Local Spin Relaxation within the Random Heisenberg Chain

Finite-temperature local dynamical spin correlations $S_{nn}(\omega )$ are studied numerically within the random spin-$1/2$ antiferromagnetic Heisenberg chain. The aim is to explain measured NMR spin-lattice relaxation times in ${\mathrm{BaCu}}_2({\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}{)}_2{\mathrm{O}}_7$, which is the realization of a random spin chain. In agreement with experiments we find that the distribution of relaxation times within the model shows a very large span similar to the stretched-exponential form. The distribution is strongly reduced with increasing $T$, but stays finite also in the high-$T$ limit. Anomalous dynamical correlations can be associated with the random singlet concept but not directly with static quantities. Our results also reveal the crucial role of the spin anisotropy (interaction), since the behavior is in contrast with the ones for the $XX$ model, where we do not find any significant $T$ dependence of the distribution.

Figure 1

Dynamical local spin correlations $S_{nn}(\omega )$ for different configurations of $J_i$. Shown are spectra for the homogeneous case $\delta J=0$ and two configurations with $\delta J=0.7$, calculated for $T=0.5$ and $L=80$ sites.

Figure 2

Probability distribution function of local relaxation rates $\mathrm{PDF}(s)$ at $T\gg 1$ evaluated using the moment expansion for different $\delta J$. Inset: Comparison of the analytical and FTD-DMRG result for $\delta J=0.5$, $L=20$ with a full basis and averaged over $N_r={10}^3$ realizations.

Figure 3

Cumulative distribution function of $s$. Shown are FTD-DMRG results for $\Delta =1$: (a) for fixed $T=0.2$ and various $\delta J$, (b) for fixed $\delta J=0.7$ and various $T\le 0.5$. Inset of (b): full diagonalization results for $\Delta =0$, $\delta J=0.7$ and various $T$.

Figure 4

(a) Exponent $\Gamma$ vs $T$ obtained from $\mathrm{PDF}(s)$ data for different $\delta J$ and isotropic case $\Delta =1$. (b) $T$ dependence of fitted $1/{\tau }_0$ for $\Delta =1$ and different $\delta J$.