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arXiv:1307.0370 · Physical Review Letters

Local Spin Relaxation within the Random Heisenberg Chain

J. Herbrych, J. Kokalj, Peter Prelovšek

Published Oct 2, 2013• 20 citations
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Abstract

Finite-temperature local dynamical spin correlations S(nn)(ω) are studied numerically within the random spin-1/2 antiferromagnetic Heisenberg chain. The aim is to explain measured NMR spin-lattice relaxation times in BaCu(2)(Si(0.5)Ge(0.5))(2)O(7), which is the realization of a random spin chain. In agreement with experiments we find that the distribution of relaxation times within the model shows a very large span similar to the stretched-exponential form. The distribution is strongly reduced with increasing T, but stays finite also in the high-T limit. Anomalous dynamical correlations can be associated with the random singlet concept but not directly with static quantities. Our results also reveal the crucial role of the spin anisotropy (interaction), since the behavior is in contrast with the ones for the XX model, where we do not find any significant T dependence of the distribution.

DOI
10.1103/physrevlett.111.147203

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Finite-temperature local dynamical spin correlations S(nn)(ω) are studied numerically within the random spin-1/2 antiferromagnetic Heisenberg chain. The aim is to explain measured NMR spin-lattice relaxation times in BaCu(2)(Si(0.5)Ge(0.5))(2)O(7), which is the realization of a random spin chain. In agreement with experiments we find that the distribution of relaxation times within the model shows a very large span similar to the stretched-exponential form. The distribution is strongly reduced with increasing T, but stays finite also in the high-T limit. Anomalous dynamical correlations can be associated with the random singlet concept but not directly with static quantities. Our results also reveal the crucial role of the spin anisotropy (interaction), since the behavior is in contrast with the ones for the XX model, where we do not find any significant T dependence of the distribution.
NMR measurements of the (29)Si spin-lattice relaxation time T(1) were used to probe the spin-1/2 random Heisenberg chain compound BaCu(2)(Si(1-x)Ge(x))(2)O(7). Remarkable differences between the pure (x=0) and the fully random (x=0.5) cases are observed, indicating that randomness generates a distribution of local magnetic relaxations. This distribution, which is reflected in a stretched exponential NMR relaxation, exhibits a progressive broadening with decreasing temperature, caused by a growing inequivalence of magnetic sites. Compelling independent evidence for the influence of randomness is also obtained from magnetization data and Monte Carlo calculations. These results suggest the formation of random-singlet states in this class of materials, as previously predicted by theory.
We study dynamical properties at finite temperature (T)(T)(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)1/{T}_{1})1/T1​). Our study reveals a continuous narrow band of low-energy excitations in $S(q,\ensuremath{\omega})$, extending throughout the qqq 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 qqq dependence further away from $q=\ensuremath{\pi}$. Furthermore we also find spin diffusion effects close to q=0q=0q=0 that are not contained within the random-singlet theory but give non-negligible contributions to the mean 1/T11/{T}_{1}1/T1​. To compare with NMR experiments, we consider the distribution of the local relaxation rates 1/T11/{T}_{1}1/T1​. We show that the local 1/T11/{T}_{1}1/T1​ values are broadly distributed, approximately according to a stretched exponential. The mean 1/T11/{T}_{1}1/T1​ first decreases with TTT, 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/T11/{T}_{1}1/T1​ has never been experimentally observed. Indeed, we show that the divergence of the mean 1/T11/{T}_{1}1/T1​ is due to rare events in the disordered chains and is concealed in experiments, where the typical 1/T11/{T}_{1}1/T1​ value is accessed.
Combining quantum Monte Carlo simulations with the maximum entropy method, we study the dynamical structure factor S(k,ω)S(k,ω)S(k,ω) of spin-1/2 antiferromagnetic Heisenberg chains with various random bond distributions. We emphasize the crossover behavior in the dynamical properties from pure chain to disorder-dominated random singlet phase due to bond randomness. For the distribution corresponding to the infinite randomness fixed point, S(k,ω)S(k,ω)S(k,ω) develops broad non-spinon excitations as well as the random-singlet peak near (k,ω)=(π,0)(k, ω) = (π, 0)(k,ω)=(π,0), consistent with the known results obtained by the real-space renormalization group method. For weak disorder, however, we find clear signature of spinon excitations, reminiscent of pure spin chains, blurred by disorder. We discuss the implication for experiments on random-bond antiferromagnetic spin chains, realizable, e.g., in BaCu2_22​(Six_xx​Ge1−x_{1-x}1−x​)2_22​O7_77​.
Recent developments in density matrix renormalization group/matrix product state techniques allow today for a quantitative computation of dynamical correlations at finite temperature in low-dimensional quantum spin systems. By combining numerical results and field theoretical asymptotic methods, e.g., bosonization, it is now possible to give a full description of the dynamical quantity one is interested in for the range of temperatures experimentally accessible. This enhances of course the possibility of one-to-one comparisons with experiments without having to resort to uncontrolled approximations. Here, the authors focus on the temperature dependence of the NMR spin-lattice relaxation rate 1/T11/{T}_{1}1/T1​ in spin-12\frac{1}{2}21​ chain systems. This quantity is directly related to local, dynamical spin-spin correlation functions. The results agree in the low-temperature regime with bosonization predictions, and show interesting deviations from this behavior at intermediate temperatures.
The ordering of weakly coupled random antiferromagnetic S=1/2S=1/2S=1/2 chains, as relevant for the recent experimentally investigated spin chain materials, is considered theoretically. The one-dimensional isotropic Heisenberg model with random exchange interactions is treated numerically on finite chains with the density-matrix renormalization-group approach as well as with the standard renormalization analysis, both within the mean-field approximation for interchain coupling ${J}_{\ensuremath{\perp}}$. Results for the ordering temperature TN{T}_{N}TN​ and for the ordered moment m0{m}_{0}m0​ are presented and are both reduced with increasing disorder, agreeing with the experimental observations. The most pronounced effect of the random singlet concept appears to be a very large span of local ordered moments, becoming wider with decreasing ${J}_{\ensuremath{\perp}}$, consistent with the $\ensuremath{\mu}$-SR experimental findings.
Heisenberg-type spin-chain materials have been extensively studied over the years, yet not much is known about their behavior in the presence of disorder. Starting from BaCu2Si2O7{\mathrm{BaCu}}_{2}{\mathrm{Si}}_{2}{\mathrm{O}}_{7}BaCu2​Si2​O7​, a typical spin-12\frac{1}{2}21​ chain system, we investigate a series of compounds with different extents of bond disorder, where the systematic replacement of Si with Ge results in a remodulation of the Cu2+{\mathrm{Cu}}^{2+}Cu2+ exchange interactions. By combining magnetometry measurements with nuclear magnetic resonance studies, we follow the evolution of the disorder-related properties from the well-ordered BaCu2Si2O7{\mathrm{BaCu}}_{2}{\mathrm{Si}}_{2}{\mathrm{O}}_{7}BaCu2​Si2​O7​ to the maximally disordered BaCu2SiGeO7{\mathrm{BaCu}}_{2}{\mathrm{SiGeO}}_{7}BaCu2​SiGeO7​. Our data indicate that already a weak degree of disorder of only 5% Ge, apart from reducing the three-dimensional magnetic ordering temperature TN{T}_{\mathrm{N}}TN​ quite effectively, induces a qualitatively different state in the paramagnetic regime. At maximum disorder our data indicate that this state may be identified with the theoretically predicted random singlet (RS) state. With decreasing disorder the extension of the RS regime at temperatures above TN{T}_{\mathrm{N}}TN​ is reduced, yet its influence is clearly manifest, particularly in the features of nuclear magnetic resonance relaxation data.