Low-dimensional spin systems, consisting of arrays of spins arranged in chains or ladders, have been investigated intensively in recent years using both exactly solvable theoretical models as well as a variety of experimental techniques. In case of random variations of the exchange couplings the renormalization group theory predicts the existence of a random-singlet (RS) state, corresponding to spins coupled at all possible distances and energy scales. The scarce availability of suitable random systems, however, has so far prevented the experimental identification of this peculiar magnetic ground state. In a combined effort using nuclear magnetic resonance (NMR), dc magnetometry and numerical simulations, we now find compelling evidence for the formation of a random-singlet state in this class of materials. Randomness seems to generate a broad distribution of local magnetic fields, in turn reflected in stretched exponential NMR relaxations. This distribution exhibits a progressive broadening with decreasing temperature, caused by the growing inequivalence of magnetic sites, as expected from the RS theory. Our findings suggest that NMR is the tool of choice for probing the low-energy physics also in other disordered magnets, where extended-scale excitations are dominant.
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.
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, a typical spin-
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+ 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 to the maximally disordered
BaCu2SiGeO7. Our data indicate that already a weak degree of disorder of only 5% Ge, apart from reducing the three-dimensional magnetic ordering temperature
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 is reduced, yet its influence is clearly manifest, particularly in the features of nuclear magnetic resonance relaxation data.
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.
Combining quantum Monte Carlo simulations with the maximum entropy method, we study the dynamical structure factor
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) develops broad non-spinon excitations as well as the random-singlet peak near
(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 BaCu
2(Si
xGe
1−x)
2O
7.
In the framework of the exactly solvable model, using nonperturbative methods, the temperature and magnetic field behavior of the NMR relaxation rate (and low-energy dynamical structure factor) for the antiferromagnetic Heisenberg spin-1/2 chain with magnetic impurities is calculated. For a single magnetic impurity, totally screened at low temperatures by spinons of the chain, the critical exponents, which determine the low-temperature and magnetic field behavior, get renormalized. For the finite concentration of impurities with the random distribution of the impurity-chain coupling the weak disorder in the distribution does not produce essential changes to the dynamical structure factor. Contrary, the strong disorder yields drastically different behavior of the low-energy dynamical characteristics, comparing to the homogeneous chain. Our results are generic, and they qualitatively agree in limiting cases with the results of perturbative calculations for disordered spin chains and with the data of experiments on quasi-one-dimensional spin compounds.