Dynamical properties of the S =1/2 random Heisenberg chain

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 ,ω ) and its ω →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 ,ω ) , extending throughout the q space, instead of being restricted to q ≈0 and q ≈π as found in the uniform system. Close to q =π , 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 =π . 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 ω₀. 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.