Singular thermodynamic properties in random magnetic chains

The nearest-neighbor random-exchange Heisenberg antiferromagnetic Heisenberg chain ($s=\frac{1}{2}$) is studied at low temperatures via an approximate renormalization-group method. This entails renormalization of the random-exchange coupling constant $J$ and of the probability law for $J$. $P_0\mathrm{}\left(J\right)\mathrm{}$. After $n$ iterations we find that the renormalized probability function $P_n\left(J^{\mathrm{}\left(n\right)\mathrm{}}\right)$, a function of the renormalized coupling $J^{\mathrm{}\left(n\right)\mathrm{}}$, develops singular behavior for small $J^{\mathrm{}\left(n\right)\mathrm{}}$, independent of the initial form of $P_0\mathrm{}\left(J\right)\mathrm{}$. This happens both for $P_0\mathrm{}\left(J\right)\mathrm{}$ that diverge or go to zero as $J\to 0$. The singular form of $P_n\left(J^{\mathrm{}\left(n\right)\mathrm{}}\right)$ is such that it leads to a specific heat and susceptibility that behave like $C\sim T^{1-{\alpha }_C\mathrm{}\left(T\right)\mathrm{}}$ and $\chi \sim T^{-{\alpha }_{\chi }\mathrm{}\left(T\right)\mathrm{}}$. ${\alpha }_C\mathrm{}\left(T\right)\mathrm{}$ and ${\alpha }_{\chi }\mathrm{}\left(T\right)\mathrm{}$ are exponents weakly dependent on temperature ($T$) that go to one as $T\to 0$ and satisfy ${\alpha }_{\chi }\mathrm{}\left(T\right)>{\alpha }_C\mathrm{}\left(T\right)\mathrm{}$, for arbitrary initial probability laws. The classical $n$-vector models, and in particular the classical Heisenberg model are also studied using a renormalization-group approach and it is found that their behavior is different than that of the quantum model: singular behavior in the thermodynamic properties is only obtained if the starting $P_0\mathrm{}\left(J\right)\mathrm{}$ is singular. An explanation of these results in terms of the scaling theory of localization is suggested. The relevance of our results to the experimental findings on the magnetic properties of tetracyanoquinodimethanide complexes is also discussed.