordered moments, becoming wider with decreasing $J_{\perp }$, consistent with the $\mu$-SR experimental findings.

Figure 1

(a) $T$ dependence of ${\chi }_{\pi }$ for various randomness $\delta J$. Black, dashed line represents RS fit, Eq. (5), for $\delta J=0.8$. Shown is also a fit for the pure case to Eq. (4). (b) Decrease of Néel temperature $T_N$ with randomness $\delta J$ for various $z_{\perp }{J}_{\perp }$. Calculated with $L=80$.

Figure 2

(a) $T=0$ staggered magnetization $m_s$ vs $h_s$ for various randomness $\delta J$. (b) $\mathrm{Log}-log$ plot of $m_s$ vs $h_s$ for $\delta J=0,0.8$. $m_s\left(h_s\right)$ for $\delta J=0$ deviates from prediction in Eq. (6) in exponent $g$ and prefactor $r$. The result for $\delta J=0.8$ shows a RS like behavior given with Eq. (7). Fits of parameters for Eqs. (6) or (7) are for regime $0.0001<h<0.01$. (c) Self-consistent solution for staggered magnetization $m_0$ vs $\delta J$ for different $z_{\perp }{J}_{\perp }$.

Figure 3

Probability distribution function of $m_i$ at $T=0$ for (a) various values of $\delta J$ and fixed $z_{\perp }{J}_{\perp }=0.05$, and (b) for fixed $\delta J=0.4$ and various $z_{\perp }{J}_{\perp }$. Thin, vertical lines represent $m_0$ for given $\delta J$ and $J_{\perp }$.

Figure 4

Dependence of the magnetic moment distribution (for fixed $J_i)$ on the magnetic field used in the mean-field approximation, as calculated for $L=800$ sites. (a) shows three distributions of staggered fields with the same average value $\langle \left|h_i\right|\rangle =0.0004$; constant one, exponential distribution, and distribution from numerical simulations (see text for details). (b) shows the corresponding cumulative distribution function (CDF) of the magnetic moment $m_i$. Vertical lines represent values of average $m_s$, which are practically the same for all three distributions of $h_i$.