Thermodynamics of random-exchange Heisenberg antiferromagnetic chains

Diagrammatic valence-bond methods are introduced for $s=\frac{1}{2}$ magnetic insulators whose random Heisenberg exchanges satisfy some distribution $f\left(x\right)\mathrm{}$. Rapid solution of arbitrary 10-spin replicas yields, for the first time, complete static thermodynamics by generating successive replicas whose exchanges match $f\left(x\right)\mathrm{}$. The usual singular distribution $J{x}^{-\gamma }$ for structurally disordered tetracyanoquinodimethane (TCNQ) salts is contrasted with a nonsingular distribution for a random concentration $c$ of weak exchanges $\epsilon J$ along a chain of uniform $\mathrm{Js}$. A renormalization procedure for interactions among odd segments at low temperature generates a wide range of couplings for a single $\epsilon >0\mathrm{}$ and yields power laws for nonsingular $f\left(x\right)\mathrm{}$. Experimental magnetic susceptibility $\chi \mathrm{}\left(T\right)\mathrm{}$, magnetization $M(T, H)$, and magnetic specific heat $C(T, H)$ results for the closely similar 1:2 complexes of quinolinium and acridinium with TCNQ are compared to $J{x}^{-\gamma }$ and to interacting segments. The usual distribution has $\gamma =0.75\mathrm{}$ and an enormous $J=2\mathrm{}\times {10}^7$ K, but fails to describe the $\chi \mathrm{}\left(T\right)\mathrm{}$ crossover around 20 K. Interacting segments fit $\chi \mathrm{}\left(T\right)\mathrm{}$ over the entire range, as well as available $M(T, H)$ and $C(T, H)$ data, for some $c=0.10\mathrm{}$ weak exchanges $\epsilon J=70\mathrm{}$ K among uniform exchanges $J=230\mathrm{}$ K. Small differences between the two complexes were not parametrized. The microscopic picture of strong disorder leads to charge disproportionation on the TCNQ stack and to $J{x}^{-\gamma }$, while weak disorder and supermolecular TCN${\mathrm{Q}}_2^{-}$ ion radicals rationalize the picture of weakly interacting segments.