Low-temperature thermodynamic properties of a random anisotropic antiferromagnetic chain

Abstract

The low-temperature thermodynamic properties of a spin-½ one-dimensional random anisotropic (Heisenberg-Ising) antiferromagnet described by the Hamiltonian $H=\Sigma i^{}{J}_i({\sigma }_x^i{\sigma }_x^{i+1\mathrm{}}+{\sigma }_y^i{\sigma }_y^{i+1\mathrm{}}+\gamma {\sigma }_z^i{\sigma }_z^{i+1\mathrm{}})$ are studied as a function of disorder and anisotropy. The $J_i>~0$ are independent random variables obeying a probability distribution $P\left(J\right)\mathrm{}$, and $0<~\gamma <~\infty$. The approach used is a numerical implementation of a real-space renormalization-group (RG) method previously introduced. The isotropic Heisenberg case ($\gamma =1\mathrm{}$), the $\mathrm{XY}$ case ($\gamma =0\mathrm{}$). and the Ising case ($\gamma =\infty$) are fixed points of the RG transformation. It is found that in the $\mathrm{XY}$ region $\gamma <~1$, including the Heisenberg point, the system exhibits singular behavior in the thermodynamic properties for arbitrary probability distributions for the couplings. For the $\mathrm{XY}$ limit ($\gamma =0\mathrm{}$) this is in agreement with known exact results. The functional form for the low-temperature susceptibility is found to be $\chi \sim \frac{1}{\left(T{\ln }^m\right(\frac{T}{T_0}\left)\right)}$ in the entire region $0<~\gamma <~1$ for arbitrary probability distributions. In the Ising region ($\gamma >1\mathrm{}$) the susceptibility shows an approximate power-law divergence for small anisotropy but goes eventually to zero as $T\to 0$. Possible relevance of these results to recent experiments on Qn${\mathrm{}\left(\mathrm{TCNQ}\right)\mathrm{}}_2$ is discussed.