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.
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) 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.
Quenched disorders can strongly influence the physical properties of quantum many-body systems. The real-space strong-disorder renormalization group (SDRG) analysis has shown that the spin-1/2 random Heisenberg chain is controlled by the infinite-randomness fixed point (IRFP) and forms a random singlet (RS) ground state. Motivated by recent thermal transport experiments on the quasi-one-dimensional antiferromagnet copper benzoate [B. Y. Pan et al., Phys. Rev. Lett. 129, 167201 (2022)], we adapt the SDRG to study the low-temperature properties of the random Heisenberg chain by assuming that its low-energy excited states are captured by the parent Hamiltonian of the RS ground state as well. We find that while the specific heat coefficient and the uniform magnetic susceptibility scale as $C/T\ensuremath{\sim}{T}^{\ensuremath{-}{\ensuremath{\alpha}}_{c}}$ and $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}{\ensuremath{\alpha}}_{s}}$ with $0<{\ensuremath{\alpha}}_{c,s}<1$, indicating a divergent low-energy density of states, the thermal and the spin conductivities scale as $\ensuremath{\kappa}/T\ensuremath{\sim}T$ and ${\ensuremath{\sigma}}_{s}\ensuremath{\sim}T$, which implies a vanishing density of extended states in the low-energy limit. We believe that such a disparity in the thermodynamic and transport properties is a common feature of random systems controlled by the IRFP.
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.
Using quantum Monte Carlo simulations, we study effects of disorder on the
S=1/2 Heisenberg model with exchange constant
J on the square lattice supplemented by multispin interactions
Q. It was found recently [L. Liu et al., Phys. Rev. X 8, 041040 (2018)] that the ground state of this $J\ensuremath{-}Q$ model with random couplings undergoes a quantum phase transition from the N\'eel antiferromagnetic state into a randomness-induced spin-liquid-like state that is a close analog to the well known random-singlet (RS) state of the Heisenberg chain with random couplings. This 2D RS state arises from a spontaneously symmetry-broken fourfold degenerate columnar valence-bond solid that is broken up by the disorder into finite domains, with spinons localized at topological defects. The interacting spinons form a critical collective many-body state without magnetic long range order but with the mean spin-spin correlations decaying with distance
r as ${r}^{\ensuremath{-}2}$, as in the one-dimensional RS state. The dynamic exponent $z\ensuremath{\ge}2$, varying continuously with the model parameters. In this work, we further investigate the properties of the RS state in the $J\ensuremath{-}Q$ model with random
Q couplings. We study the temperature dependence of the specific heat and various susceptibilities for large enough systems to reach the thermodynamic limit. We also analyze the size dependence of the critical magnetic order parameter and its susceptibility in the ground state. For all these quantities, we find consistency with the conventional quantum-critical scaling laws when the condition implied by the ${r}^{\ensuremath{-}2}$ form of the spin correlations is imposed. In particular, all the different quantities can be explained by the same value of the dynamic exponent
z at fixed model parameters. We argue that the RS state identified in the $J\ensuremath{-}Q$ model corresponds to a generic renormalization group fixed point that can be reached in many quantum magnets with random couplings and that it has already been observed experimentally.
The ordering of weakly coupled random antiferromagnetic
S=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 and for the ordered moment
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.