The low-temperature thermodynamic properties of a random antiferromagnetic Heisenberg chain in zero magnetic field are studied approximately by a renormalisation group method. The couplings J obey a probability distribution function P(J) approximately 1/J alpha , with alpha a constant. It is found that the susceptibility chi and the specific heat C behave like chi approximately T - alpha chi and C approximately T 1- alpha c at low temperatures alpha chi and alpha c are weakly varying functions of temperature and slowly approach unity as T goes to zero. Comparison of these results with other theoretical analysis as well as experimental results on organic charge transfer compounds is also discussed.
The low-temperature thermodynamic properties of a spin-\textonehalf{} one-dimensional random anisotropic (Heisenberg-Ising) antiferromagnet described by the Hamiltonian $H=\ensuremath{\Sigma}{i}^{}{J}_{i}({\ensuremath{\sigma}}_{x}^{i}{\ensuremath{\sigma}}_{x}^{i+1}+{\ensuremath{\sigma}}_{y}^{i}{\ensuremath{\sigma}}_{y}^{i+1}+\ensuremath{\gamma}{\ensuremath{\sigma}}_{z}^{i}{\ensuremath{\sigma}}_{z}^{i+1})$ are studied as a function of disorder and anisotropy. The
Ji> 0 are independent random variables obeying a probability distribution
P(J), and $0<~\ensuremath{\gamma}<~\ensuremath{\infty}$. The approach used is a numerical implementation of a real-space renormalization-group (RG) method previously introduced. The isotropic Heisenberg case ($\ensuremath{\gamma}=1$), the
XY case ($\ensuremath{\gamma}=0$). and the Ising case ($\ensuremath{\gamma}=\ensuremath{\infty}$) are fixed points of the RG transformation. It is found that in the
XY region $\ensuremath{\gamma}<~1$, including the Heisenberg point, the system exhibits singular behavior in the thermodynamic properties for arbitrary probability distributions for the couplings. For the
XY limit ($\ensuremath{\gamma}=0$) this is in agreement with known exact results. The functional form for the low-temperature susceptibility is found to be $\ensuremath{\chi}\ensuremath{\sim}\frac{1}{(T{\mathrm{ln}}^{m}(\frac{T}{{T}_{0}}))}$ in the entire region $0<~\ensuremath{\gamma}<~1$ for arbitrary probability distributions. In the Ising region ($\ensuremath{\gamma}>1$) the susceptibility shows an approximate power-law divergence for small anisotropy but goes eventually to zero as $T\ensuremath{\rightarrow}0$. Possible relevance of these results to recent experiments on Qn
(TCNQ)2 is discussed.
An Ising model with an arbitrary distribution
P(J) of exchange interactions
J is solved exactly for the annealed case in which the system is allowed to reach complete thermal equilibrium at each temperature. The solution is expressed in terms of an Ising model on the same lattice with a single exchange parameter, allowing an exact solution to be obtained in one and two dimensions. Some feeling for the effect of annealing can be found by examining correlations between exchange interactions on neighboring bonds. Some special distributions
P(J) are examined in detail.
We have carried out Monte Carlo studies of a random-exchange antiferromagnetic spin-(1/2) chain. For systems with XY-like (anisotropic) and with Heisenberg (isotropic) coupling, our results confirm the existence of a disorder-induced low-temperature (T) divergence in the long-wavelength
Sz-
Sz susceptibility \ensuremath{\chi} which was previously predicted by real-space renormalization-group (RSRG) treatments. Over the finite temperature range studied, these results are consistent with a 1/(T
ln2T) behavior of \ensuremath{\chi}, and hence in qualitative agreement with the RSRG results. As in the XY-Heisenberg regime, we also find a disorder-induced enhancement of the low-T susceptibility for a system with Ising-like exchange coupling which, over the finite temperature range studied, is again consistent with RSRG results. However, there are inconsistencies between the RSRG predictions in the Ising-like regime at very low temperatures, and the exact results for the random-exchange Ising chain and the low-temperature behavior of \ensuremath{\chi} in the Ising-like regime may in fact be more complicated than predicted by RSRG. Finally, we also present results for the antiferromagnetic susceptibility and structure factor. For both Heisenberg and Ising-like systems, we find that disorder suppresses the long-range antiferromagnetic correlations at low T.
The nearest-neighbor random-exchange Heisenberg antiferromagnetic Heisenberg chain (
s=21) 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.
P0(J). After
n iterations we find that the renormalized probability function
Pn(J(n)), a function of the renormalized coupling
J(n), develops singular behavior for small
J(n), independent of the initial form of
P0(J). This happens both for
P0(J) that diverge or go to zero as $J\ensuremath{\rightarrow}0$. The singular form of
Pn(J(n)) is such that it leads to a specific heat and susceptibility that behave like $C\ensuremath{\sim}{T}^{1\ensuremath{-}{\ensuremath{\alpha}}_{C}(T)}$ and $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}{\ensuremath{\alpha}}_{\ensuremath{\chi}}(T)}$. ${\ensuremath{\alpha}}_{C}(T)$ and ${\ensuremath{\alpha}}_{\ensuremath{\chi}}(T)$ are exponents weakly dependent on temperature (
T) that go to one as $T\ensuremath{\rightarrow}0$ and satisfy ${\ensuremath{\alpha}}_{\ensuremath{\chi}}(T)>{\ensuremath{\alpha}}_{C}(T)$, 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
P0(J) 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.
The authors consider the random-field ferromagnetic Ising chain, for a class of continuous symmetric probability distributions of the random magnetic fields, of a diluted power-times-exponential type. This class of distribution is 'exactly solvable', in the sense that the disorder can be integrated explicitly, at any temperature. A detailed analysis of the low-temperature thermodynamics is presented. Exact expressions are obtained for the ground-state energy, the zero-temperature entropy, and the amplitude of the specific heat, which vanishes linearly at low temperature. The diluted symmetric binary distribution, where the magnetic fields can only assume the values +or-H B or zero, can be viewed as a limiting case of the class of exactly solvable distributions. The low-temperature physics of this discrete model is investigated in detail, including the exponential fall-off of the specific heat.