Level curvatures and many-spin quantum systems

For quantum systems depending on several parameters, we define the tensor of level curvatures as the second derivatives of the discrete energy eigenvalues with respect to the parameters: . For quantum systems with irregular spectra of random-matrix type, we show that both the diagonal and the off-diagonal components of the tensor of level curvatures obey a similar statistical distribution with a long tail of universal character within the orthogonal, the unitary, and the symplectic ensembles. The curvature distribution is numerically calculated for a finite and planar square cluster of spins in dipolar interaction like in nuclear magnetism. The tail of the curvature distribution is shown to agree with the theoretical prediction. Moreover, we give a partial list of possible quantities which could behave like level curvatures.