NMR relaxation rate of (quasi-)one-dimensional spin gapped systems in magnetic fields

The NMR relaxation rates of (quasi-)one-dimensional spin gapped systems such as the $S=1/2$ spin ladder with a diagonal interaction and the Haldane-gap system are investigated at low temperatures in magnetic fields. We show that in the regime where the gap is collapsed by magnetic fields, the NMR relaxation rate diverges with decreasing temperature and its exponent shows a behavior characteristic of the model as a function of magnetic field. We discuss our results in comparison with recent experimental findings for the spin ladder system ${\rm Cu_{2}(C_{5}H_{12}N_{2})_{2}Cl_{4}}$ and for the Haldane-gap system ${\rm (CH_3)_4NNi(NO_2)_3}$.