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Physical review. B, Condensed matter

Random exchange effects in antiferromagnetic quantum spin chains: A Monte Carlo study

H.-B. Schüttler, Douglas J. Scalapino, Paul Michael Grant

Published Mar 1, 1987• 8 citations
Publisher

Abstract

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 ${S}^{z}$-${S}^{z}$ 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 ${\mathrm{ln}}^{2}$T) 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.

DOI
10.1103/physrevb.35.3461

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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{S}^{z}Sz-Sz{S}^{z}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 ln2{\mathrm{ln}}^{2}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 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{J}_{i}>~0Ji​> 0 are independent random variables obeying a probability distribution P(J)P(J)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\mathrm{XY}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\mathrm{XY}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\mathrm{XY}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{(\mathrm{TCNQ})}_{2}(TCNQ)2​ is discussed.
The low energy properties of the spin-1/2 random Heisenberg chain with ferromagnetic and antiferromagnetic interactions are studied by means of the density matrix renormalization group (DMRG) and real space renormalization group (RSRG) method for finite chains. The results of the two methods are consistent with each other. The deviation of the gap distribution from that of the random singlet phase and the formation of the large-spin state is observed even for relatively small systems. For a small fraction of the ferromagnetic bond, the effect of the crossover to the random singlet phase on the low temperature susceptibility and specific heat is discussed. The crossover concentration of the ferromagnetic bond is estimated from the numerical data.
Using the continuous time quantum Monte Carlo loop algorithm, we calculate the temperature dependence of the uniform susceptibility, and the specific heat 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 both quantities and strongly support the conjecture drawn from the approximative 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.
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.
We study the Heisenberg S=1/2 chain with random ferro- and antiferromagnetic couplings using quantum Monte Carlo simulations at ultra-low temperatures, converging to the ground state. Finite-size scaling of correlation functions and excitation gaps demonstrate an exotic critical state in qualitative agreement with previous strong-disorder renormalization group calculations but with scaling exponents depending on the coupling distribution. We find dual scaling regimes of the transverse correlations versus the distance, with an L independent form C(r)=r^{-μ} for r≪L and C(r,L)=L^{-η}f(r/L) for r/L>0, where μ>η and the scaling function is delivered by our analysis. These results are at variance with previous spin-wave and density-matrix renormalization group calculations, thus highlighting the power of unbiased quantum Monte Carlo simulations.
A renormalization-group study of the effects of short-range correlations in the thermodynamics of a random-exchange Heisenberg antiferromagnetic chain model with spin S=(1/2) is presented. The temperature dependence of the exponent ${\ensuremath{\alpha}}_{\ensuremath{\chi}}$ of the low-temperature diverging susceptibility is significantly weakened by the correlation effects. The universality in the thermodynamic properties is preserved in the correlated case, but it is related to a weaker decay of correlations than in the uncorrelated model which decays exponentially. The weaker temperature dependence of ${\ensuremath{\alpha}}_{\ensuremath{\chi}}$ is in closer agreement with experimental results on charge-transfer salts.