Exchange-coupled pair model for the random-exchange Heisenberg antiferromagnetic chain

An approximate model for the $s=\frac{1}{2}$ one-dimensional Heisenberg antiferromagnet with random exchange is presented. A probability for exchange ($J$) of the form $P\left(J\right)=\mathrm{const}{J}^{-\alpha }$, with the constant $\alpha$ in the range $0<\mathrm{}\alpha <1\mathrm{}$, is considered. The approximation is to decouple alternate spins, which leads to isolated exchange-coupled spin pairs having the same $P\left(J\right)\mathrm{}$. Explicit results for the magnetization ($M$), susceptibility ($\chi$), entropy ($S$), and specific heat ($C_H$) are given. Several limiting cases are worked out. The low-field ($\mathrm{Zeeman}\mathrm{energy}\ll k_{B}T$) behavior is $\chi =\mathrm{const}{T}^{-\alpha }$, whereas at intermediate fields ($\mathrm{Zeeman}\mathrm{energy}\gg k_{B}T$), $M=\mathrm{const}{H}^{1-\alpha }$, and $C_{H}H=\mathrm{const} T{H}^{-\alpha }$. These results are in agreement with the observed behavior of several tetracyanoquinodimethane (TCNQ) charge-transfer salts with asymmetric donors, such as quinolinium-${\mathrm{}\left(\mathrm{TCNQ}\right)\mathrm{}}_2$. Comparison with other approximations shows this model to give results almost indistinguishable from the spinless fermion model of Bulaevskii et al. and the cluster model of Theodorou and Cohen. It is argued that the work of Alexander, Bernasconi, and Orbach shows that there is a relationship between the density of states and $P\left(J\right)\mathrm{}$ for the Heisenberg chain, and that they are not the same. This leads to a value of $\alpha$ for the thermodynamic properties which differs from that of $P\left(J\right)\mathrm{}$ for the fully coupled Heisenberg chain.