We investigate the magnetic susceptibility $\ensuremath{\chi}(T)$ of quantum spin chains of
N=1280 spins with power-law long-range antiferromagnetic couplings as a function of their spatial decay exponent $\ensuremath{\alpha}$ and cutoff length $\ensuremath{\xi}$. The calculations are based on the strong disorder renormalization method which is used to obtain the temperature dependence of $\ensuremath{\chi}(T)$ and distribution functions of couplings at each renormalization step. For the case with only algebraic decay $(\ensuremath{\xi}=\ensuremath{\infty})$ we find a crossover at ${\ensuremath{\alpha}}^{*}=1.066$ between a phase with a divergent low-temperature susceptibility $\ensuremath{\chi}(T\ensuremath{\rightarrow}0)$ for $\ensuremath{\alpha}>{\ensuremath{\alpha}}^{*}$ to a phase with a vanishing $\ensuremath{\chi}(T\ensuremath{\rightarrow}0)$ for $\ensuremath{\alpha}<{\ensuremath{\alpha}}^{*}$. For finite cutoff lengths $\ensuremath{\xi}$, this crossover occurs at a smaller ${\ensuremath{\alpha}}^{*}(\ensuremath{\xi})$. Additionally, we study the localization of spin excitations for $\ensuremath{\xi}=\ensuremath{\infty}$ by evaluating the distribution function of excitation energies, and we find a delocalization transition that coincides with the opening of the pseudogap at ${\ensuremath{\alpha}}_{c}={\ensuremath{\alpha}}^{*}$.
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.
Using the quantum Monte Carlo loop algorithm, we calculate the temperature dependence of the uniform susceptibility, the specific heat, the correlation length, the generalized staggered susceptibility, and magnetization of a spin-1/2 chain with random antiferromagnetic and ferromagnetic couplings, down to very low temperatures. Our data show a consistent scaling behavior in all the quantities and support strongly the conjecture drawn from the approximate real-space renormalization group treatment. A statistical analysis scheme is developed which will be useful for the search of scaling behavior in numerical and experimental data of random spin chains.
The thermodynamics of finite open antiferromagnetic XXZ chains is studied using field theory, Bethe ansatz and quantum Monte Carlo methods. For the susceptibility a parameter-free result as a function of the number of sites L and temperature T beyond the scaling limit is derived. The limiting cases ( J being the exchange constant), where the boundary correction shows a logarithmically suppressed Curie behaviour, and , where a crossover to the ground state behaviour takes place, are discussed in detail. On the basis of this analysis we present a simple formula for the averaged susceptibility of a spin-1/2 chain doped with non-magnetic impurities. We show that the effective Curie constant has a highly non-trivial temperature dependence and shows scaling in the low temperature limit. Finally, corrections due to intrachain and interchain couplings and implications for experiments on Sr 2 Cu 1− x Pd x O 3+δ and related compounds are discussed.
We study dynamical properties at finite temperature
(T) of Heisenberg spin chains with random antiferromagnetic exchange couplings, which realize the random singlet phase in the low-energy limit, using three complementary numerical methods: exact diagonalization, matrix-product-state algorithms, and stochastic analytic continuation of quantum Monte Carlo results in imaginary time. Specifically, we investigate the dynamic spin structure factor $S(q,\ensuremath{\omega})$ and its $\ensuremath{\omega}\ensuremath{\rightarrow}0$ limit, which are closely related to inelastic neutron scattering and nuclear magnetic resonance (NMR) experiments (through the spin-lattice relaxation rate
1/T1). Our study reveals a continuous narrow band of low-energy excitations in $S(q,\ensuremath{\omega})$, extending throughout the
q space, instead of being restricted to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as found in the uniform system. Close to $q=\ensuremath{\pi}$, the scaling properties of these excitations are well captured by the random-singlet theory, but disagreements also exist with some aspects of the predicted
q dependence further away from $q=\ensuremath{\pi}$. Furthermore we also find spin diffusion effects close to
q=0 that are not contained within the random-singlet theory but give non-negligible contributions to the mean
1/T1. To compare with NMR experiments, we consider the distribution of the local relaxation rates
1/T1. We show that the local
1/T1 values are broadly distributed, approximately according to a stretched exponential. The mean
1/T1 first decreases with
T, but below a crossover temperature it starts to increase and likely diverges in the limit of a small nuclear resonance frequency ${\ensuremath{\omega}}_{0}$. Although a similar divergent behavior has been predicted and experimentally observed for the static uniform susceptibility, this divergent behavior of the mean
1/T1 has never been experimentally observed. Indeed, we show that the divergence of the mean
1/T1 is due to rare events in the disordered chains and is concealed in experiments, where the typical
1/T1 value is accessed.
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.