Low-temperature thermodynamic properties of a random anisotropic antiferromagnetic chain

The low-temperature thermodynamic properties of a spin-{} one-dimensional random anisotropic (Heisenberg-Ising) antiferromagnet described by the Hamiltonian $H={Σ}{i}^{}{J}ᵢ({{σ}}ₓⁱ{{σ}}ₓⁱ⁺¹+{{σ}}yⁱ{{σ}}yⁱ⁺¹+{γ}{{σ}}zⁱ{{σ}}zⁱ⁺¹)$ are studied as a function of disorder and anisotropy. The ${J}ᵢ>~0$ are independent random variables obeying a probability distribution $P(J)$, and $0<~{γ}<~{∞}$. The approach used is a numerical implementation of a real-space renormalization-group (RG) method previously introduced. The isotropic Heisenberg case (${γ}=1$), the $XY$ case (${γ}=0$). and the Ising case (${γ}={∞}$) are fixed points of the RG transformation. It is found that in the $XY$ region ${γ}<~1$, including the Heisenberg point, the system exhibits singular behavior in the thermodynamic properties for arbitrary probability distributions for the couplings. For the $XY$ limit (${γ}=0$) this is in agreement with known exact results. The functional form for the low-temperature susceptibility is found to be ${χ}{~}{1}{(T{ln}ᵐ({T}{{T}₀}))}$ in the entire region $0<~{γ}<~1$ for arbitrary probability distributions. In the Ising region (${γ}>1$) the susceptibility shows an approximate power-law divergence for small anisotropy but goes eventually to zero as $T{→}0$. Possible relevance of these results to recent experiments on Qn${(TCNQ)}₂$ is discussed.