Magnon-induced nuclear relaxation in the quantum critical region of a Heisenberg linear chain

The low-temperature properties of spin-1/2 one-dimensional (1D) Heisenberg antiferromagnetic (HAF) chains which have relatively small exchange couplings between the spins can be tuned using laboratory-scale magnetic fields. Magnetization measurements, made as a function of temperature, provide phase diagrams for these systems and establish the quantum critical point (QCP). The evolution of the spin dynamics behavior with temperature and applied field in the quantum critical (QC) region, near the QCP, is of particular interest and has been experimentally investigated in a number of 1D HAFs using neutron scattering and nuclear magnetic resonance as the preferred techniques. In the QC phase both quantum and thermal spin fluctuations are present. As a result of extended spin correlations in the chains, magnon excitations are important at finite temperatures. An expression for the NMR spin-lattice relaxation rate $1/T_1$ of probe nuclei in the QC phase of 1D HAFs is obtained by considering Raman scattering processes which induce nuclear spin flips. The relaxation rate expression, which involves the temperature and the chemical potential, predicts scaling behavior of $1/T_1$ consistent with recent experimental findings for quasi-1D HAF systems. A simple relationship between $1/T_1$ and the deviation of the magnetization from saturation ($M_S-M$) is predicted for the QC region.

Figure 1

Representative phase diagram for a 1D HAF in a dimensionless representation $k_{B}T/J$ vs $\mu /\phantom{\mu k_B}{k}_{B}T$, where $J$ is the exchange coupling between NN spins and $\mu =g{\mu }_0{\mu }_B(H_C-H)$ is the chemical potential. The QCP occurs at $\mu =0$, and the straight lines which meet at this point represent crossover boundaries between TLL, QC, and gapped phases as indicated. For $k_{B}T/J>0.5$ the system transitions to its paramagnetic phase with rise in temperature.

Figure 2

Predicted behavior, based on Eq. (7), of the scaled NMR spin-lattice relaxation rate $C/T_1$ as a function of temperature in the QC phase for a 1D HAF. A Raman scattering mechanism is used in deriving Eq. (8). The parameter $C=1/(4{\omega }_I^2/\phantom{1/(4{\omega }_I^2\pi {\omega }_E)}\pi {\omega }_E)$, where ${\omega }_I=A_{\perp }/\phantom{A_{\perp }\hslash }\hslash$, with $A_{\perp }$ the amplitude of transverse component of the fluctuating hyperfine interaction and ${\omega }_E=J/\phantom{J\hslash }\hslash$. In the plot it is assumed, for convenience, that $C=1$ and $J/\phantom{J{k}_B}{k}_B=10\mathrm{K}$. If NMR probe nuclei do not lie in superexchange paths the hyperfine coupling is likely to be dominated by the dipolar interaction between $S$ and $I$ spins.

Figure 3

The predicted NMR spin-lattice relaxation rate $1/T_1$ as a function of $\mu /\phantom{\mu k_B}{k}_{B}T$ for magnon scattering processes in the QC phase of a 1D HAF obtained using Eq. (8). A particular value of $\mu$ just below the QCP is used, corresponding to $(H_C-H)/\phantom{(H_C-H)H_C}{H}_C=0.01$, with $C=1$ and $J/\phantom{J{k}_B}{k}_B=10\mathrm{K}$. The inset shows a plot of the scaling function $F(\mu /\phantom{F(\mu k_B}{k}_{B}T)$ vs $\mu /\phantom{\mu k_B}{k}_{B}T$ for the same $\mu$ value.