Finite-temperature spin dynamics near the quantum critical point of transverse field Ising chain with a small longitudinal field

When the transverse-field Ising chain at its quantum critical point is subjected to a small longitudinal field, the perturbed conformal field theory led to a field theory with an exotic E8 symmetry [1]. Recent neutron scattering experiments have provided evidence for the lightest two particles in this E8 model in the quasi-1D Ising ferromagnet CoNb$_2$O$_6$ [2]. While the zero temperature dynamic of the model is well known, its finite-temperature counterpart has not yet been systematically studied. We study the low-frequency dynamical spin structure factor at finite temperatures using the form-factor method. We show that the dominant contribution to the spin dynamics comes from the channel between two lightest particles, and demonstrate how the spin dynamics differ from a diffusion form. Using these results, we determine the temperature dependence of the NMR relaxation rate. We suggest that, for CoNb$2$O$6$, measurements of the NMR relaxation rate provide a means to further test the applicability of the E8 model. \\[4pt] [1] A. B. Zamolodchikov, Int. J. Mod. Phys. A4, 4235(1989).\\[0pt] [2] R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, K. Kiefer2, Science, 327, 177 (2010).